Unique Determination of a Magnetic Schrödinger Operator with Unbounded Magnetic Potential from Boundary Data

Unique Determination of a Magnetic Schrödinger Operator with Unbounded Magnetic Potential from... Abstract We consider the Gel’fand–Calderón problem for a Schrödinger operator of the form $$-(\nabla + iA)^2 + q$$, defined on a ball $$B$$ in $$\mathbb{R}^3$$. We assume that the magnetic potential $$A$$ is small in $$W^{s,3}$$ for some $$s>0$$, and that the electric potential $$q$$ is in $$W^{-1,3}$$. We show that, under these assumptions, the magnetic field $$\text{curl} A$$ and the potential $$q$$ are both determined by the Dirichlet–Neumann relation at the boundary $$\partial B$$. The assumption on $$q$$ is critical with respect to homogeneity, and the assumption on $$A$$ is nearly critical. Previous uniqueness theorems of this type have assumed either that both $$A$$ and $$q$$ are bounded or that $$A$$ is zero. 1 Boundary data for Schrödinger operators Consider a Schrödinger Hamiltonian of the form   LA,qu=−(∇+iA)2u+qu. Here $$A$$ represents a magnetic vector potential and $$q$$ represents an electric scalar potential. Let $$\Omega \subset \mathbb{R}^n$$ be an bounded open set. Define the Dirichlet–Neumann relation by   ΛA,qΩ={(u|∂Ω,(∂ν+iν⋅A)u|∂Ω):u∈H1(B)andLA,qu=0}, where $$\nu$$ is the outward unit normal to $$\partial \Omega$$. The Gel’fand–Calderón problem [10, 19] is to determine the magnetic field $$\text{curl} A$$ and the electric potential $$q$$ from the Dirichlet–Neumann relation $$\Lambda_{A,q}$$. In principle, this is possible if $$\Lambda_{A,q}$$ uniquely determines $$\text{curl} A$$ and $$q$$. We are interested in proving uniqueness under minimal a priori regularity assumptions on $$A$$ and $$q$$. To avoid unnecessary technical complications, we take $$\Omega$$ to be a ball in $$\mathbb{R}^3$$ and assume that the coefficients $$A$$ and $$q$$ are supported away from $$\partial \Omega$$. Theorem 1.1. Fix $$s>0$$. Let $$B = B(0,1)$$ be the unit ball in $$\mathbb{R}^3$$. Suppose that $$A_i$$ and $$q_i$$ are supported in the smaller ball $$\tfrac{1}{2} B$$. If, for each $$i = 1,2$$, the magnetic potential $$A_i$$ is small in the $$W^{s,3}$$ norm, the electric potential $$q_i$$ is in $$W^{-1,3}$$, and $$\Lambda_{A_1,q_1} = \Lambda_{A_2,q_2}$$, then $$\text{curl} A_1 = \text{curl} A_2$$ and $$q_1 = q_2$$. □ One physical motivation for studying this problem comes from quantum mechanics. For compactly supported potentials, the map $$\Lambda_{A,q}$$ contains the same information as the scattering matrix at a fixed energy level. The scattering matrix contains observations, made at spatial infinity, of a localized (short-range) potential. This can be defined for potentials that are exponentially decreasing rather than compactly supported, so inverse scattering at fixed energy is a generalization of the Gel’fand–Calderón problem. Unique determination of a bounded electric potential $$q$$ from the Dirichlet-to-Neumann map in the absence of a magnetic potential was proven by Sylvester and Uhlmann [51] (see also [34]). The proof is based on a density argument using complex geometrical optics (CGO) solutions, inspired by Calderón’s treatment of the linearized inverse conductivity problem in [10]. The Sylvester–Uhlmann method was adapted to the case of a nonzero magnetic potential by Sun in [48], where uniqueness was proven for $$A\in C^2$$ and $$q \in L^\infty$$, subject the requirement that $$\lVert{\text{curl} A}\rVert_\infty$$ be small. The basic method we use in this article is the same as that in [48]; in particular, we retain a smallness condition on the magnetic potential. Uniqueness for large smooth $$A$$ was proved by Nakamura et al. [37] using a pseudodifferential conjugation technique from [36]. This was improved to $$A \in C^1$$ in [52] using symbol smoothing. In [44], it was shown that, by imposing a Coulomb gauge, the result of [48] could be extended to small Dini continuous $$A$$ (this includes the case where $$A$$ is $$\alpha$$-Hölder for some $$\alpha>0$$). The smallness condition was removed in [45] using pseudodifferential conjugation. In [31], the Coulomb gauge condition and pseudodifferential conjugation were eliminated using an argument based on Carleman estimates with slightly convex weights, and uniqueness was proven for $$A \in L^\infty$$. Our result requires that $$A$$ is small and slightly more differentiable than in [31]. On the other hand, we require much less integrability for $$A$$ and $$q$$, so that our conditions on $$A$$ and $$q$$ are much closer to being scale-invariant. It does not seem that the method in [31] for removing the smallness condition on $$A$$ extends to the case of unbounded potentials. However, we believe that a pseudodifferential conjugation argument could be used to remove the smallness condition for the result in this paper. Another approach to the problem in the spirit of Faddeev’s pioneering work in [17] is based on the $$\bar\partial$$ method of Beals and Coifman [3]. Using this approach, the inverse scattering problem for small $$A$$ and $$q$$ in $${\rm e}^{-\gamma\langle x \rangle} C^\infty$$ was solved by Khenkin and Novikov in [29]. Uniqueness for $$A = 0$$ and large $$q \in {\rm e}^{-\gamma\langle {x} \rangle } L^\infty$$ was proven by Novikov [41]. Uniqueness for large $$(A,q) \in {\rm e}^{-\gamma\langle {x} \rangle} C^\infty$$ was proven by Eskin and Ralston [16]. A proof of uniqueness for $$A = 0$$ and $$q \in {\rm e}^{-\gamma\langle x \rangle} L^\infty$$ using a density argument more similar to the Sylvester–Uhlmann approach detailed above is also possible [33, 54]. This density argument was modified to include $$A \in {\rm e}^{-\gamma\langle {x} \rangle} W^{1,\infty}$$ in [43]. Since the Laplacian has units $$(\text{length})^{-2}$$, the $$L^\infty$$ norms of $$A$$ and $$q$$ are not dimensionless quantities. This is undesirable from a physical point of view. Assuming that $$A$$ and $$q$$ are bounded excludes even subcritical potentials with $$\lvert{x}\rvert^{-1}$$ singularities at the origin (for example, a localized Coulomb-type potential). A scale-invariant assumption in $$n$$ dimensions is that $$A$$ be in $$L^n$$ and that $$q$$ be in $$L^{n/2}$$ or $$W^{-1,n}$$ (by Sobolev embedding, $$L^{n/2} \subset W^{-1,n}$$). Chanillo [12], using the weighted inequalities of Chanillo and Sawyer [13], proved uniqueness in the inverse boundary value problem for $$A = 0$$ and compactly supported $$q$$ with small norm in the scale-invariant Fefferman–Phong classes $$F_{>(n-1)/2}$$ (including, in particular, potentials of small weak $$L^{n/2}$$ norm). Chanillo’s article also includes an argument of Jerison and Kenig proving uniqueness for $$q \in L^{n/2+}$$ with no smallness condition. This was extended to include the scale-invariant case $$q \in L^{n/2}$$ by Lavine and Nachman (see [15] for details). A closely-related problem is Calderón’s problem, which is to recover the coefficient in the equation $$\text{div} (\gamma \nabla u) = 0$$ from the Dirichlet-to-Neumann map $$\Lambda_\gamma$$. In Sylvester and Uhlmann’s work [51], this problem is reduced to the problem of recovering a Schrödinger potential $$q$$, where $$q = \gamma^{-1/2} \Delta \gamma$$. Unless $$\gamma$$ has two derivatives, the potential $$q$$ will end up having negative regularity. In [6, 7, 42] it was shown that the Sylvester–Uhlmann argument carries through for conductivities with $$3/2$$ derivatives. In [23], the author and Tataru showed uniqueness for $$\gamma \in C^1$$ or $$\gamma$$ with small Lipschitz norm using an averaging argument. In [38], a more involved averaging argument was used to prove uniqueness in three dimensions for $$\gamma \in H^{3/2+}$$. In [22], the author used arguments similar to those of [38], combined with the $$L^p$$ Carleman estimates of [28], to show uniqueness for $$\gamma \in W^{1,n}$$ in dimensions $$n=3,4$$. This corresponds to recovering a Schrödinger potential $$q \in W^{-1,n}$$. In two dimensions, the problem has a fairly different character, and we refer the reader to [1, 4, 8, 9, 20, 25, 29, 32, 35, 40, 49, 50]. The main contribution of this paper is in the construction of CGO solutions. These are solutions to the Schrödinger equation $$L_{A,q} u = 0$$ of the form $$u = {\rm e}^{x \cdot \zeta}(a + \psi)$$. To construct such solutions, we need to understand the conjugated Laplacian $$\Delta_\zeta$$, defined by   Δζ=e−x⋅ζΔex⋅ζ, where $$\zeta \in \mathbb{C}^n$$ and $$\tau = \lvert{\text{Re} \zeta}\rvert$$ is large. In particular, we would like to show that the operator $$L_{A,q,\zeta}$$, defined by   LA,q,ζ=e−x⋅ζLA,qex⋅ζ=−Δζ−2iA⋅(ζ+∇)−i∇⋅A−A⋅A+q, (1) is invertible on some function spaces. To do this, we need a lower bound for $$\Delta_\zeta$$ that can absorb the lower order terms. Estimates for operators like $$\Delta_\zeta$$ arise in the unique continuation problem for operators of the form $$-\Delta + A \cdot \nabla + V$$. For example, the weak unique continuation property for operators of the form $$-\Delta + V$$, where $$V \in L^{n/2}_{\mathrm{loc}}$$, follows from the $$L^p$$ Carleman estimate of Kenig et al. [28], which states that   ‖e−τx1u‖p′≲‖e−τx1Δu‖p, with $$1/p+1/p'=1$$ and $$1/p-1/p'=2/n$$. This estimate is equivalent to an estimate of the form   ‖v‖p′≲‖Δτe1v‖p for the conjugated Laplacian $$\Delta_{\tau e_1}$$. These $$L^p$$ Carleman estimates are similar to Strichartz estimates [47] for dispersive equations, and can be proven in a similar way, using Fourier restriction theorems (in particular, the Stein–Tomas theorem [46, 53] and its variants). The first application of the Fourier restriction theory to unique continuation appears in the work of Hörmander [24], who attributes the idea to Sjölin. Hörmander did not believe that this approach could be used to prove a unique continuation theorem for potentials in the critical space $$L^{n/2}$$. However, this is precisely what was achieved by Kenig, Ruiz, and Sogge in [28]. The strong unique continuation property for these potentials had been established by Jerison and Kenig in [27]. The connection between the $$L^p$$ Carleman estimate in [27] and Fourier restriction theorems was clarified in [26]. Given $$A\in L^q_{\mathrm{loc}}$$, unique continuation for the operator $$-\Delta + A \cdot \nabla$$ would follow from a gradient Carleman estimate of the form   ‖eτϕ∇u‖Lr≲‖eτϕΔu‖Lp, (2) where $$1/p-1/r = 1/q$$. Barcelo et al. [2] showed that no such gradient Carleman estimate can hold for linear weights of the form $$\phi = x_1$$ unless $$r = p = 2$$. In contrast, they proved unique continuation for $$A \in L^{(3n-2)/2}_{\mathrm{loc}}$$ and $$V \in L^{n/2+}_{\mathrm{loc}}$$ by establishing a gradient Carleman estimate (2) for the convex weight $$\phi = x_1 + x_1^2$$. They showed that their results are sharp, in the sense that for any weight $$\phi$$, the estimate (2) cannot hold uniformly in $$\tau$$ and $$u$$ unless the exponents $$p$$ and $$r$$ satisfy the condition $$1/p-1/r \leq 2/(3n-2)$$. This means that the Carleman method cannot be directly applied when $$1/q > 2/(3n-2)$$. Nevertheless, Wolff showed in [55] that the operator $$\Delta + A \cdot \nabla + q$$ has the weak unique continuation property for $$A \in L^{n}$$ and $$V \in L^{n/2}$$. Subsequently, Wolff’s method was adapted to the corresponding strong unique continuation problem by Koch and Tataru [30]. The idea is that the gradient Carleman estimate (2) can be rescued by localizing it to a very small set. In fact, if $$u$$ is supported on a set $$E$$ of volume $$\lvert{E}\rvert$$, then the gradient Carleman estimate (2) holds with a loss of $$\tau\lvert{E}\rvert^{1/n}$$. We will attempt to very roughly outline how one might exploit this fact. Wolff’s point of departure is the following observation: if $$\mu$$ is any compactly supported measure and $$v$$ is a unit vector, then multiplying $$\mu$$ by an exponential weight $${\rm e}^{\tau x \cdot v}$$ tends to push the mass of $$\mu$$ to the boundary $$\partial K$$, where $$K$$ is the convex hull of $$\text{supp} \mu$$. More concretely, if we let $$x_{v}^{\max}$$ denote the maximum value assumed by the weight $$x_v = x\cdot v$$ on $$K$$, then the measure $$e^{\tau x\cdot v} \mu$$ should concentrate on the set $$\{\lvert{x_v - x_v^{\max}}\rvert \lesssim \tau^{-1}\}$$. Now, this does not look very useful, as in general this set appears to have volume closer to $$\tau^{-1}$$ than to $$\tau^{-d}$$. In particular, this is the case if $$\mu$$ concentrates on a portion of $$\partial K$$ which is close to a plane $$\{x \cdot v =\text{const}\}$$. On the other hand, the Carleman method allows considerable freedom in choosing the parameters $$v$$ and $$\tau$$. It is only necessary to establish Carleman estimates for a single measure $$\mu$$, which depends on the function $$u$$ whose existence one wants to rule out. Thus one must somehow exploit the geometrical fact that $$\partial K$$, as the boundary of a convex body, cannot efficiently approximate planes $$\{x \cdot v = \text{const}\}$$ in too many distinct directions $$v\in S^{n-1}$$. For example, if we try to approximate all of the planes simultaneously by taking $$\mu$$ to be a uniform measure on the unit sphere, then $${\rm e}^{\tau x\cdot v} \mu$$ concentrates on a relatively large set (a rectangle of volume $$\tau^{-(n-1)/2-1}$$), but this set only contains a $$\tau^{-(n-1)/2}$$th part of the total mass of $$\mu$$. It turns out that in this case we can glue together different Carleman estimates and gain the required factor of $$\tau^{-1}$$. Unfortunately, it is not clear how this approach could be used to establish uniform bounds for an operator of the form (1), as Wolff’s argument in [55] only works for a single function $$u$$. We will establish an estimate that holds uniformly in $$u$$ as long as the vector $$\zeta$$ avoids certain “bad” directions. These bad directions depend only on the magnetic potential $$A$$ and not (as in Wolff’s method) on the function $$u$$. Our method of selecting the good directions is based on a much simpler Fourier-theoretic orthogonality argument, which is “dual” in a sense to Wolff’s idea. The use of Fourier analysis and orthogonality makes it unclear whether the method generalizes to higher dimensions. This remains an open problem. That localization has a smoothing effect is consistent with the uncertainty principle, since localization in physical space at the scale $$\mu^{-1}$$ corresponds to averaging in Fourier space at the scale $$\mu$$. This averaging smooths out the singular behavior of $$\Delta_\zeta$$ at the characteristic set, since the distance in Fourier space from the characteristic set $$\text{char} \Delta_\zeta$$ is effectively bounded below by $$\mu$$. When the modulation $$d(\xi,\text{char} \Delta_\zeta)$$ is large, the operator $$\Delta_\zeta$$ has good lower bounds. Instead of localizing in physical space and taking advantage of this fact indirectly, we use this high-modulation gain directly in order to overcome the failure of the gradient Carleman estimate. To keep track of the modulation, we use Bourgain-type spaces [5] with norm $$\lVert{\cdot}\rVert_{\dot X^{b}_\zeta}$$ given by   ‖u‖X˙ζb=‖|Δζ|bu‖L2. Solvability of (1) will follow from a bilinear estimate of the form   |⟨A⋅(∇+ζ)u,v|⟩≲‖u‖X˙ζ1/2‖v‖X˙ζ1/2. The $$\dot X^{1/2}_\zeta$$ norm localizes the Fourier transforms $$\hat u$$ and $$\hat v$$ near the characteristic set $$\Sigma_\zeta$$, which lies in the plane   (Reζ)⊥={ξ:ξ⋅Re(ζ)=0}. Thus, the worst-case scenario occurs when $$\hat A(\xi)$$ also concentrates on the plane $$(\text{Re} \zeta)^\perp$$. Using an averaging argument based on Plancherel’s theorem, we will show that $$\hat A$$ cannot concentrate on too many planes through the origin. This will show that $$A\cdot (\nabla+\zeta)$$ is a bounded map from $$\dot X^{1/2}_\zeta$$ to $$\dot X^{-1/2}_\zeta$$ for most values of $$\text{Re} \zeta$$. Since $$\text{Re} \zeta$$ is, to a large extent, a free parameter, this is enough to obtain many CGO solutions and prove uniqueness. We now give an outline of the paper. Sections 2–4 contain standard material due to [16, 48, 51]. In Section 5, we define a dyadic decomposition in frequency and modulation, which will be used extensively throughout the paper. In Section 6, we introduce the Bourgain spaces $$\dot X^b_\zeta$$ and $$X^b_\zeta$$ and recall some basic estimates for these spaces from [22, 23]. In Section 5, we review some averaging estimates from [22, 23] and prove an additional averaging estimate which follows from the Carleson–Sjölin theorem. In Section 8, we prove new estimates for the amplitude $$a$$ of the CGO solutions. The amplitude has the form   a=exp⁡(∂¯e−1A), where $$\bar\partial_e = (e_1 + i e_2) \cdot \nabla$$ for some orthonormal vectors $$\{e_1,e_2\}$$ in $$\mathbb{R}^3$$. Since $$A$$ is only assumed to be in $$W^{s,3}$$, the amplitude $$a$$ may behave very badly. However, if $$\bar\partial^{-1}_e$$ were replaced by $$\lvert{\nabla}\rvert^{-1}$$, then $$a$$ would be bounded in $$L^\infty\cap W^{1,3}_{\mathrm{loc}}$$. By averaging, we show that, for many choices of $$e_1$$ and $$e_2$$, the behavior of $$a$$ is acceptable. In particular, we show that expressions of the form $$a q$$ (where $$q \in W^{-1,3}$$) and $$\Delta a$$ are bounded in $$X^{-1/2}_\zeta$$. Establishing this is a bit delicate and constitutes the main technical difficulty in this paper relative to previous work. We note that it is not too difficult to produce CGO solutions with remainders $$\psi$$ whose $$\dot X^{1/2}_\zeta$$ norm grows like $$o(\tau^{1/2})$$. This was accomplished in the author’s dissertation [21] and is sufficient to show that the magnetic potential $$A$$ is determined by $$\Lambda_{A,q}$$. This is because the main term in the integral identity (3) has size $$\tau$$, so errors of order $$o(\tau)$$ are acceptable. However, once it is shown that the magnetic potentials $$A_1$$ and $$A_2$$ coincide, the main term in the integral identity (3) has size 1, so the error terms should be of order $$o(1)$$. If the $$\dot X^{1/2}_\zeta$$ norm of the remainders $$\psi$$ were to grow like $$o(\tau^{1/2})$$, we would not be able to control the error terms in the integral identity without assuming that $$q$$ is bounded. In Section 9, which contains results from the author’s dissertation [21], we prove estimates for the operator norms of the terms in $$L_{A,q,\zeta}+\Delta_\zeta$$. Since $$\Delta_\zeta$$ maps $$\dot X^{1/2}_\zeta$$ isometrically to $$\dot X^{-1/2}_\zeta$$, the operator $$L_{A,q,\zeta}$$ has a bounded inverse $$L_{A,q,\zeta}^{-1}: \dot X^{-1/2}_\zeta \to \dot X^{1/2}_\zeta$$ as long as the operator norm $$\lVert{L_{A,q,\zeta} + \Delta_\zeta}\rVert_{\dot X^{1/2}_\zeta \to \dot X^{-1/2}_\zeta}$$ is sufficiently small. In [22], the author showed that multiplication by a potential $$q$$ in $$W^{-1,3}$$ is bounded in this operator norm by combining the $$L^p$$ Carleman estimates of [24] with an averaging argument. In the present work, we also consider first-order terms such as $$A \cdot \nabla$$. These terms are more difficult to control, since the behavior of $$A \cdot \nabla$$ is worse than the behavior of $$q$$, particularly when $$A$$ concentrates at low frequencies. To remedy this, we use the fact that when the frequency of $$A$$ is sufficiently low, the curvature of the characteristic set does not play an important role. In this section, we encounter some logarithmic divergences, which is why we need a regularity assumption on $$A$$. It is likely that this limitation can be removed, at least for $$A \in L^{3+}$$, by using a refined version of the pseudodifferential conjugation technique in [36]. This technique should also eliminate the smallness assumption on $$A$$. We hope to address this problem in future work. In Section 10, we show that our averaged estimates are sufficient to run Sun’s version of the Sylvester–Uhlmann argument. In [22], the author concluded the proof of uniqueness in the case $$A = 0$$ using a compactness argument from [38]. This argument relies on the decay of the operator norm $$\lVert{q}\rVert_{X^{1/2}_\zeta \to X^{-1/2}_\zeta}$$ as $$\tau \to \infty$$. It fails for the magnetic Schrödinger equation, because the operator norm $$\lVert{A}\rVert_{X^{1/2}_\zeta \to X^{-1/2}_\zeta}$$ does not decay as $$\tau \to \infty$$, even for smooth $$A$$. Instead, we use the Banach–Alaoglu theorem to show that the Fourier transforms of $$\text{curl} A$$ and $$q$$, which are analytic, vanish on a set of positive measure. 2 An integral identity We now give a very rough outline of how to show that $$\Lambda_{A_1,q_1}\,{=}\,\Lambda_{A_2,q_2}$$ implies that $$(\text{curl}A_1,q_1)\,{=}\,(\text{curl} A_2,q_2)$$ using the Sylvester–Uhlmann strategy. The first step is to write the condition that $$\Lambda_{A_1,q_1} = \Lambda_{A_2,q_2}$$ as an integral identity. Lemma 2.1. Let $$B$$ be the unit ball in $$\mathbb{R}^n$$. Suppose that $$A_i \in L^n$$ and $$q_i \in W^{-1,n}$$ have support in $$\tfrac{1}{2} B$$. If $$\Lambda^B_{A_1, q_2} = \Lambda^B_{A_2,q_2}$$, then the integral identity   ∫[i(A1−A2)⋅(u1∇u2−u2∇u1)+(A12−A22+q1−q2)u1u2]dx=0 (3) holds for any $$u_i \in H^1(B)$$ solving $$L_{A_1,q_1}u_1 = 0$$ and $$L_{- A_2, q_2} u_2 = 0$$ in $$B$$. □ Proof. Define the bilinear form $$Q_{A,q}$$ by   QA,q(u,v)=∫B(∇u⋅∇v+iA⋅(u∇v−v∇u)+(A2+q)uv)dx. If $$u$$ and $$v$$ are functions in $$H^1(B)$$ and $$u$$ is a weak solution to the equation $$L_{A,q}u =0$$ in $$B$$, then   0=∫B(−div∇u−idiv(Au)−iA⋅∇u+A2+q)vdx=QA,q(u,v)−∫∂B∂νu⋅vdx, (4) where $$\nu$$ is the outward unit normal to $$\partial B$$. Thus we have the identity   QA,q(u,v)=⟨∂νu|∂B,v¯|∂B⟩L2(∂B). (5) Suppose we are given functions $$u_1$$ and $$u_2$$ in $$H^1(B)$$ satisfying the equations $$L_{A_1,q_1} u_1 =0$$ and $$L_{-A_2,q_2} u_2 =0$$. The assumption that $$\Lambda_{A_1,q_2} \,{=}\, \Lambda_{A_2,q_2}$$ implies that there is some $$v_2$$ in $$H^1(B)$$ such that $$L_{A_2,q_2} v_2 = 0$$ and   u1−v2|∂B=0,∂ν(u1−v2)|∂B=0. Thus, by the identity (5), we derive that   QA1,q1(u1,u2)=⟨∂νu1|∂B,u¯2|∂B⟩L2(∂B)=⟨∂νv2|∂B,u¯2|∂B⟩L2(∂B)=QA2,q2(v2,u2) On the other hand, by the definition of $$Q_{A,q}(u,v)$$, we have $$Q_{A,q}(v,u) = Q_{-A,q}(u,v)$$. Thus, using the identity (5) again, we derive   QA2,q2(v2,u2)=Q−A2,q2(u2,v2)=Q−A2,q2(u2,u1)=QA2,q2(u1,u2). We conclude that $$Q_{A_1,q_1}(u_1,u_2) - Q_{A_2,q_2}(u_1,u_2) = 0$$, which is (3). ■ To use this integral identity, we construct CGO solutions $$u_1$$ and $$u_2$$ to the equations $$L_{A_1,q_1} u_1 = 0$$ and $$L_{-A_2,q_2} u_2 = 0$$. The CGO solutions $$u_i$$ are approximately complex exponentials $${\rm e}^{x\cdot \zeta_i}$$, where the $$\zeta_i$$ are chosen such that   ζ1=τ(e1+ie2)+O(|k|)ζ2=−τ(e1+ie2)+O(|k|)ζ1+ζ2=ik for some arbitrary vectors $$e_1, e_2, k \in \mathbb{R}^3$$ satisfying   e1⊥e2⊥k|e1|=|e2|=1. Substituting the CGO solutions $$u_i \sim e^{x\cdot \zeta_i}$$ into the integral identity (3) gives   0∼−2iτ(e1+ie2)⋅∫B(A1−A2)eik⋅xdx+∫B(A12−A22+q1−q2)eik⋅xdx. Taking the limit as $$\tau \to \infty$$, we have   0=(e1+ie2)⋅A1−A2^(k) for every pair $$\{e_1, e_2\}$$ of orthonormal vectors perpendicular to $$k$$. This implies that $$\text{curl} A_1 = \text{curl} A_2$$. In particular, by Poincaré’s lemma, there is a gauge transform $$\psi$$ such that $$A_1 - A_2 = \nabla \psi$$. The Dirichlet–Neumann relation is invariant under such gauge transforms. Lemma 2.2. Suppose $$\psi$$ is a function supported in $$\tfrac{1}{2} B$$ such that $$\psi \in W^{1,3}(B)\cap L^\infty(B)$$. Then $$\Lambda_{A + \nabla \psi, q} = \Lambda_{A, q}$$. □ Proof. We have   e−iψLA,qeiψ=−(∇+iA+i∇ψ)2+q. Thus the map $$u \mapsto e^{-i \psi} u$$ is a bijection between solutions to $$L_{A,q} u = 0$$ and solutions to $$L_{A + \nabla \psi,q} u =0$$. Since $$\psi$$ is supported in $$\tfrac{1}{2} B$$, multiplication by $${\rm e}^{i\psi}$$ does not change the boundary data, so the conclusion of the lemma follows. ■ By the gauge-invariance of the Dirichlet–Neumann relation, we have $$\Lambda_{A_2,q_2} = \Lambda_{A_1,q_2}$$. Since we assumed that $$\Lambda_{A_2,q_2} = \Lambda_{A_1,q_1}$$, this implies that $$\Lambda_{A_1,q_1} = \Lambda_{A_1,q_2}$$. Now construct CGO solutions to the equations $$L_{A_1,q_1} u_1 = L_{-A_1,q_2}u_2 = 0$$. Substituting the $$u_i$$ into the integral identity (3) again gives   0∼∫B(q1−q2)eik⋅xdx, and we can conclude that $$q_1 = q_2$$. 3 A transport equation When the magnetic potential $$A$$ is nonzero, the form of the CGO solutions will depend on $$A$$. We construct solutions of the form   u=ex⋅ζ(a+ψ), (6) where $$a = e^{-i\phi}$$ for a suitable function $$\phi$$ depending on $$\zeta$$. The remainder $$\psi$$ must solve the equation   LA,q,ζψ=−Δa−2ζ⋅∇a−i(∇⋅A)a−2iA⋅∇a−2iζ⋅Aa+A2a+qa, (7) where the operator $$L_{A,q,\zeta}$$ is defined by   LA,q,ζ=e−x⋅ζLA,qex⋅ζ=−(∇+iA+ζ)2+q. In order to eliminate the terms of order $$\tau$$ on the right hand side of (7), we choose $$\phi$$ so that $$a$$ solves (roughly speaking) a transport equation of the form   ζ⋅∇a=−iζ⋅Aa. Equivalently, the function $$\phi$$ satisfies an equation of the form   ζ⋅∇ϕ=ζ⋅A. Since $$\zeta = \tau (e_1 + i e_2)$$, where $$e_1$$ and $$e_2$$ are orthonormal vectors, this a $$\bar\partial$$ equation for $$\phi$$ in the plane determined by $$e_1$$ and $$e_2$$. Given $$e=e_1 + i e_2$$, where $$e_1$$ and $$e_2$$ are orthogonal unit vectors, define   ∂¯e=e1⋅∇+ie2⋅∇. We now assume for simplicity that $$e_1$$ and $$e_2$$ are the standard basis vectors. In this case, the operator $$\bar\partial$$ is given by   ∂¯=∂1+i∂2. Let $$f$$ be a function defined on the complex plane, which we identify with $$\mathbb{R}^2$$ by writing $$z = z_1 + iz_2$$. The equation   ∂¯u=f is of Cauchy–Riemann type. If $$f$$ is smooth and compactly supported, then it has a solution given by the formula   ∂¯−1f(w)=12π∫f(w−z)zdz1dz2. The kernel $$(2 \pi z)^{-1}$$ is locally integrable, so it has good mapping properties. Lemma 3.1. If $$f: \mathbb{C} \to \mathbb{R}$$ is supported in $$B(0,1/2)$$, then   ‖⟨w⟩∂¯e−1f(w)‖L∞≲‖f‖L∞. □ Proof. Write   |∂¯e−1f(w)|≲‖f‖L∞∫χB(0,1/2)(z−w)|z|−1dz1dz2 When $$\lvert{w}\rvert \leq 1$$, we estimate the integral by   ∫B(0,3/2)|z|−1dz1dz2∼1. When $$\lvert{w}\rvert > 1$$, we have $$\lvert{z}\rvert \geq \lvert{w}\rvert/2$$ in the region of integration, so we estimate instead by   |w|−1∫χB(0,1/2)(z−w)dz1dz2∼|w|−1. ■ When we substitute CGO solutions of the form $$u_i = {\rm e}^{x \cdot \zeta_i} ({\rm e}^{i \phi_i} + \psi)$$ into the integral identity (3), the main term has the form   −i(ζ1−ζ2)⋅∫B(A1−A2)ei(ϕ1−ϕ2)eix⋅kdx. The next lemma, due to [16] says that we can remove the factor $${\rm e}^{i (\phi_1-\phi_2)}$$ from this integral and recover the Fourier transform. Lemma 3.2. Let $$e_1, e_2, k \,{\in}\, \mathbb{R}^n$$ be arbitrary vectors satisfying $$\lvert{e_1}\rvert\,{=}\,\lvert{e_2}\rvert\,{=}\,1$$ and $$e_1\cdot e_2 = e_1\cdot k= e_2\cdot k=0$$. Let $$A \in C_0^\infty(\mathbb{R}^n)$$, and let $$\phi = \bar\partial_e^{-1}(e\cdot A)$$. Then   (e1+ie2)⋅∫Ae−iϕeix⋅kdx=(e1+ie2)⋅∫Aeix⋅kdx. □ Proof. We first prove the lemma in the case $$n =2$$, where by necessity $$k=0$$. Without loss of generality, we may assume that $$e_1$$ and and $$e_2$$ are the standard basis vectors. Since   (e1+ie2)⋅Ae−iϕ=i∂¯e(e−iϕ), we may write   (e1+ie2)⋅∫Ae−iϕ(x)dx=i∫∂¯e(e−iϕ)dx. (8) By the divergence theorem, we have   ∫(∂1+i∂2)(e−iϕ)dx=limR→∞∫∂B(0,R)(ν1+iν2)e−iϕdS, (9) where $$\nu$$ is the outward unit normal on the circle $$\partial B(0,R)$$. By Lemma 3.1, we have $$\lvert{\phi}\rvert = O(1/\langle {x} \rangle)$$. Thus we have a Taylor expansion of the form   e−iϕ=1−iϕ+O(⟨x⟩−2). Substituting the Taylor series into the right hand side of (9) and applying the divergence theorem again, we find that   ∫∂B(0,R)(ν1+iν2)eiϕdS=∫∂B(0,R)(ν1+iν2)dS−i∫∂B(0,R)(ν1+iν2)ϕdS+∫∂B(0,R)O(R−2)dS=∫B(0,R)∂¯e(1)dx−i∫B(0,R)∂¯eϕdx+O(R−1) Taking the limit as $$R \to \infty$$ we obtain the identity   ∫(∂1+i∂2)(eiϕ)dx=−i∫∂¯eϕdx=−i∫e⋅Adx. Substituting this identity into (8) proves the lemma in the case $$n=2$$. To prove the general case, we assume without loss of generality that $$e_1$$ and $$e_2$$ are the first standard basis vectors. Write $$x = (z, x')$$, where $$z \in \mathbb{R}^2$$ and $$x'\in \mathbb{R}^{n-2}$$. By the two-dimensional case, we have   (e1+ie2)⋅∫A(z,x′)eiϕ(z,x′)dz=(e1+ie2)⋅∫A(z,x′)dz. Since $$k$$ is orthogonal to $$e_1$$ and $$e_2$$, the function $${\rm e}^{ik\cdot x}$$ depends only on $$x'$$. Thus we can multiply both sides by $${\rm e}^{ik\cdot x}={\rm e}^{ik'\cdot x'}$$ and integrate in $$x'$$ to obtain the general case. ■ 4 The operator $$\Delta_\zeta$$ In order to construct solutions to the equation (7) for the remainder $$\psi$$, we consider operators of the form   Δζ=e−x⋅ζΔex⋅ζ. The complex vector $$\zeta \in \mathbb{C}^3$$ is given by   ζ=τ(e1+iη), where $$\tau > 0$$, $$\lvert{e_1}\rvert = 1$$, $$\lvert{\eta}\rvert \leq 1$$ and $$\eta \perp e_1$$. The symbol of $$\Delta_\zeta$$ is   pζ(ξ)=(iξ+ζ)2=−(ξ+τη)2+2iτe1⋅ξ+τ2. The characteristic set $$\Sigma_\zeta$$ is the intersection of the plane perpendicular to $$e_1$$ and a sphere centered at $$-\tau \eta$$.   Σζ={ξ:ξ⋅e1=0,|ξ+τη|=τ}. We will refer to the distance from this set as the modulation. The symbol $$p_\zeta$$ is elliptic at high modulation and vanishes simply on $$\Sigma_\zeta$$, as the reader can easily check (or see [23]).   |pζ|∼{τd(ξ,Σζ) when d(ξ,Σζ)≤τ/8τ2+|ξ|2 when d(ξ,Σζ)≥τ/8. (10) 5 Dyadic projections If $$m$$ is a smooth function on $$\mathbb{R}^n$$, then $$m(D)$$ will denote the Fourier multiplier with symbol $$m(\xi)$$. Let $$\chi \in C^\infty_0([0,1])$$ be a smooth function such that $$\chi = 1$$ on $$[0,3/4]$$. For each dyadic integer $$\lambda = 2^k$$, define the Littlewood–Paley projection $$P_{\leq \lambda}$$ on to frequencies of magnitude $$\lvert{\xi}\rvert \leq \lambda$$ by $$P_{\leq \lambda} = \chi(\lvert{D}\rvert/\lambda)$$. Similarly, define the projection $$P_{>\lambda}$$ on to frequencies of magnitude $$\lvert{\xi}\rvert \gtrsim \lambda$$ by $$P_{>\lambda} = I-P_{\leq \lambda}$$, and define the projection $$P_{\lambda}$$ on to frequencies of magnitude $$\lvert{\xi}\rvert \sim \lambda$$ by $$P_\lambda = P_{\leq \lambda} - P_{\leq \lambda/2}$$. Note that $$I = \sum_{\lambda} P_\lambda$$. Thus we can decompose a function $$f$$ into a sum of dyadic pieces $$f_\lambda = P_\lambda f$$. We can use the Littlewood–Paley decomposition to characterize the Besov spaces $$B^s_{p,q}$$. Given $$s \in \mathbb{R}$$, $$p \in (1,\infty)$$, and $$q \in [1,\infty]$$, the Besov space $$B^s_{p,q}$$ is characterized by the norm   ‖u‖Bp,qs=‖u≤1‖p+(∑λ>1(λs‖uλ‖p)q)1/q. For any integer $$k$$, the Littlewood–Paley square function estimate implies that $$B^k_{p,2} \subset W^{k,p} \subset B^k_{p,p}$$ when $$p\geq 2$$ and that $$B^k_{p,p} \subset W^{k,p} \subset B^k_{p,2}$$ when $$p \leq 2$$. When $$s$$ is not an integer, the Sobolev space $$W^{s,p}$$ is usually defined in such a way that $$W^{s,p} = B^s_{p,p}$$. We will make frequent use of Nikol’skii’s inequality [39] (which is usually referred to as Bernstein’s inequality for some reason). If $$\lambda$$ is any dyadic integer and $$q \geq p$$, then   ‖f≤λ‖q≲λn(1/p−1/q)‖f‖p. (11) Given a pair $$\{e_1,e_2\}$$ of orthonormal vectors, we set $$e = e_1 + i e_2$$ and define partial Littlewood–Paley projections   P≤λe1=χ(|D⋅e1|/λ)P≤λe=χ(|D⋅e|/λ). We define $$P^{e_1}_\lambda$$ and $$P^e_{\lambda}$$ in a similar way. Next, we define projections $$Q_\nu^\zeta$$ to regions where $$d(\xi,\Sigma_\zeta) \sim \nu$$. Let $$\zeta$$ be a complex vector of the form $$\zeta = \tau(e_1 + i \eta)$$. We first define the projection $$C_{\leq \nu}^\zeta$$ by   C≤νζ=χ((|D⊥+τη|−τ)/ν), where $$\xi^\perp = \xi - (\xi\cdot e_1) e_1$$. We then define the projection $$Q_{\leq \nu}^\zeta$$ by   Q≤νζ=P≤νe1C≤νζ. Finally, we define the projection $$Q_\nu^\zeta$$ by $$Q_\nu^\zeta = Q_{\leq \nu}^\zeta - Q_{\leq \nu/2}^\zeta$$ as before. Similarly, we define $$Q_{>\nu} = 1- Q_{\leq \nu}$$. When the choice of $$\zeta$$ is clear from context, we will suppress the dependence of the $$Q$$ projections on $$\zeta$$. Similarly, we will write $$P^1$$ instead of $$P^{e_1}$$ or $$P^{Ue_1}$$. Define projections $$Q_l$$ and $$Q_h$$ on to low and high modulation by   Qlζ=Q≤τ/8ζQhζ=Q>τ/8ζ. Note that the projection $$Q_h^\zeta$$ projects to the region where $$\Delta_\zeta$$ is elliptic. 6 The $$X_\zeta^{b}$$ spaces Given $$b \in (-1,1)$$, define the homogeneous $$\lVert{\cdot}\rVert_{\dot X^{b}_\zeta}$$ norm by   ‖u‖X˙ζb=‖|Δζ|bu‖2, and define the inhomogeneous $$\lVert{\cdot}\rVert_{X^b_\zeta}$$ norm by   ‖u‖Xζb=‖(|Δζ|+τ)bu‖2. By the symbol estimates (10), we have the low-modulation $$L^2$$ estimate   ‖Qμu‖X˙ζb∼(μτ)b‖Qμu‖2, (12) which holds for $$\mu \leq \tau/8$$, and the high-modulation $$L^2$$ estimate   ‖Qhu‖Xζb∼‖u‖Hτb/2, (13) where the semiclassical $$\lVert{\cdot}\rVert_{H^s_\tau}$$ norm is defined by   ‖u‖Hτs=‖(|D|+τ)su‖2. The $$X^b_\zeta$$ spaces behave well under localization, as we see from the following lemma. Lemma 6.1. ([23 Lemma 2.2]). If $$\phi$$ is a Schwartz function, then   ‖ϕu‖Xζ1/2≲ϕ‖u‖X˙ζ1/2 (14)  ‖ϕu‖X˙ζ−1/2≲ϕ‖u‖Xζ−1/2. (15) □ Let $$\phi$$ be a fixed Schwartz function which is identically equal to one on the unit ball. Then   τ1/2‖u‖L2(B(0,1))≤τ1/2‖ϕu‖L2≤‖ϕu‖Xζ1/2≲‖u‖X˙ζ1/2. Replacing $$u$$ with $$u(x/R)$$, we obtain the Agmon–Hörmander-type estimate   τ1/2R−1/2‖u‖L2(B(0,R))≲‖u‖X˙ζ1/2. (16) Using the estimate (16), it is not hard to show that the space $$\dot X^{1/2}_\zeta$$ is a Banach space and embeds continuously into $$\tau^{-1/2} L^2(\mathbb{R}^3, \langle x \rangle^{-1-\delta}\,dx)$$ for any $$\delta>0$$. The next lemma gives Strichartz-type estimates for the $$X^b_\zeta$$ spaces. Lemma 6.2. ([24 Proposition 6.3]). Suppose $$\nu\leq \tau/8$$. Then for any $$f \in \dot X^{1/2}_\zeta(\mathbb{R}^3)$$, we have   ‖Qνf‖6≲(ν/τ)1/3‖f‖X˙ζ1/2. (17)  ‖f‖6≲‖f‖X˙ζ1/2. (18) □ We also have the dual estimates   ‖Qνf‖Xζ−1/2≲(ν/τ)1/3‖f‖6/5 (19)  ‖f‖Xζ−1/2≲‖f‖6/5. (20) 7 Averaging estimates We will need to average various norms with respect to parameters $$(\tau, U)$$, which will be chosen from the set $$[2,\infty) \times O(3)$$. In order to distinguish this averaging from integration over physical space, we will use probabilistic notation. Let $$(X,\sigma)$$ be a finite measure space such that $$\sigma(X)>0$$. Let $$Z$$ be an integrable function on $$X$$. We write the average of $$Z$$ over $$X$$ as   E[Z∣X]=σ(X)−1∫XZdσ. Similarly, for a measurable subset $$Y$$ of $$X$$, we write   P[Y∣X]=σ(Y∩X)σ(X). Define the $$L^p$$ average of $$Z$$ over $$X$$ by   Ep[Z∣X]=‖Z‖Lp(X,dσ)‖1‖Lp(X,dσ). Unless otherwise specified, the set $$X$$ will be the orthogonal group $$O(3)$$, and $$\sigma$$ will be normalized Haar measure on $$O(3)$$. Given a measurable function $$Z$$ on $$[\tau_*,2\tau_*] \times O(3)$$, define   Eτ∗p[Z]=Ep[Z∣[τ∗,2τ∗]×O(3)], where the average here is taken with respect to the measure $$m$$ on $$[2, \infty) \times O(3)$$ given by   dm(τ,U)=(τlog⁡τ)−1dτdσ(U). For a positive integer $$K$$, define   E~Kp[Z]=Ep[Z∣[2K,2K2]×O(3)] and   P~K[Y]=P[Y∣[2K,2K2]×O(3)]. The quantity $$m([2^K,2^{K^2}] \times O(3))$$ is given by   ∫2K2K2(τlog⁡τ)−1dτdσ(U)∼log⁡K. (21) On each dyadic interval $$[\tau^*,2\tau^*]$$, the weight $$(\tau \log \tau)^{-1}$$ is approximately constant. Thus we can estimate $$\widetilde{\mathbb{E}}_{k}^{p}[Z]$$ by   E~Kp[Z]p ~(log⁡K)−1∑2K≤τ∗<2K2(log⁡τ∗)−1Eτ∗p[Z]p. We will need the following property of the Haar measure: if $$f$$ is an integrable function on $$S^{2}$$, then for any fixed $$\theta\in S^{2}$$, we have the identity   E[f(U⋅ θ)∣O(3)]=E[f(ω)∣S2]. (22) We will use the averaging in $$\tau$$ to take advantage of the extra decay in expressions of the form $$(\lambda /\tau)^\alpha \lVert{P_\lambda f}\rVert_p$$, where $$\lambda$$ is a dyadic integer less than or comparable to $$\tau$$. Namely, if $$\alpha > 0$$ and $$p \in [2,\infty)$$, then we have frequency convolution estimate   E~Kp[∑λ≲τ(λ/τ)α(log⁡τ)1/p‖Pλf‖p]≲(log⁡K)−1/p‖f‖p. (23) To see this, we recall the normalization (21) and use Young’s inequality, which gives   ‖∑λ≲τ(λ/τ)α‖Pλf‖p‖Lp([2K,2K2],τ−1dτ)≲(∑λ‖Pλf‖pp)1/p. That is, the right-hand side has the form $$\lVert{T(\{a_\lambda\})}\rVert_{L^p}$$, where $$T: l^p \to L^p([2^K,2^{K^2}],\tau^{-1}\,{\rm d}\tau)$$ is given by $$T(\{a_\lambda\})(\tau) = \sum_{\lambda} K(\lambda,\tau) a_\lambda$$ and $$\sup_\lambda \lVert{K(\lambda,\tau)}\rVert_{L^1([2^K,2^{K^2}],\tau^{-1}\,d\tau)} + \sup_\tau \lVert{K(\lambda,\tau)}\rVert_{l^1_\lambda}$$ is bounded. By the Besov embedding $$L^p \subset B^0_{p,p}$$, the right hand side is bounded by $$\lVert{f}\rVert_p$$. Lemma 7.1. Let $$p \in [2,\infty]$$ and let $$1/p'=1-1/p$$. Let $$\mu,\nu,\lambda$$ be dyadic integers, such that $$\mu,\nu\lesssim \lambda$$. Then   Ep[⟨λ/ν⟩1/p⟨λ/μ⟩1/p‖P≤μUe1P≤νUe2Pλu‖p]≲‖u‖p (24)  Ep′[⟨λ/ν⟩1/p⟨λ/μ⟩1/p‖P≤μUe1P≤νUe2Pλu‖p′]≲‖u‖p′. (25) As usual, the averages are taken over $$U$$ in $$O(3)$$. If $$p \in [2,4]$$, then we also have   Eτ∗p[(1+log+⁡(τ/ν))5(1/p−1/2)⟨λ/ν⟩3/p−1/2‖Q≤ντ(Ue1+iUe2)Pλu‖p]≲‖u‖p (26) □ Proof. When $$p=2$$, all of these estimates follow from Plancherel’s theorem and Fubini. To prove the first estimate (24) when $$p=2$$, we write   E2[‖P≤νUe1P≤μUe2Pλu‖2]2∼∫RnE|ϕ(ξλ)χ(ξ⋅(Ue1)μ,ξ⋅(Ue2)ν)|2|u^(ξ)|2dξ Here $$\phi$$ is supported on an annulus, and $$\chi$$ is supported on a square. Since $$U$$ is orthogonal, we have $$\xi \cdot (Ue_i) = (U^{-1} \xi) \cdot e_i$$. Thus we can compute the last integral using the identity (22):   E|ϕ(ξλ)χ((U−1ξ)⋅e1μ,(U−1ξ)⋅e2ν)|2≲sup|ξ|∼λE[|χ(|ξ|ω⋅e1μ,|ξ|ω⋅e2ν)|2|ω∈S2] The quantity on the right is bounded by the area of the intersection of the unit sphere with a rectangle centered at the origin of size proportional to $$(\mu/\lambda) \times (\nu/\lambda)\times 1$$. Since the area of such a region is bounded by $$\langle{\lambda/\mu}\rangle^{-1} \langle{\lambda/\nu}\rangle^{-1}$$, we have   E2[‖P≤νUe1P≤μUe2Pλu‖2]2≲⟨λ/μ⟩−1⟨λ/ν⟩−1‖u‖22. The $$p=2$$ case of the last estimate (26) is proven in the same way. Since $$\lvert {p_\zeta(\xi)} \rvert \lesssim \tau \nu$$ on the Fourier support of $$Q_\nu$$, it suffices to show that   Eτ∗[Z(τ,U)]≲⟨λ/ν⟩−1, (27) where   Z(τ,U)=sup|ξ|∼λ|χ(|−|ξ|2−2τξ⋅(Ue2)+2τ|ξ⋅(Ue1)|τν)|2 and $$\chi$$ is compactly supported. Using the identity (22) again yields   Eτ∗[Z]≲sup|ξ|∼λ1τ∗∫τ∗2τ∗∫S2|χ(|−|ξ|2−2τ|ξ|ω⋅e2|+2τ||ξ|ω⋅e1τν)|2dS(ω)dτ. (28) View $$(\tau,\omega)$$ as polar coordinates on $$\mathbb{R}^3$$, and change variables to $$u = \tau\omega$$. In the annular region $$\{\lvert u \rvert\in [\tau_*,2\tau_*]\}$$, the volume element $$du$$ is bounded below by $$\tau_*^{2} \, {\rm d}S(\omega) \,{\rm d}\tau$$. Thus the integral on the right hand side of (28) is bounded by   1τ∗3∫|u|∈[τ∗,2τ∗]|χ(|ξ|(|−|ξ|−2u⋅e2|+2|u⋅e1|)τν)|2du. The integrand is supported on a rectangle of size proportional to $$\tau(\langle{\lambda/\nu}\rangle^{-1} \times \langle{\lambda/\nu}\rangle^{-1} \times 1)$$. So the integral is bounded by the quantity $$\langle{\lambda/\nu}\rangle^{-2}$$, which establishes (27). This shows that   Eτ∗2[‖Q≤νPλu‖2]2≲⟨λ/ν⟩2‖u‖22. To prove the $$p \neq 2$$ case of the first two estimates (24) and (25), we define an operator $$T$$ by   Tu(U,x):=P≤μUe1P≤νUe2Pλu(x). We have shown that $$T$$ satisfies the $$L^2$$ bound   ‖T‖L2(R3)→L2(O(3)×R3)≲⟨λ/ν⟩−1/2⟨λ/μ⟩−1/2. On the other hand, since the Littlewood–Paley projections $$P^{Ue_i}_{\leq \mu}$$ and $$P_\lambda$$ are all bounded on every $$L^p$$ space, the operator $$T$$ also satisfies the bounds   ‖T‖L∞(R3)→L∞(O(3)×R3)+‖T‖L1(R3)→L1(O(3)×R3)≲1. By interpolation, we obtain the $$L^p$$ bounds (24) and (25). To prove the $$p\neq 2$$ case of the last estimate (26) for $$Q_{\leq \nu}^{\zeta(\tau,U)}$$, we interpolate with an $$L^4$$ bound. The operator $$Q_{\leq \nu}$$ factors as   Q≤νζ(τ,U)=C≤νζ(τ,U)P≤νUe1, where the operator $$C_{\leq \nu}^{\zeta(\tau,U)}$$, defined by   C≤νζ(τ,U)=χ((|D⊥+τe2|−τ)/ν), localizes the vector $$\xi^\perp = (0,\xi \cdot U e_2, \xi\cdot Ue_3)$$ to a neighborhood of a circle of radius $$\tau$$ and center $$(\xi\cdot (Ue_1), \tau e_2)$$. The Carleson–Sjölin theorem ([11]) implies that $$C^{\zeta(\tau,U)}_{\leq \nu}$$ satisfies the $$L^4$$ bound   ‖C≤νζ(τ,U)‖L4(R3)→L4(R3)≲(1+log+⁡(τ/ν))5/4. (29) This estimate (modulo rescaling and modulation) is explicit in [14]. Thus, by combining the $$L^4$$ bound (29) with the case $$p=4$$ of the bound (24) that we have already established, we obtain the $$L^4$$ bound   Eτ∗4[‖Q≤νζ(τ,U)u‖4]≲(1+log+⁡(τ/ν))5/4Eτ∗4[‖P≲νUe1u‖4]≲(1+log+⁡(τ/ν))5/4⟨λ/ν⟩−1/4‖u‖4. Interpolating this $$L^4$$ estimate with the $$L^2$$ estimate we have already established, we obtain the $$L^p$$ estimate (26). ■ 8 Estimates for the amplitude To analyze the behavior of $$\bar\partial^{-1}$$, we introduce an auxiliary function $$\eta$$ to use as a mollifier. Let $$\eta: \mathbb{R}^2 \to \mathbb{R}$$ be a smooth compactly supported bump function, such that $$\int_{\mathbb{R}^2} \eta = 1$$ and   ∫z1α1z2α2η(z)dz1dz2=0, (30) for every pair $$(\alpha_1,\alpha_2)$$ of nonnegative integers such that $$1 \leq \alpha_1 + \alpha_2 \leq 2M$$, where $$M$$ is some large number to be determined later. The vanishing moment condition (30) ensures that the Fourier transform $$\hat \eta$$ satisfies   η^(ξ)=1+O(|ξ|2M+1). (31) Define the operator $$\tilde P$$ (acting on functions on $$\mathbb{R}^2$$) by   P~u=η^(D)u, Let $$\chi_\nu$$ be the symbol of the Littlewood–Paley projection $$P_{\nu}$$. Since $$\hat \eta$$ is Schwartz and $$\chi_\nu$$ is supported in the set $$\{\xi: \lvert {\xi} \rvert \lesssim \nu\}$$, we have   ⟨ξ⟩l|∇ξk(χνη^)(ξ)|≲k,l,Nν−N for all non-negative integers $$k,l , N$$ and uniformly in $$\nu \geq 1$$. This implies that the integral kernel $$K_\nu$$ of $$\tilde P P_\nu$$ is Schwartz, and more precisely that $$ \lvert {K_\nu(x)} \rvert \lesssim_{N} \nu^{-N}\langle {x} \rangle^{-10}$$. Thus for any $$p \in [1,\infty]$$ and $$N > 0$$, the operator $$\tilde P$$ is almost orthogonal to $$\{P_\nu\}_{\nu \geq 1}$$, in the sense that   ‖P~Pν‖Lp(R2)→Lp(R2)≲Nν−N (32) for all $$N> 0$$, and uniformly in $$\nu \geq 1$$. On the other hand, the vanishing property (31) implies that   ⟨ξ⟩l|∇ξk(χν(1−η^))(ξ)|≲k,lν2M+1−k for all non-negative integers $$k, l\leq M$$ and uniformly in $$\nu \leq 1$$. It follows that for $$M$$ sufficiently large, the operator $$(1-\tilde P)$$ is almost orthogonal to $$\{P_\nu\}_{\nu \leq 1}$$, in the sense that   ‖(1−P~)Pν‖Lp(R2)→Lp(R2)≲νM (33) for all $$p \in [1,\infty]$$ and uniformly in $$\nu \leq 1$$. It is easy to see that the same estimates hold on $$L^p(\mathbb{R}^3)$$ if $$\tilde P$$ and $$ P_\nu$$ are replaced by $$\tilde P^e = \hat \eta(D \cdot e)$$ and $$P_\nu^e = \chi_\nu(D\cdot e)$$. We apply this machinery to show that the behavior of $$\bar\partial$$ near its the characteristic set can be ignored if everything is localized. Lemma 8.1. Let $$u$$ be a function on $$\mathbb{R}^2$$ whose support lies in the unit ball. Let $$E\subset \mathbb{R}^2$$ be a set of finite measure. Then for $$p \in [1,\infty]$$ we have   ‖∂¯−1u‖Lp(E)≲(1+|E|1/p)(‖P≤1u‖p+‖∂¯−1P>1u‖p). (34) □ Proof. Decompose $$u$$ as $$u = \tilde P u + (1-\tilde P) u$$. By Hölder’s inequality, we have $$\lVert{\bar\partial^{-1} \tilde P u}\rVert_{L^p(E)} \leq \lvert {E} \rvert^{1/p} \lVert{\bar\partial^{-1} \tilde Pu }\rVert_\infty$$. By Lemma 3.1 and Nikol’skii’s inequality (11)   ‖∂¯−1P~u‖∞≲‖P~u‖∞≲‖P~P≤1u‖∞+∑ν>1‖P~Pνu‖∞≲‖P≤1u‖p+∑ν>1ν2/p‖P~Pνu‖p. By the almost orthogonality bound (32), we have   ∑ν>1ν2/p‖P~Pνu‖p≲‖P≤1u‖p+∑ν>1ν2/p−N‖Pνu‖p≲‖P≤1u‖p+∑ν>1ν2/p+1−N‖Pν∂¯−1u‖p≲‖P≤1u‖p+‖∂¯−1P>1u‖p. For $$(1-\tilde P)u$$ we use the almost orthogonality bound (33) and the fact that $$(1-\tilde P)$$ is bounded on $$L^p$$ for any $$p$$. Thus   ‖∂¯−1(1−P~)u‖p≲∑ν≤1ν−1‖(1−P~)Pνu‖p+‖∂¯−1P>1u‖p≲∑ν≤1νM−1‖P≤1u‖p+‖∂¯−1P>1u‖p≲‖P≤1u‖p+‖∂¯−1P>1u‖p. ■ Using the localization estimate (34), we show that $$\bar\partial_{Ue}^{-1} \nabla f$$ is bounded on average in $$L^2(B)$$ with a slight loss of regularity. Lemma 8.2. Let $$f \in H^s(\mathbb{R}^3)$$, where $$s>0$$, and suppose that $$\text{supp} f \subset B$$, where $$B = B(0,1)$$. Then   E2[‖∂¯Ue−1∇f‖L2(B)]≲s‖f‖Hs. □ Proof. First we apply the localization estimate (34), to obtain   ‖∂¯Ue−1∇f‖L2(B)≲‖P≤1Ue∇f‖L2+‖P>1Ue∂¯Ue−1∇f‖L2. We bound both terms using the averaging estimate (24).   E2[‖P≤1Ue∇f‖L2]≲‖∇f≤1‖2+∑λ>1λE2[‖P≤1Uefλ‖2]≲‖f‖2+∑λ>1‖fλ‖2≲‖f‖Hs. Here $$f_{\leq 1}$$ and $$f_\lambda$$ are defined using the Littlewood–Paley projections $$P_\lambda$$ on $$\mathbb{R}^3$$. Similarly, since $$P^{Ue}_\nu f_\lambda = 0$$ unless $$\nu \lesssim \lambda$$, we have   E2[‖P>1Ue∂¯Ue−1∇f‖L2]≲∑1≤ν≲λ(λ/ν)E2[‖PνUefλ‖2]≲∑λ>1(log⁡λ)‖fλ‖2≲‖f‖Hs. ■ We now show that the $$\bar\partial^{-1}_{U e}$$ operator takes compactly supported functions in the Besov space $$B^0_{3,1}(\mathbb{R}^3)$$ to bounded functions. If the $$\bar\partial^{-1}_{Ue}$$ operator was replaced by $$\lvert {D} \rvert^{-1}$$, then this property would hold without any averaging. Lemma 8.3. Let $$f \in B^0_{3,1}(\mathbb{R}^3)$$, and suppose that $$\text{supp} f \subset B(0,1)$$. Then   E3[‖∂¯Ue−1f‖∞]≲‖f‖B3,10. □ Proof. First, we show that for such $$f$$, we have the estimate   ‖∂¯Ue−1f‖∞≲Z(U), where   Z(U)=‖f‖3+∑1≤ν,λ(λ/ν)1/3‖P≤νUePλf‖3. To this end, we apply the localization estimate (34), which gives   ‖∂¯Ue−1f‖∞≲‖P≤1Uef‖∞+‖∂¯Ue−1P>1Uef‖∞. Next, we decompose $$f$$ into Littlewood-Paley pieces and apply Nikol’skii’s inequality (11). We estimate $$P^{Ue}_{\leq 1} f$$ by   ‖P≤1Uef‖∞≲‖P≤1Uef≤1‖∞+∑ν,λ>1‖P≤1Uefλ‖∞≲‖f‖3+∑ν,λ>1λ1/3‖P≤1Uefλ‖3.≤Z(U). We estimate $$\bar\partial^{-1}_{Ue}P^{Ue}_{>1} f$$ in the same way. Note that $$P_\nu^{Ue} f_\lambda$$ vanishes unless $$\nu \lesssim \lambda$$, so   ‖∂¯Ue−1P>1Uef‖∞≲∑1<ν≲λν−1‖PνUefλ‖∞≲∑1<ν≲λν−1/3λ1/3‖PνUefλ‖3≲Z(U). Finally, we show that $$Z(U)$$ is bounded on average. This follows from the averaging estimate (24), which gives   E3[Z(U)]≲‖f‖3+∑1≤ν≤λ(ν/λ)1/3‖Pλf‖+∑1≤λ≤ν(λ/ν)1/3‖Pλ‖3≲‖f‖3+∑λ≥1‖Pλf‖3≲‖f‖B3,10. ■ To state the next lemma, we introduce the mixed-norm notation   ‖f‖Lx1p1Lx2p2Lx3p3=‖‖‖f(x1,x2,x3)‖Lx3p3‖Lx2p2‖Lx1p1. Given an orthonormal frame $$\{e_1,e_2,e_3\}$$, we will also use the notation   ‖f‖Le1p1Le2p2Le3p3=‖f(x)‖Ly1p1Ly2p2Ly3p3, where the $$L^p$$ norms on the right hand side are taken with respect to the coordinates $$y_i = x \cdot e_i$$. Sometimes we will write $$L^p_e$$ for $$L^p_{e_1} L^p_{e_2}$$, where $$e = e_1 + i e_2$$. We will also omit to specify all of the directions $$e_i$$ when they can be inferred from context. Observe that the norms $$L^\infty L^1_{e}$$ and $$L^{3/2}$$ scale identically under the isotropic dilations $$x \mapsto \lambda x$$, where $$\lambda$$ is a positive real number. Of course, there cannot possibly be a straightforward relationship between these norms, since they scale differently with respect to the anisotropic dilations $$x_e + x_{e_3} \mapsto \lambda x_e + \mu x_{e_3}$$, where $$x_e$$ and $$x_{e_3}$$ are the projections of $$x$$ on to the $$e$$ and $$e_3$$ directions, respectively. The next lemma, due to Falconer [18], states that we can control one norm with respect to the other (with an $$\epsilon$$ loss of integrability) if we average over all of the different frames $$\{e_1,e_2,e_3\}$$. Since we will apply it to $$A$$ in the subcritical space $$W^{s,3}$$, this will suffice for our purposes. We will not use Lemma 8.4 as stated. Instead we will use an easy consequence: if $$q$$ lies in the range $$(3,3/(1-s))$$, then   Eq[‖A‖L∞LUe2]≲‖A‖q≲‖A‖Ws,3. (35) We prove the above estimate by applying Lemma 8.4 to the function $$A^2$$, which lies in $$L^{q/2}$$. Lemma 8.4. Let $$f \in L^{p}(\mathbb{R}^3)$$, where $$p> 3/2$$. Assume $$f$$ is supported in a ball $$B(0,1)$$. Then   Ep[‖f‖L∞LUe1]≲‖f‖p. □ Proof. Let $$g = \lvert {f} \rvert$$, and let $$\eta$$ be a mollifier as defined above. Since $$g$$ is non-negative, we have   ‖g‖Lx3∞Lx1,x21=esssupx3∫g(x1,x2,x3)dx1dx2=esssupx3∫∫η((y1−x1)+i(y2−x2))dy1dy2g(x1,x2,x3)dx1dx2=esssupx3∫η∗g(y1,y2,x3)dy1dy2 Thus we can replace $$g$$ by $$\tilde P g$$, since   ‖g‖Lx3∞Lx1,x21≤‖P~g‖Lx3∞Lx1,x21. More generally, for $$U \in O(3)$$, we have   ‖g‖L∞LUe1≤‖P~Ueg‖L∞LUe1. Now, since $$\tilde P^{Ue} g$$ is supported in the ball $$B(0,2)$$, Hölder’s inequality implies that   ‖P~Ueg‖L∞LUe1≲‖P~Ueg‖L∞LUep. Decompose $$g$$ as $$g = \sum_{\nu,\lambda\geq 1} P^{Ue}_\nu P_\lambda g$$. By abuse of notation, we redefine $$P_1$$ and $$P^{Ue}_1$$ as $$P_1 = P_{\leq 1}$$ and $$P^{Ue}_1 = P^{Ue}_{\leq 1}$$. For each of these pieces, we apply Nikol’skii’s inequality (11) in the $$U e_3$$ direction, which gives   ‖P~UePνUePλg‖L∞LUep≲λ1/p‖P~UePνUePλg‖p By the almost orthogonality bound (32), this implies that   ∑ν,λ≥1‖P~UePνUePλg‖L∞LUep≲∑ν,λ≥1λ1/pν−N‖PνUePλg‖p Averaging over $$O(3)$$ using the averaging bound (25), we obtain   ∑ν,λ≥1λ1/pν−NE3[‖P~UePνUePλg‖L∞LUep]≲∑ν,λ≥1λ1/p⟨λ/ν⟩2/p−2ν−N‖Pλg‖p≲∑λ≥1(λ1/p−N+λ3/p−2)‖g‖p≲‖g‖p. ■ We are now ready to prove estimates in the space $$X^{-1/2}_\zeta$$ for some dangerous terms that will appear as inhomogeneous terms in the equation (7), which we will use to construct the remainder term $$\psi$$. The next lemma is fairly straightforward to prove if $$a = 1$$; in that case it follows from the fact that the $$X^{-1/2}_\zeta$$ norm is controlled, on average, by the $$W^{-1,2}$$ norm. Since we control the $$W^{-1,3}$$ norm, there is some slack here. When $$a$$ is nontrivial, we will have to work harder, but this extra slack will help us get the required estimates. Lemma 8.5. Let $$q \in W^{-1,3}(\mathbb{R}^3)\cap W^{-1,2} (\mathbb{R}^3)$$, and suppose that for each $$(\tau, U)$$ in $$\mathbb{R}_+ \times O(3)$$ we are given a function $$a_{\tau,U}$$, such that   M=supτ,U(‖a‖∞+τ−1‖∇a‖∞+τ−2‖∇2a‖∞+‖∇a‖2)<∞. Then   E~K2‖a⋅q‖XτU(e1+ie2)−1/2 ≲M(log⁡K)−1/3(‖q‖W−1,2+‖q‖W−1,3)+‖q>2K‖W−1,2. (36) □ Proof. Let $$\zeta = \tau U(e_1+ie_2)$$. In what follows, we will use $$\lVert{\cdot}\rVert$$ to denote the $$X^{-1/2}_\zeta$$ norm. Since we are working with homogeneous norms, it is convenient to redefine all of our dyadic projections by $$P_1 = P_{\leq 1}$$, $$Q_1 = Q_{\leq 1}$$ and so on. At high modulation, we use the high-modulation estimate (13)   ‖Qh(aq)‖≲‖aq‖Hτ−1. We estimate this using the definition of $$H^{-1}_\tau$$. For a test function $$u$$, we have   |(aq,u)|=|(q,a¯u)|≲∑1≤λ≤τ|(qλ,a¯u)|+|(q>τ,a¯u)|≲∑1≤λ≤τ(λ/τ)‖qλ‖W−1,2‖a‖∞τ‖u‖2+‖q>τ‖W−1,2‖au‖H1≲M(∑1≤λ≤τ(λ/τ)‖qλ‖W−1,2+‖q>τ‖W−1,2)‖u‖Hτ1. Thus by duality, we have   ‖aq‖Hτ−1≲M∑1≤λ≤τ(λ/τ)‖qλ‖W−1,2+M‖q>τ‖W−1,2. Applying the frequency convolution estimate (23), we have   E~K2[‖aq‖Hτ−1] ≲M(log⁡K)−1/2‖q‖W−1,2+M‖q>2K‖W−1,2. We decompose the low-modulation part as   Ql(a⋅q)=LL+HH, (37) where the low–low part is given by   LL=Ql(a≲τq≲τ) and the high–high part is given by   HH=∑λ1,λ2≫τQl(aλ1qλ2)=∑λ≫τQl(aλq∼λ). For each $$\lambda$$, the expression $$q_{\sim \lambda}$$ denotes a sum of Littlewood–Paley projections of $$q$$ with frequencies comparable to $$\lambda$$. Here we use that fact that if the ratio between $$\lambda_1$$ and $$\lambda_2$$ is very large or very small, then $$a_{\lambda_1} q_{\lambda_2}$$ has Fourier support in $$\{\xi: \lvert {\xi} \rvert \sim \max\{\lambda_1,\lambda_2\} \gg \tau\}$$. We further decompose the low-low part as   LL=I+II+III+IV+V, where   I=∑1≤λ≲τ∑μ≥A(λ,τ)Qμ(a≲τqλ)II=∑1≤λ≲τ∑μ<A(λ,τ)Qμ(a≤μqλ)III=∑1≤λ≲τ∑μ<A(λ,τ)Qμ(a≥λqλ)IV=∑1≤λ≲τ∑μ<A(λ,τ)Qμ(a(μ,B(μ,λ,τ))qλ)V=∑1≤λ≲τ∑μ<A(λ,τ)Qμ(a[B(μ,λ,τ),λ)qλ). The cut-offs $$A(\lambda,\tau)$$ and $$B(\mu,\lambda,\tau)$$ will be chosen later. We estimate $$\lVert{I}\rVert_{X^{-1/2}_\zeta}$$ by the $$L^2$$ estimate (12).   ‖I‖≲∑1≤λ≲τ∑μ≥A(λ,τ)(μτ)−1/2λ‖a‖∞‖qλ‖W−1,2≲‖a‖∞∑λ≲τ(λ2/τ)1/2A(λ,τ)−1/2‖qλ‖W−1,2. Taking $$A(\lambda,\tau) = \lambda^{2-2\epsilon} \tau^{-1+2\epsilon}$$, we apply the frequency convolution estimate (23) again to obtain   ‖I‖≲‖a‖∞∑λ≲τ(λ/τ)ϵ‖qλ‖W−1,2E~K2[‖I‖]≲(log⁡K)−1/2‖a‖∞‖q‖W−1,2. For $$\lVert{II\rVert}$$ multiplication by $$a_{\lesssim \mu}$$ shifts the Fourier support by at most $$\mu$$. Thus we have $$Q_\mu(a_{\leq \mu}q_{>\mu}) = Q_\mu(a_{\leq \mu} Q_{\lesssim \mu}q_{>\mu})$$. By the $$L^2$$ estimate (12) and the averaging estimate (26), we have   Eτ∗2[‖II‖]≲∑μ≲A(λ,τ∗)(μτ∗)−1/2‖a‖∞Eτ∗2[‖Q≲μqλ‖2]≲M∑λ≲τ∗∑μ≲A(λ,τ∗)(μ/τ∗)1/2‖qλ‖W−1,2≲M∑λ≲τ∗(λ/τ∗)1−ϵ‖qλ‖W−1,2E~K2[‖II‖]≲M(log⁡K)−1/2‖q‖W−1,2. For $$\lVert{III}\rVert$$, we use the Strichartz estimate (19):   ‖III‖≲∑λ≲τ∑μ<A(λ,τ)∑ν≥λ(μ/τ)1/3‖aνqλ‖6/5≲∑λ≲τ∑μ<A(λ,τ)∑ν≥λ(μ/τ)1/3(λ/ν)‖∇aν‖2‖qλ‖W−1,3≲∑λ≲τ(λ/τ)(2−2ϵ)/3‖∇a‖2‖qλ‖W−1,3E~K3[‖III‖]≲M(log⁡K)−1/3‖q‖W−1,3. For the terms in $$\lVert {IV_\mu} \rVert$$ we can use the identity $$Q_\mu(a_{(\mu,B)} q_\lambda) = Q_\mu(a_{(\mu,B)} Q_{\lesssim B} q_\lambda)$$. Thus by the $$L^2$$ estimate (12) and the averaging estimate (26) we have, with $$B = \min{\{\lambda, \mu^{1/2}\tau^{1/2-2\epsilon} \lambda^{2\epsilon}\}}$$,   ‖Qμ(a(μ,B)qλ)‖≲(μτ)−1/2‖a‖∞‖Q≲Bqλ‖2Eτ∗2[‖Qμ(a(μ,B)qλ)‖]≲M(λ/τ∗)2ϵ‖qλ‖W−1,2. Summing over $$\mu$$, we obtain   Eτ∗2[‖IV‖]≲ M(log⁡τ∗)1/2∑λ≲τ∗(λ/τ∗)2ϵ‖qλ‖W−1,2E~K2[‖IV‖]≲M(log⁡K)−1/2‖q‖W−1,2. For $$\lVert {V_\mu} \rVert$$, we use the identity $$Q_\mu(a_\nu q_\lambda) = Q_\mu(a_\nu Q_{\lesssim \nu} q_\lambda)$$. Using the Strichartz estimate (19) and the averaging estimate (26), we obtain   ‖Qμ(aνqλ)‖≲(μ/τ)1/3‖aνQ≲ν‖qλ6/5≲(μ/τ)1/3‖aν‖2‖Q≲νqλ‖3≲(μ/τ)1/3ν−1‖∇aν‖2‖Q≲ν‖qλ3Eτ∗3[‖Qμ(aνqλ)‖]≲(μ/τ∗)1/3(λ/ν)1/2(τ∗/ν)2ϵM‖qλ‖W−1,3. Summing over $$\nu$$, we have   Eτ∗3[‖Qμ(a[B,λ)qλ)‖]≲(μ/τ∗)1/3(λ/B)1/2(τ∗/B)2ϵM‖qλ‖W−1,3≲μ1/12−ϵτ∗−7/12+2ϵ+4ϵ2λ1/2−ϵ−4ϵ2M‖qλ‖W−1,3≲(μ/λ)1/12−ϵ(λ/τ∗)7/12−2ϵ−4ϵ2M‖qλ‖W−1,3. Summing over $$\mu$$ and applying the frequency convolution estimate (23), this gives   Eτ∗3[‖V‖]≲ ∑λ≲τ∗(λ/τ∗)αM‖qλ‖W−1,3E~K3[‖V‖]≲M(log⁡K)−1/3‖q‖W−1,3. Finally, we estimate the high-high terms. When the modulation is sufficiently small, we use the Strichartz estimate (19)   ‖Q≤C(aλq∼λ)‖≲(C/τ)1/3λ‖aλ‖2‖q∼λ‖W−1,3≲(C/τ)1/3M‖q‖W−1,3. (38) When the modulation is large, we use the $$L^2$$ estimate (12) and then estimate $$\lVert{A}\rVert_6$$ by interpolation.   ‖Q>C(aλq∼λ)‖≲(Cτ)−1/2λ‖aλ‖6‖q∼λ‖W−1,3≲(Cτ)−1/2λ‖aλ‖∞2/3‖aλ‖21/3‖q∼λ‖W−1,3≲(Cτ)−1/2λ2/3(τ/λ)4/3M‖q‖W−1,3. (39) Here we use that $$\lVert {\nabla^2 a} \rVert_\infty \lesssim \tau^2 M$$. Let $$C = \tau \lambda^{-\epsilon}$$. Summing the inequalities (38) and (39) over $$\lambda \gtrsim \tau$$, we obtain   ‖HH‖≲ ∑λ≳τ(λ−ϵ/3+τ1/3λ−2/3+ϵ)M‖q‖W−1,3E~K[‖HH‖]≲2−ϵK/3M‖q‖W−1,3. ■ In the next lemma, we make use of the relationship between the operator $$\bar\partial_e$$ and the operator $$\Delta_\zeta$$. Lemma 8.6. Fix $$s>1$$. Let $$B = B(0,1)$$. Let $$A$$ be a smooth function supported in $$\frac{1}{2} B$$, and let $$\chi$$ be a cut-off supported in $$B$$ such that $$\chi = 1$$ on $$\tfrac{1}{2} B$$. Let $$a = \exp(\bar\partial_e^{-1} A)$$. Then   ‖Δ(χa)‖Xζ−1/2≲s(1+‖∇∂¯e−1A‖L2(B)+e‖∂¯e−1A‖∞+‖A‖L∞Le2+‖⟨∇1⟩−1/2+s⟨∇2⟩−1/2+s∇A‖2+‖⟨∇1,2⟩−1+s∇A‖2)4. □ Proof. As in the previous lemma, we redefine $$P_1$$ as $$P_{\leq 1}$$ and so on. At high modulation we use the high-modulation estimate (13).   ‖QhΔ(χa)‖X−1/2≲‖χa‖H1≲‖a‖H1(B). It remains to consider the low-modulation part of $$\chi a$$. By the $$L^2$$ estimate (12),   ‖QlΔ(χa)‖X−1/22≲∑1≤μ≤τ/8∑λ≲τ(μτ)−1‖QμPλeΔ(χa)‖22. Now we observe that at low modulation, the symbol bounds (10) give   |ξ|2=2iζ⋅ξ−pζ(ξ)≲τ|ξ⋅e|+τd(ξ,Σ). Thus, when $$\lambda \leq \mu$$, the symbol of $$Q_\mu P_\lambda^e \nabla$$ is bounded by $$(\mu\tau)^{1/2}$$. It follows that   ∑1≤μ≤τ/8∑λ≤μ(μτ)−1‖QμPλeΔ(χa)‖22≲∑1≤μ≤τ/8∑λ≤μ‖QμPλe∇(χa)‖22≲‖∇(χa)‖22≲‖A‖H1(B)2. It remains to control the terms where $$\lambda > \mu$$. In this case the symbol of $$Q_\mu P_\lambda^e\nabla$$ is bounded by $$(\lambda\tau)^{1/2}$$, so we have   ∑1≤μ≤min{λ,τ/8}(μτ)−1/2‖QμPλeΔ(χa)‖2≲∑1≤μ≤min{λ,τ/8}μ−1/2λ−1/2‖QμPλe∇∂¯e(χa)‖2. (40) Since the commutator $$[\nabla \bar\partial_e,\chi]$$ satisfies the bound   ‖[∇∂¯e,χ]a‖2≲‖a‖H1(B), we may replace $$\nabla \bar\partial_e(\chi a)$$ with $$\chi \nabla\bar\partial_e a$$ on the right-hand side of (40). Now we use the definition of $$a$$ to write   ∇∂¯ea=∇(Aa)=∇Aa+A∇a. For $$A \nabla a$$, we apply Nikol’skii’s inquality (11) in the $$e_1$$ and $$e_2$$ directions and use the identity $$\nabla a = \nabla \bar\partial_e^{-1} A \cdot a$$.   μ−1/2λ−1/2‖QμPλe(χA∇a)‖2≲(μλ)−s/2‖QμPλe(χ⋅A∇a)‖L2Le1/(1−s/2)≲(μλ)−s/2‖A‖L∞Le2‖χ∇a‖L2Le2/(1−s)≲(μλ)−s/2‖A‖L∞Le2‖a‖∞‖χ∇∂¯e−1A‖L2Le2/(1−s). Since $$s>0$$, we can sum the right-hand side over $$\mu$$ and $$\lambda$$ as long as the last factor is bounded. To check this, we use the localization estimate (34) and Sobolev embedding.   ‖χ∇∂¯e−1A‖L2Le2/(1−s)≲‖∇P≤1eA‖L2Le2/(1−s)+‖∇∂¯e−1P>1eA‖L2Le2/(1−s)≲‖∇P≤1eA‖2+‖∇⟨∇1,2⟩s∂¯e−1P>1eA‖2≲‖⟨∇1,2⟩s−1∇A‖2. For $$(\chi a)\nabla A $$, we decompose using the Littlewood–Paley dichotomy, as we did with (37):   Pλe((χa)∇A)=∑κ≲λPλe(Pκe(χa)⋅P≲λe∇A)+Pλe(∑η≫λPηe(χa)⋅P∼ηe∇A). For the low-low terms, we have two cases. When $$\kappa \,{\leq}\, \mu$$, we use the identity $$Q_\mu (P_{\leq \mu}^e f\cdot g) = Q_\mu (P_{\leq \mu}^e f \cdot P^{e_1}_{\lesssim \mu} g)$$. Thus we have   μ−1/2λ−1/2‖QμPλe(P≤μe(χa)⋅P≲λeP≲μe1∇A)‖2≲‖a‖∞(μλ)−1/2‖P≲λe2P≲μe1∇A‖2. Summing over $$\mu$$ and $$\lambda$$, we obtain   ∑1≤μ,λ≤τ/8⋯≲‖a‖∞‖⟨∇1⟩−1/2+s⟨∇2⟩−1/2+s∇A‖2. When $$\kappa > \mu$$, we have instead $$Q_\mu(P_\kappa^e f \cdot g) = Q_\mu(P_\kappa^e f \cdot P^{e_1}_{\lesssim \kappa} g)$$. Then   μ−1/2λ−1/2‖Qμ(Pκe(χa)P≲λeP≲κe1∇A)‖2≲λ−1/2‖Pκe(χa)P≲λeP≲κe1∇A)‖Le3,e22Le11≲λ−1/2‖Pκe(χa)‖Le3∞Le2∞Le12‖P≲λe2P≲κe1∇A‖L2≲λ−1/2κ−1/2‖Pκe∂¯e(χa)‖L∞Le2‖P≲λe2P≲κe1∇A‖L2. Summing over $$\kappa$$, $$\mu$$, and $$\lambda$$, we obtain   ∑1≤μ<κ≤λ≤τ/8⋯≲‖a‖∞(1+‖A‖L∞Le2)‖⟨∇1⟩−1/2+s⟨∇2⟩−1/2+s∇A‖L2. For the high–high terms, we use Nikol’skii’s inequality and then transfer the $$\bar\partial^{-1}_e$$ from $$a$$ to $$A$$:   μ−1/2λ−1/2‖QμPλe(Pηe(χa)⋅P∼ηe∇A)‖2≲‖Pηe(χa)⋅P∼ηe∇A‖L2Le1≲‖Pηe(χa)‖L∞Le2‖P∼ηe∇A‖2≲η−1‖Pηe∂¯(χa)‖L∞Le2‖P∼ηe∇A‖2≲η−s‖a‖∞(1+‖A‖L∞Le2)‖⟨∇1,2⟩−1+s∇A‖L2. The sum of the right hand side over $$\eta \geq \lambda \geq \mu\geq 1$$ is bounded, and the proof is complete. ■ 9 Solvability of $$L_{A,q,\zeta}$$ Now we show that on average, the terms in $$L_{A,q,\zeta} + \Delta_\zeta$$ are all perturbative. Here we note an important difference between the estimate for $$q$$ and the estimate for $$A$$: as the parameter $$K$$ gets large, the right-hand side of the estimate (43) for $$q$$ goes to zero. However, this does not hold for the estimate (41) for $$A$$, and for this reason we can only handle the case where $$A$$ is small. Lemma 9.1. Let $$e_1$$ be a fixed unit vector in $$\mathbb{R}^3$$, and let $$\eta$$ be a vector in $$\mathbb{R}^3$$ such that $$\lvert {\eta} \rvert\leq 1$$. Define the operator norm $$\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert{\cdot}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert_{\tau,U }$$ by $$\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert{T}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert = \lVert{\cdot}\rVert_{X^{1/2}_{\zeta(\tau,U)} \to X_{\zeta(\tau,U)}^{-1/2}}$$, where $$\zeta(\tau,U) = \tau U(e_1 + i \eta)$$. Suppose $$A \in L^3(\mathbb{R}^3)$$. For every dyadic integer $$\lambda$$ such that $$1\leq \lambda \leq 100\tau$$ we have   E3[τ|||Aλ|||τ,U+|||Aλ⋅∇|||τ,U]≲min{⟨log+⁡λ⟩,⟨log+⁡τ/λ}⟩1/3‖Aλ‖L3. (41) On the other hand, we have the high-frequency estimate   |||∇A>100τ|||τ,U+|||A>100τ⋅∇|||τ,U≲‖A>100τ‖L3. (42) Finally, for $$q \in W^{-1,3}$$ we have   E~K3[‖|q|‖τ,U]≲(log⁡K)−1/3‖q‖W−1,3+‖q≥2K‖W−1,3. (43) □ Proof. It is convenient to use a bilinear characterization of the $$\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert{\cdot}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert$$ norm   |||A|||=sup{|⟨Au,v⟩|:‖u‖Xζ1/2=‖v‖Xζ1/2=1}. Decompose $$u$$ and $$v$$ into low and high modulation parts:   ⟨Au,v⟩=⟨AQhu,v⟩+⟨AQlu,Qhv⟩+⟨AQlu,Qlv⟩. The terms with $$Q_h$$ can be estimated by the high-modulation estimate (13) and the Strichartz estimate (18). For example,   τ|⟨AQhu,v⟩|≲τ‖A‖3‖Qhu‖2‖v‖6≲‖A‖3‖Qhu‖Xζ1/2‖v‖Xζ1/2. It remains to estimate the low modulation terms. Write   ⟨Aλ⋅Qlu,Qlv⟩=∑τ/8≥μ,ν≥1∫Aλ⋅Qμu⋅Qνv¯dx, (44) where we make the notational convention that $$Q_1 = Q_{\leq 1}$$. Note that when $$\lambda\geq 100\tau$$ these terms are all zero, so for the high-frequency estimate (42) there is nothing left to prove. Set   aμ=‖Qμu‖Xζ1/2bμ=‖Qμv‖Xζ1/2, and   Bμ,ν=τ|∫Aλ⋅Qμu⋅Qνv¯dx|. We claim that   ∑μ(aμ2+bμ2)≲1 implies that ∑μ,νBμ,ν≲Z(U), (45) where   E3[Z(U)]≲‖Aλ‖L3. By symmetry, it suffices to treat the terms where $$\mu \leq \nu$$. Since $$Q_\mu u \cdot \bar{Q_\nu v}$$ has Fourier support in the set $$\{\lvert {\xi\cdot( Ue_1)} \rvert \leq 2\nu\}$$, we have   ∫Aλ⋅Qμu⋅Qνv¯dx=∫P≤8νUe1Aλ⋅Qμu⋅Qνv¯dx. Suppose first that $$\lambda^2 > \mu\tau$$. In this case we use Hölder’s inequality and estimate $$Q_\mu u$$ by using the Strichartz estimate (18) and the $$L^2$$ estimate (12):   Bμ,ν≲τ‖P≤8νUe1Aλ‖3‖Qμu‖6‖Qνv‖2≲‖P≤8νUe1Aλ‖3τ(μ/τ)1/3(ντ)−1/2aμbν. By Young’s inequality, we have   ∑μ≤ν(μ/ν)1/12aμbν≲1, so that the sum over $$\lambda^2 > \mu\tau$$ is bounded by   ∑μ≤νλ2>μτBμ,ν≲supν≥μλ2>μτ(λ/ν)1/3(μ/ν)1/12(μτ/λ2)1/6‖P≤8νUe1Aλ‖3≲Z1(U)+Z2(U)+‖Aλ‖3, where $$Z_1(U)$$ and $$Z_2(U)$$ are given by   Z1(U)3=∑max{1,λ2/τ}≤ν≤λ(λ/ν)(λ2/ντ)1/12‖P≤8νUe1Aλ‖33Z2(U)3=∑ν≤λ2/τ(λ/ν)(ντ/λ2)1/2‖P≤8νUe1Aλ‖33. Now we check that $$Z_1$$ and $$Z_2$$ are bounded on average by applying the averaging estimate (24).   E3[Z1(U)]3≲∑max{1,λ2/τ}≤ν≤λ(λ2/ντ)1/12‖Aλ‖33≲‖Aλ‖33E3[Z2(U)]3≲∑ν≤λ2/τ(ντ/λ)1/2/λ‖Aλ‖33≲‖Aλ‖33. Next, we treat the case $$\lambda \leq (\mu\tau)^{1/2}$$. Note that   ⟨Au,v⟩=⟨Ae−iv⋅xu,e−iv⋅xv⟩, and that the $$X^{1/2}_\zeta$$ spaces have the modulation invariance   ‖e−iv⋅xu‖Xζ1/2∼‖u‖Xζ+iv1/2. Thus we may as well assume that $$\eta$$ is zero. We subdivide the set   El={ξ:d(ξ,Σζ)≤τ/8} into $$M=\left \lfloor {(\tau/\mu)^{1/2}} \right \rfloor$$ sectors $$S_k$$, defined for $$k=0,\dotsc,M-1$$ by   Sk=El∩{(ξ1,rcos⁡θ,rsin⁡θ):θ∈(2π/M)[k,k+1),r∈R+}. Here we recall that Let $$R_k$$ be Fourier projection on to $$S_k$$. The distance between two points in $$E_l$$ is bounded below by $$\tau \theta$$, where $$\theta$$ is the angular separation between the points. Thus for any two sectors $$S_j$$ and $$S_k$$, we have   d(Sj,Sk)≳(μτ)1/2dM(j,k), where $$d_M(j,k) = \min \{\lvert {j-k} \rvert,M-\lvert {j-k} \rvert\}$$. Since $$A_\lambda \cdot R_k f$$ has Fourier support in the set $$\{S_k + B(0, 2\lambda)\}$$, we find that the inner product $$\langle{A_\lambda \cdot R_k f, R_j g}\rangle$$ vanishes unless $$\lvert {d_M(j,k)} \rvert \leq C$$, so that   Bμ,ν≲τ∑dM(j,k)≤C|⟨P≤8νUe1Aλ⋅RkQμu,RjQνv⟩|. The Fourier support of $$R_k Q_\mu u$$ is contained in a rectangle of size proportional to $$\mu^{1/2}\tau^{1/2}\times \mu \times \mu$$. Thus, applying Hölder and Nikol’skii’s inequality (11) in each direction separately, we obtain   ‖P≤8ν1Aλ⋅RkQμu‖L2≲‖P≤8νUe1Aλ‖L∞L3L3‖RkQμu‖L2L6L6≲λ1/3μ2/3‖P≤8νUe1Aλ‖L3‖RkQμu‖L2. Now apply Cauchy–Schwarz to the sum over $$j$$ and $$k$$. This gives   Bμ,ν≲λ1/3μ2/3‖P≤8νUe1Aλ‖3(∑k‖RkQμu‖22)1/2(∑k‖RkQνv‖22)1/2≲μ2/3λ1/3‖P≤8νUe1Aλ‖L3‖Qμu‖L2‖Qνv‖L2≲(μ/ν)1/6(λ/ν)1/3aμbν‖P≤8νUe1Aλ‖L3. Thus   ∑λ2≤μτμ≤νBμ,ν≲supν≥max{λ2/τ,1}(λ/ν)1/3‖P≤8νUe1Aλ‖3≲Z3+‖Aλ‖3, where $$Z_3$$ is given by   Z3(U)3=∑max{λ2/τ,1}≤ν≤λ(λ/ν)1/3‖P≤8νUe1Aλ‖3. Applying the averaging estimate (24), we have   E3[Z3(U)]3≲∑max{λ2/τ,1}≤ν≤λ‖Aλ‖3≲min{⟨log+⁡λ⟩,⟨log+⁡τ/λ⟩}1/3‖Aλ‖3. This proves the claim (45), which shows that   E3[τ|||Aλ|||τ,U]≲min{⟨log+⁡λ⟩,⟨log+⁡τ/λ⟩}1/3‖Aλ‖3. The estimate for $$\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert{A\cdot \nabla}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert$$ now follows, since   ‖A⋅∇u‖Xζ−1/2≲|||A|||‖∇Qlu‖Xζ1/2+‖A⋅∇Qhu‖Xζ−1/2. Since $$Q_l$$ localizes to frequencies $$\lvert \xi \rvert \lesssim \tau$$, we can estimate the first term using the bound   ‖∇Qlu‖Xζ1/2≲τ‖u‖Xζ1/2. For second term, we apply the Strichartz estimate (20) and the high-modulation estimate (13)   ‖A⋅∇Qhu‖Xζ−1/2≲‖A⋅∇Qhu‖L6/5≲‖A‖3‖∇Qhu‖L2≲‖A‖3‖u‖Xζ1/2. Thus we have   E3[|||Aλ⋅∇|||]≲E3[τ|||Aλ|||]+‖Aλ‖3≲⟨log+⁡τ/λ⟩1/3‖Aλ‖3. Finally we derive the estimate (43) for $$\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert{q}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert$$. By Nikol’skii’s inequality, we can control $$\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert{q_{\leq 1}}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert$$ by   |||q<1|||≲τ−1‖q≤1‖∞≲τ−1‖q‖W−1,3. On the other hand, the estimate (41) for $$\tau \left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert{A_\lambda}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert$$ applied to $$q_\lambda$$ gives   E3[|||q[1,100τ]|||]≲∑1≤λ≤100τ(λ/τ)⟨log+⁡τ/λ⟩1/3‖qλ‖W−1,3≲∑1≤λ≤100τ(λ/τ)1−ϵ‖qλ‖W−1,3. Thus the frequency convolution estimate (23) gives   E˜K3[‖|q[1,100τ]|‖]≲(logK)−1/3‖q‖W−1,3. Together with the high-frequency estimate (42), this gives the desired bound (43) for $$\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert{q}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert$$. ■ 10 Proof of the main theorem We will need the following Poincaré lemma: Lemma 10.1. Suppose that $$A \in L^3(\mathbb{R}^3)$$ is compactly supported, and that   curlA=0. Then there exists $$\psi \in W^{1,3}(\mathbb{R}^3)$$ supported in $$\text{supp} A$$ such that   ∇ψ=A. □ Proof. If $$\psi$$ exists then   Δψ=div(∇ψ)=div(A). Thus we set   ψ(x)=c∫∑i∂iK(y)Ai(x−y)dy, where $$K(y) \sim \lvert{y}\rvert^{-1}$$ is the fundamental solution of the Laplacian. Since $$\lvert{\partial_i K}\rvert \lesssim \lvert{y}\rvert^{-2}$$, the kernel is locally integrable. By Young’s inequality and the fact that $$A$$ is compactly supported, we have $$\psi \in L^3_{{\mathrm{loc}}}$$. Furthermore, the function $$\psi$$ is smooth away from $$\text{supp} A$$ and decays to zero at infinity. The vector Laplacian is given by   Δ=grad∘div−curl∘curl. Since $$\text{curl} A$$ and $$\text{curl}\nabla \psi$$ are both zero, we have   Δ(A−∇ψ)=∇(divA−Δψ)=0. Since $$A - \nabla \psi$$ vanishes at infinity, this implies, by the maximum principle, that $$\nabla \psi = A$$. In particular, $$\nabla \psi = 0$$ away from $$\text{supp} A$$, and since $$\psi$$ decays at infinity we conclude that $$\psi = 0$$ away from $$\text{supp} A$$. ■ Proof. (of Theorem 1.1) Let $$B = B(0,1)$$. We construct solutions in $$H^1(B)$$ to the Schrödinger equation $$L_{A,q} u = 0$$ of the form   u=ex⋅ζ~(a+ψ). (46) Let $$\chi,\tilde \chi \in C^\infty_0(B)$$ be cutoff functions satisfying $$\chi = \tilde \chi = 1$$ in $$\tfrac{1}{2} B$$ and $$\chi \tilde \chi = \tilde \chi$$. We construct $$\psi$$ by solving the equation   LA,q,ζ~ψ=χ~(F+G), (47) where   LA,q,ζ~=−Δζ~−2iζ~⋅A+D⋅A+2A⋅D+q, (48)  F=−Δ(χa)+(D⋅A)a+2A⋅Da+A2a+qa, (49) and   G=−2ζ~⋅∇a−2iζ~⋅Aa. (50) In order to eliminate the terms of order $$\tau$$ in $$G$$, we let $$a = {\rm e}^{-i\phi_\zeta}$$, where   ϕζ=∂¯ζ−1(χζ⋅A≤100τ), (51) where $$\zeta \in \mathbb{C}^n$$ will be chosen such that $$\lvert{\zeta - \tilde \zeta}\rvert = O(1)$$. With this choice of $$\phi$$, the function $$G$$ satisfies   χ~G=χ~(−2iζ⋅A>100τa−(ζ~−ζ)⋅(2∇a+2iAa)). We choose the parameters $$\zeta_i,\tilde \zeta_i$$ as follows: Fix a radius $$r\in [1,2]$$ and an orthonormal frame $$\{e_1,e_2, e_3\}$$, and define   ζ1(τ,U)=τU(e1+ie2)ζ2(τ,U)=−ζ1(τ,U)ζ~1(τ,U):=τUe1+i(r2Ue3+τ2−r2Ue2)ζ~2(τ,U):=−τUe1+i(r2Ue3−τ2−r2Ue2). Note that $$\lvert{\zeta_i - \tilde \zeta_i}\rvert \lesssim 1$$. In particular, the spaces $$X^b_{\zeta_i}$$ and $$X^b_{\tilde \zeta_i}$$ have equivalent norms. Let $$\tilde A_1 = A_1$$ and $$\tilde A_2 = -A_2$$, define $$F_i,G_i,\phi_{i,\zeta}$$ as in (49), (50), (51) by replacing $$A$$ with $$\tilde A_i$$. Let   M=∑i(‖Ai‖Ws,3+‖qi‖W−1,3)Z0=∑i(∑k=0,1,2τ−k‖∇kϕi,ζi‖∞+‖∇ϕi,ζi‖L2(B)+‖Ai‖L∞Le2+‖⟨∇1⟩−1/2+s⟨∇2⟩−1/2+s∇Ai‖2+‖⟨∇1,2⟩−1+s∇Ai‖2)Z1=∑i,l,m(τ‖Ai‖Xζj1/2→Xζj−1/2+‖Ai∇‖Xζj1/2→Xζj−1/2+‖ql,m‖Xζj1/2→Xζj−1/2)Z2=∑i,j,l,m‖χaiqj,l‖Xζm−1/2. Here the $$q_{j,l}$$ are all the terms that are bounded in $$W^{-1,3}$$, namely   qj,1=Aj2qj,2=qjqj,3=τP>100τAjqj,4=∇Aj. By the localization estimate (14) and the fact that $$\lvert{\zeta - \tilde \zeta}\rvert \lesssim 1$$, we have   ‖LA~i,q,ζ~i+Δζ~i‖X˙ζ~i1/2→X˙ζ~i−1/2≲Z1. If $$Z_1$$ is sufficiently small, then by the contraction mapping principle there are $$u_i = e^{x\cdot \tilde \zeta_i} (e^{-i\phi_i} + \psi_i)$$ solving $$L_{\tilde A_i,q_i} u_i = 0$$ in $$B$$ such that   ‖ψi‖X˙ζ~i1/2≲‖χ~(Fi+Gi)‖Xζi−1/2. By Lemma 8.6, the Strichartz estimate (20) and Hölder’s inequality, we have   ‖Fi‖Xζi−1/2≲(1+Z0)4+Z2+‖Ai‖3‖∇ai‖L2(B)≲(1+Z0)4+Z2+MeZ0Z0. On the other hand we can simply estimate $$\tilde \chi G$$ by   ‖χ~Gi‖Xζi−1/2≲Z2+eZ0(Z0+M).  Thus we have   ‖ψi‖X˙ζ~i1/2≲g(M,Z0,Z2), where $$g$$ is continuous. Now we apply the integral identity (3) to the solutions $$u_1$$ and $$u_2$$ to obtain   0=∫[i(A1−A2)⋅(u1∇u2−u2∇u1)+(A12−A22+q1−q2)u1u2]dx=I+II+III+IV, where   I=i(ζ2−ζ1)∫χ(A1−A2)≤100τe−i(ϕ1+ϕ2)eik⋅xdxII=∫[(ζ−ζ~)χA≤100τ+χζ~A>100τ+A∇ϕ+χq]e−iϕeik⋅xdxIII=∫[ζ~Aaψ+A∇aψ+Aa∇ψ+χqaψ]eik⋅xdxIV=∫qψ1ψ2eik⋅xdx. Here $$k = rUe_3$$. The error terms $$II$$–$$IV$$ are in schematic form. For example, the notation $$\int \chi qa \psi {\rm e}^{ik\cdot x}$$ represents a linear combination of terms $$\int\chi q_{l,m} a_i \psi_j {\rm e}^{ik \cdot x}$$. The first expression $$I$$ contains the main term. We remove the exponential factor $${\rm e}^{-i(\phi_1+\phi_2)}$$ using Lemma 3.2. We use the Littlewood–Paley commutator estimate $$\lVert{[\chi,P_{\leq 100\tau}]}\rVert_{L^1\to L^1} \lesssim \tau^{-1}$$ to control some of the errors.   I=i(ζ2−ζ1)⋅(χ(A1−A2)≤100τ)∧(k)=i(ζ2−ζ1)⋅(P≤100τ(A1−A2)+[χ,P≤100τ](A1−A2))∧(k)=i(ζ2−ζ1)⋅(A1−A2)∧(k)+O(‖A1−A2‖1). Next we estimate the terms in $$II$$.   |II|≲‖aA‖L1+τ‖A>100τ‖H−1‖χa‖H1+‖A‖2‖χa∇ϕ‖2+‖q‖H−1‖χa‖H1≲eZ0Z0M. For the terms in $$III$$ we use the the duality between $$\dot X^{1/2}_\zeta$$ and $$\dot X^{-1/2}_\zeta$$, the localization estimate (15), the trivial estimate $$\lVert{u}\rVert_{X^{-1/2}_\zeta} \lesssim \tau^{-1/2} \lVert{u}\rVert_2$$, the Strichartz estimate (20), and the fact that multiplication by $$e^{ik\cdot x}$$ is bounded in $$X^{-1/2}_\zeta$$:   |III|≲(τ‖Aa‖Xζ−1/2+‖A∇a‖Xζ−1/2+‖∇Aχa‖Xζ−1/2+‖qχa‖Xζ−1/2)‖ψ‖X˙ζ1/2≲(τ1/2‖A‖2eZ0+‖A‖3‖∇a‖L2(B)+Z2)g(M,Z0,Z2)≲(τ1/2MeZ0+MeZ0Z0+Z2)g(M,Z0,Z2). For $$IV$$, we estimate by $$\lVert{q}\rVert_{X^{1/2}_\zeta\to X^{-1/2}_\zeta}\lVert{\psi_1}\rVert_{X^{1/2}_\zeta} \lVert{\psi_2}\rVert_{X^{1/2}_\zeta}$$. Thus   |IV|≲Z1g(M,Z0,Z2)2. Combining all these estimates, we obtain   |i(ζ2−ζ1)⋅(A1−A2)∧(k)|≤τ1/2f(M,Z0,Z1,Z2), (52) where $$f$$ is continuous. To conclude, we must select $$\zeta$$ such that all of the constants $$Z_i$$ are bounded uniformly in $$\zeta$$. Let $$\epsilon = \sum_i \lVert{A_i}\rVert_{W^{s,3}}$$, which we assume to be small. By Lemmas 7.1, 8.2, 8.3 and the corollary (35) of Falconer’s maximal function estimate, we have (using Hölder’s inequality to control $$\mathbb{E}$$ by $$\mathbb{E}^p$$), that   E[Z0∣SO(3)]≤Cϵ. Define the set $$V$$ by   V={U∈SO(3):Z0<2Cϵ}, By Chebyshev’s inequality, we have $$\mathbb P[V] \geq \frac{1}{2}$$. Similarly, Lemma 9.1 implies that   E~K[Z1∣V]≲∑1≤λ≲τ⟨log+λ⟩λ−s‖Aλ‖Ws,3+‖A>2K‖L3+(log⁡K)−1/3‖q‖W−1,3+‖q≥2K‖W−1,3. Since the $$q$$ terms decay to zero as $$K \to \infty$$, we have   lim supK→∞E~K[Z1∣V]≤Cϵ. Lemma 9.1 implies that   E~K[Z2∣[2K,2K2]×V]≤h(Z0,M), where $$h$$ is continuous. Thus for sufficiently large $$K$$, it follows that the inequality   (2Cϵ)−1Z1+(2h(Z0,M))−1Z2≤2 holds on a set $${{\tilde{V}}_{K}}\subset [{{2}^{K}},{{2}^{{{K}^{2}}}}]\times V$$ with $${{\widetilde{\mathbb{P}}}_{K}}[{{\tilde{V}}_{K}}]\ge \frac{1}{4}$$. By choosing $$\epsilon$$ small, we can ensure that $$Z_1$$ is sufficiently small on $$\tilde V_K$$ that we can use the contraction mapping principle to construct $$\psi_i$$ as above. Furthermore, for $$(\tau,U) \in \tilde V_K$$, the quantities $$Z_i$$ are all bounded independently of $$\tau, U, K$$. Let   J(r,U)=U(e1+ie2)⋅(A1−A2)∧(rUe3). By (52), we have $$\lvert{J(r,U)}\rvert \lesssim \tau^{-1/2}$$ for all $$(\tau,U)$$ in the set $$\tilde V_K$$. Integrating this inequality over all $$(\tau, U)$$ in $$\tilde V_K$$ and $$r$$ in $$[1,2]$$, we have   ∫SO(3)∫12∫[2K,2K2]1V~K|J(r,U)|(log⁡K)−1(τlog⁡τ)−1dτdrdσ(U)=O(e−K/2) Let   ηK=∫[2K,2K2]1V~K(log⁡K)−1(τlog⁡τ)−1dτ, and note that   ∫SO(3)∫12ηV~Kdrdσ(U) ~P~K[V~K]≥14. By the Banach–Alaoglu theorem, there is a sequence $$K_i\to \infty$$ and a function $$\eta \in L^\infty(SO(3) \times [1,2])$$ such that $$\eta_{K_i} \rightharpoonup \eta$$. Since $$\int \eta = \lim \int \eta_{K_i} \geq \tfrac{1}{4}$$, it is clear that $$\eta \neq 0$$. On the other hand,   ∫η(r,U)|J(r,U)|drdσ(U)=limi→∞∫ηKi(r,U)|J(r,U)|drdσ(U)=0. It follows that $$J(r,U)$$ vanishes on a set of positive measure. But $$A_1 - A_2$$ is a compactly supported function, which implies that $$J(r,U)$$ is analytic in $$r$$ and $$U$$. Thus we can conclude that $$J(r,U) = 0$$ in $$\mathbb{R}_+\times SO(3)$$. By replacing $$SO(3)$$ by its complement throughout the argument, we find that $$J(r,U) = 0$$ in $$\mathbb{R}_+ \times (O(3)\setminus SO(3))$$ as well. Let $$H = A_1 - A_2$$. Since $$J(r,U)$$ vanishes uniformly, we must have $$v \cdot \hat H(k)=0$$ whenever $$v \cdot k = 0$$. In particular, $$0 = (w \times k) \cdot \hat H(k) = (\text{curl} H)^\wedge(k) \cdot w$$ for any $$w,k\in \mathbb{R}^n$$, so $$\text{curl} H = 0$$. By Lemma 10.1, there is a gauge transformation $$\psi$$ such that $$A_2 = A_1 + \nabla \psi$$, which implies that $$\Lambda_{A_1,q_1} = \Lambda_{A_2,q_2} = \Lambda_{A_1,q_2}$$. We can repeat the whole argument to obtain   0=∫((q1−q2)eik⋅x+χqaψeik⋅x+χqψ1ψ2eik⋅x)dx. Since $$\tilde E_K[\lVert{\chi a q}\rVert_{X^{-1/2}_\zeta} + \lVert{q}\rVert_{X^{1/2}_\zeta \to X^{-1/2}_\zeta}\mid \tilde V_K] \to 0$$ as $$K \to \infty$$ by Lemma 9.1 and Lemma 9.1, we can repeat the arguments above to show that that $$(q_1 -q_2)^\wedge(k) = 0$$ for all $$k$$. 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Unique Determination of a Magnetic Schrödinger Operator with Unbounded Magnetic Potential from Boundary Data

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Abstract

Abstract We consider the Gel’fand–Calderón problem for a Schrödinger operator of the form $$-(\nabla + iA)^2 + q$$, defined on a ball $$B$$ in $$\mathbb{R}^3$$. We assume that the magnetic potential $$A$$ is small in $$W^{s,3}$$ for some $$s>0$$, and that the electric potential $$q$$ is in $$W^{-1,3}$$. We show that, under these assumptions, the magnetic field $$\text{curl} A$$ and the potential $$q$$ are both determined by the Dirichlet–Neumann relation at the boundary $$\partial B$$. The assumption on $$q$$ is critical with respect to homogeneity, and the assumption on $$A$$ is nearly critical. Previous uniqueness theorems of this type have assumed either that both $$A$$ and $$q$$ are bounded or that $$A$$ is zero. 1 Boundary data for Schrödinger operators Consider a Schrödinger Hamiltonian of the form   LA,qu=−(∇+iA)2u+qu. Here $$A$$ represents a magnetic vector potential and $$q$$ represents an electric scalar potential. Let $$\Omega \subset \mathbb{R}^n$$ be an bounded open set. Define the Dirichlet–Neumann relation by   ΛA,qΩ={(u|∂Ω,(∂ν+iν⋅A)u|∂Ω):u∈H1(B)andLA,qu=0}, where $$\nu$$ is the outward unit normal to $$\partial \Omega$$. The Gel’fand–Calderón problem [10, 19] is to determine the magnetic field $$\text{curl} A$$ and the electric potential $$q$$ from the Dirichlet–Neumann relation $$\Lambda_{A,q}$$. In principle, this is possible if $$\Lambda_{A,q}$$ uniquely determines $$\text{curl} A$$ and $$q$$. We are interested in proving uniqueness under minimal a priori regularity assumptions on $$A$$ and $$q$$. To avoid unnecessary technical complications, we take $$\Omega$$ to be a ball in $$\mathbb{R}^3$$ and assume that the coefficients $$A$$ and $$q$$ are supported away from $$\partial \Omega$$. Theorem 1.1. Fix $$s>0$$. Let $$B = B(0,1)$$ be the unit ball in $$\mathbb{R}^3$$. Suppose that $$A_i$$ and $$q_i$$ are supported in the smaller ball $$\tfrac{1}{2} B$$. If, for each $$i = 1,2$$, the magnetic potential $$A_i$$ is small in the $$W^{s,3}$$ norm, the electric potential $$q_i$$ is in $$W^{-1,3}$$, and $$\Lambda_{A_1,q_1} = \Lambda_{A_2,q_2}$$, then $$\text{curl} A_1 = \text{curl} A_2$$ and $$q_1 = q_2$$. □ One physical motivation for studying this problem comes from quantum mechanics. For compactly supported potentials, the map $$\Lambda_{A,q}$$ contains the same information as the scattering matrix at a fixed energy level. The scattering matrix contains observations, made at spatial infinity, of a localized (short-range) potential. This can be defined for potentials that are exponentially decreasing rather than compactly supported, so inverse scattering at fixed energy is a generalization of the Gel’fand–Calderón problem. Unique determination of a bounded electric potential $$q$$ from the Dirichlet-to-Neumann map in the absence of a magnetic potential was proven by Sylvester and Uhlmann [51] (see also [34]). The proof is based on a density argument using complex geometrical optics (CGO) solutions, inspired by Calderón’s treatment of the linearized inverse conductivity problem in [10]. The Sylvester–Uhlmann method was adapted to the case of a nonzero magnetic potential by Sun in [48], where uniqueness was proven for $$A\in C^2$$ and $$q \in L^\infty$$, subject the requirement that $$\lVert{\text{curl} A}\rVert_\infty$$ be small. The basic method we use in this article is the same as that in [48]; in particular, we retain a smallness condition on the magnetic potential. Uniqueness for large smooth $$A$$ was proved by Nakamura et al. [37] using a pseudodifferential conjugation technique from [36]. This was improved to $$A \in C^1$$ in [52] using symbol smoothing. In [44], it was shown that, by imposing a Coulomb gauge, the result of [48] could be extended to small Dini continuous $$A$$ (this includes the case where $$A$$ is $$\alpha$$-Hölder for some $$\alpha>0$$). The smallness condition was removed in [45] using pseudodifferential conjugation. In [31], the Coulomb gauge condition and pseudodifferential conjugation were eliminated using an argument based on Carleman estimates with slightly convex weights, and uniqueness was proven for $$A \in L^\infty$$. Our result requires that $$A$$ is small and slightly more differentiable than in [31]. On the other hand, we require much less integrability for $$A$$ and $$q$$, so that our conditions on $$A$$ and $$q$$ are much closer to being scale-invariant. It does not seem that the method in [31] for removing the smallness condition on $$A$$ extends to the case of unbounded potentials. However, we believe that a pseudodifferential conjugation argument could be used to remove the smallness condition for the result in this paper. Another approach to the problem in the spirit of Faddeev’s pioneering work in [17] is based on the $$\bar\partial$$ method of Beals and Coifman [3]. Using this approach, the inverse scattering problem for small $$A$$ and $$q$$ in $${\rm e}^{-\gamma\langle x \rangle} C^\infty$$ was solved by Khenkin and Novikov in [29]. Uniqueness for $$A = 0$$ and large $$q \in {\rm e}^{-\gamma\langle {x} \rangle } L^\infty$$ was proven by Novikov [41]. Uniqueness for large $$(A,q) \in {\rm e}^{-\gamma\langle {x} \rangle} C^\infty$$ was proven by Eskin and Ralston [16]. A proof of uniqueness for $$A = 0$$ and $$q \in {\rm e}^{-\gamma\langle x \rangle} L^\infty$$ using a density argument more similar to the Sylvester–Uhlmann approach detailed above is also possible [33, 54]. This density argument was modified to include $$A \in {\rm e}^{-\gamma\langle {x} \rangle} W^{1,\infty}$$ in [43]. Since the Laplacian has units $$(\text{length})^{-2}$$, the $$L^\infty$$ norms of $$A$$ and $$q$$ are not dimensionless quantities. This is undesirable from a physical point of view. Assuming that $$A$$ and $$q$$ are bounded excludes even subcritical potentials with $$\lvert{x}\rvert^{-1}$$ singularities at the origin (for example, a localized Coulomb-type potential). A scale-invariant assumption in $$n$$ dimensions is that $$A$$ be in $$L^n$$ and that $$q$$ be in $$L^{n/2}$$ or $$W^{-1,n}$$ (by Sobolev embedding, $$L^{n/2} \subset W^{-1,n}$$). Chanillo [12], using the weighted inequalities of Chanillo and Sawyer [13], proved uniqueness in the inverse boundary value problem for $$A = 0$$ and compactly supported $$q$$ with small norm in the scale-invariant Fefferman–Phong classes $$F_{>(n-1)/2}$$ (including, in particular, potentials of small weak $$L^{n/2}$$ norm). Chanillo’s article also includes an argument of Jerison and Kenig proving uniqueness for $$q \in L^{n/2+}$$ with no smallness condition. This was extended to include the scale-invariant case $$q \in L^{n/2}$$ by Lavine and Nachman (see [15] for details). A closely-related problem is Calderón’s problem, which is to recover the coefficient in the equation $$\text{div} (\gamma \nabla u) = 0$$ from the Dirichlet-to-Neumann map $$\Lambda_\gamma$$. In Sylvester and Uhlmann’s work [51], this problem is reduced to the problem of recovering a Schrödinger potential $$q$$, where $$q = \gamma^{-1/2} \Delta \gamma$$. Unless $$\gamma$$ has two derivatives, the potential $$q$$ will end up having negative regularity. In [6, 7, 42] it was shown that the Sylvester–Uhlmann argument carries through for conductivities with $$3/2$$ derivatives. In [23], the author and Tataru showed uniqueness for $$\gamma \in C^1$$ or $$\gamma$$ with small Lipschitz norm using an averaging argument. In [38], a more involved averaging argument was used to prove uniqueness in three dimensions for $$\gamma \in H^{3/2+}$$. In [22], the author used arguments similar to those of [38], combined with the $$L^p$$ Carleman estimates of [28], to show uniqueness for $$\gamma \in W^{1,n}$$ in dimensions $$n=3,4$$. This corresponds to recovering a Schrödinger potential $$q \in W^{-1,n}$$. In two dimensions, the problem has a fairly different character, and we refer the reader to [1, 4, 8, 9, 20, 25, 29, 32, 35, 40, 49, 50]. The main contribution of this paper is in the construction of CGO solutions. These are solutions to the Schrödinger equation $$L_{A,q} u = 0$$ of the form $$u = {\rm e}^{x \cdot \zeta}(a + \psi)$$. To construct such solutions, we need to understand the conjugated Laplacian $$\Delta_\zeta$$, defined by   Δζ=e−x⋅ζΔex⋅ζ, where $$\zeta \in \mathbb{C}^n$$ and $$\tau = \lvert{\text{Re} \zeta}\rvert$$ is large. In particular, we would like to show that the operator $$L_{A,q,\zeta}$$, defined by   LA,q,ζ=e−x⋅ζLA,qex⋅ζ=−Δζ−2iA⋅(ζ+∇)−i∇⋅A−A⋅A+q, (1) is invertible on some function spaces. To do this, we need a lower bound for $$\Delta_\zeta$$ that can absorb the lower order terms. Estimates for operators like $$\Delta_\zeta$$ arise in the unique continuation problem for operators of the form $$-\Delta + A \cdot \nabla + V$$. For example, the weak unique continuation property for operators of the form $$-\Delta + V$$, where $$V \in L^{n/2}_{\mathrm{loc}}$$, follows from the $$L^p$$ Carleman estimate of Kenig et al. [28], which states that   ‖e−τx1u‖p′≲‖e−τx1Δu‖p, with $$1/p+1/p'=1$$ and $$1/p-1/p'=2/n$$. This estimate is equivalent to an estimate of the form   ‖v‖p′≲‖Δτe1v‖p for the conjugated Laplacian $$\Delta_{\tau e_1}$$. These $$L^p$$ Carleman estimates are similar to Strichartz estimates [47] for dispersive equations, and can be proven in a similar way, using Fourier restriction theorems (in particular, the Stein–Tomas theorem [46, 53] and its variants). The first application of the Fourier restriction theory to unique continuation appears in the work of Hörmander [24], who attributes the idea to Sjölin. Hörmander did not believe that this approach could be used to prove a unique continuation theorem for potentials in the critical space $$L^{n/2}$$. However, this is precisely what was achieved by Kenig, Ruiz, and Sogge in [28]. The strong unique continuation property for these potentials had been established by Jerison and Kenig in [27]. The connection between the $$L^p$$ Carleman estimate in [27] and Fourier restriction theorems was clarified in [26]. Given $$A\in L^q_{\mathrm{loc}}$$, unique continuation for the operator $$-\Delta + A \cdot \nabla$$ would follow from a gradient Carleman estimate of the form   ‖eτϕ∇u‖Lr≲‖eτϕΔu‖Lp, (2) where $$1/p-1/r = 1/q$$. Barcelo et al. [2] showed that no such gradient Carleman estimate can hold for linear weights of the form $$\phi = x_1$$ unless $$r = p = 2$$. In contrast, they proved unique continuation for $$A \in L^{(3n-2)/2}_{\mathrm{loc}}$$ and $$V \in L^{n/2+}_{\mathrm{loc}}$$ by establishing a gradient Carleman estimate (2) for the convex weight $$\phi = x_1 + x_1^2$$. They showed that their results are sharp, in the sense that for any weight $$\phi$$, the estimate (2) cannot hold uniformly in $$\tau$$ and $$u$$ unless the exponents $$p$$ and $$r$$ satisfy the condition $$1/p-1/r \leq 2/(3n-2)$$. This means that the Carleman method cannot be directly applied when $$1/q > 2/(3n-2)$$. Nevertheless, Wolff showed in [55] that the operator $$\Delta + A \cdot \nabla + q$$ has the weak unique continuation property for $$A \in L^{n}$$ and $$V \in L^{n/2}$$. Subsequently, Wolff’s method was adapted to the corresponding strong unique continuation problem by Koch and Tataru [30]. The idea is that the gradient Carleman estimate (2) can be rescued by localizing it to a very small set. In fact, if $$u$$ is supported on a set $$E$$ of volume $$\lvert{E}\rvert$$, then the gradient Carleman estimate (2) holds with a loss of $$\tau\lvert{E}\rvert^{1/n}$$. We will attempt to very roughly outline how one might exploit this fact. Wolff’s point of departure is the following observation: if $$\mu$$ is any compactly supported measure and $$v$$ is a unit vector, then multiplying $$\mu$$ by an exponential weight $${\rm e}^{\tau x \cdot v}$$ tends to push the mass of $$\mu$$ to the boundary $$\partial K$$, where $$K$$ is the convex hull of $$\text{supp} \mu$$. More concretely, if we let $$x_{v}^{\max}$$ denote the maximum value assumed by the weight $$x_v = x\cdot v$$ on $$K$$, then the measure $$e^{\tau x\cdot v} \mu$$ should concentrate on the set $$\{\lvert{x_v - x_v^{\max}}\rvert \lesssim \tau^{-1}\}$$. Now, this does not look very useful, as in general this set appears to have volume closer to $$\tau^{-1}$$ than to $$\tau^{-d}$$. In particular, this is the case if $$\mu$$ concentrates on a portion of $$\partial K$$ which is close to a plane $$\{x \cdot v =\text{const}\}$$. On the other hand, the Carleman method allows considerable freedom in choosing the parameters $$v$$ and $$\tau$$. It is only necessary to establish Carleman estimates for a single measure $$\mu$$, which depends on the function $$u$$ whose existence one wants to rule out. Thus one must somehow exploit the geometrical fact that $$\partial K$$, as the boundary of a convex body, cannot efficiently approximate planes $$\{x \cdot v = \text{const}\}$$ in too many distinct directions $$v\in S^{n-1}$$. For example, if we try to approximate all of the planes simultaneously by taking $$\mu$$ to be a uniform measure on the unit sphere, then $${\rm e}^{\tau x\cdot v} \mu$$ concentrates on a relatively large set (a rectangle of volume $$\tau^{-(n-1)/2-1}$$), but this set only contains a $$\tau^{-(n-1)/2}$$th part of the total mass of $$\mu$$. It turns out that in this case we can glue together different Carleman estimates and gain the required factor of $$\tau^{-1}$$. Unfortunately, it is not clear how this approach could be used to establish uniform bounds for an operator of the form (1), as Wolff’s argument in [55] only works for a single function $$u$$. We will establish an estimate that holds uniformly in $$u$$ as long as the vector $$\zeta$$ avoids certain “bad” directions. These bad directions depend only on the magnetic potential $$A$$ and not (as in Wolff’s method) on the function $$u$$. Our method of selecting the good directions is based on a much simpler Fourier-theoretic orthogonality argument, which is “dual” in a sense to Wolff’s idea. The use of Fourier analysis and orthogonality makes it unclear whether the method generalizes to higher dimensions. This remains an open problem. That localization has a smoothing effect is consistent with the uncertainty principle, since localization in physical space at the scale $$\mu^{-1}$$ corresponds to averaging in Fourier space at the scale $$\mu$$. This averaging smooths out the singular behavior of $$\Delta_\zeta$$ at the characteristic set, since the distance in Fourier space from the characteristic set $$\text{char} \Delta_\zeta$$ is effectively bounded below by $$\mu$$. When the modulation $$d(\xi,\text{char} \Delta_\zeta)$$ is large, the operator $$\Delta_\zeta$$ has good lower bounds. Instead of localizing in physical space and taking advantage of this fact indirectly, we use this high-modulation gain directly in order to overcome the failure of the gradient Carleman estimate. To keep track of the modulation, we use Bourgain-type spaces [5] with norm $$\lVert{\cdot}\rVert_{\dot X^{b}_\zeta}$$ given by   ‖u‖X˙ζb=‖|Δζ|bu‖L2. Solvability of (1) will follow from a bilinear estimate of the form   |⟨A⋅(∇+ζ)u,v|⟩≲‖u‖X˙ζ1/2‖v‖X˙ζ1/2. The $$\dot X^{1/2}_\zeta$$ norm localizes the Fourier transforms $$\hat u$$ and $$\hat v$$ near the characteristic set $$\Sigma_\zeta$$, which lies in the plane   (Reζ)⊥={ξ:ξ⋅Re(ζ)=0}. Thus, the worst-case scenario occurs when $$\hat A(\xi)$$ also concentrates on the plane $$(\text{Re} \zeta)^\perp$$. Using an averaging argument based on Plancherel’s theorem, we will show that $$\hat A$$ cannot concentrate on too many planes through the origin. This will show that $$A\cdot (\nabla+\zeta)$$ is a bounded map from $$\dot X^{1/2}_\zeta$$ to $$\dot X^{-1/2}_\zeta$$ for most values of $$\text{Re} \zeta$$. Since $$\text{Re} \zeta$$ is, to a large extent, a free parameter, this is enough to obtain many CGO solutions and prove uniqueness. We now give an outline of the paper. Sections 2–4 contain standard material due to [16, 48, 51]. In Section 5, we define a dyadic decomposition in frequency and modulation, which will be used extensively throughout the paper. In Section 6, we introduce the Bourgain spaces $$\dot X^b_\zeta$$ and $$X^b_\zeta$$ and recall some basic estimates for these spaces from [22, 23]. In Section 5, we review some averaging estimates from [22, 23] and prove an additional averaging estimate which follows from the Carleson–Sjölin theorem. In Section 8, we prove new estimates for the amplitude $$a$$ of the CGO solutions. The amplitude has the form   a=exp⁡(∂¯e−1A), where $$\bar\partial_e = (e_1 + i e_2) \cdot \nabla$$ for some orthonormal vectors $$\{e_1,e_2\}$$ in $$\mathbb{R}^3$$. Since $$A$$ is only assumed to be in $$W^{s,3}$$, the amplitude $$a$$ may behave very badly. However, if $$\bar\partial^{-1}_e$$ were replaced by $$\lvert{\nabla}\rvert^{-1}$$, then $$a$$ would be bounded in $$L^\infty\cap W^{1,3}_{\mathrm{loc}}$$. By averaging, we show that, for many choices of $$e_1$$ and $$e_2$$, the behavior of $$a$$ is acceptable. In particular, we show that expressions of the form $$a q$$ (where $$q \in W^{-1,3}$$) and $$\Delta a$$ are bounded in $$X^{-1/2}_\zeta$$. Establishing this is a bit delicate and constitutes the main technical difficulty in this paper relative to previous work. We note that it is not too difficult to produce CGO solutions with remainders $$\psi$$ whose $$\dot X^{1/2}_\zeta$$ norm grows like $$o(\tau^{1/2})$$. This was accomplished in the author’s dissertation [21] and is sufficient to show that the magnetic potential $$A$$ is determined by $$\Lambda_{A,q}$$. This is because the main term in the integral identity (3) has size $$\tau$$, so errors of order $$o(\tau)$$ are acceptable. However, once it is shown that the magnetic potentials $$A_1$$ and $$A_2$$ coincide, the main term in the integral identity (3) has size 1, so the error terms should be of order $$o(1)$$. If the $$\dot X^{1/2}_\zeta$$ norm of the remainders $$\psi$$ were to grow like $$o(\tau^{1/2})$$, we would not be able to control the error terms in the integral identity without assuming that $$q$$ is bounded. In Section 9, which contains results from the author’s dissertation [21], we prove estimates for the operator norms of the terms in $$L_{A,q,\zeta}+\Delta_\zeta$$. Since $$\Delta_\zeta$$ maps $$\dot X^{1/2}_\zeta$$ isometrically to $$\dot X^{-1/2}_\zeta$$, the operator $$L_{A,q,\zeta}$$ has a bounded inverse $$L_{A,q,\zeta}^{-1}: \dot X^{-1/2}_\zeta \to \dot X^{1/2}_\zeta$$ as long as the operator norm $$\lVert{L_{A,q,\zeta} + \Delta_\zeta}\rVert_{\dot X^{1/2}_\zeta \to \dot X^{-1/2}_\zeta}$$ is sufficiently small. In [22], the author showed that multiplication by a potential $$q$$ in $$W^{-1,3}$$ is bounded in this operator norm by combining the $$L^p$$ Carleman estimates of [24] with an averaging argument. In the present work, we also consider first-order terms such as $$A \cdot \nabla$$. These terms are more difficult to control, since the behavior of $$A \cdot \nabla$$ is worse than the behavior of $$q$$, particularly when $$A$$ concentrates at low frequencies. To remedy this, we use the fact that when the frequency of $$A$$ is sufficiently low, the curvature of the characteristic set does not play an important role. In this section, we encounter some logarithmic divergences, which is why we need a regularity assumption on $$A$$. It is likely that this limitation can be removed, at least for $$A \in L^{3+}$$, by using a refined version of the pseudodifferential conjugation technique in [36]. This technique should also eliminate the smallness assumption on $$A$$. We hope to address this problem in future work. In Section 10, we show that our averaged estimates are sufficient to run Sun’s version of the Sylvester–Uhlmann argument. In [22], the author concluded the proof of uniqueness in the case $$A = 0$$ using a compactness argument from [38]. This argument relies on the decay of the operator norm $$\lVert{q}\rVert_{X^{1/2}_\zeta \to X^{-1/2}_\zeta}$$ as $$\tau \to \infty$$. It fails for the magnetic Schrödinger equation, because the operator norm $$\lVert{A}\rVert_{X^{1/2}_\zeta \to X^{-1/2}_\zeta}$$ does not decay as $$\tau \to \infty$$, even for smooth $$A$$. Instead, we use the Banach–Alaoglu theorem to show that the Fourier transforms of $$\text{curl} A$$ and $$q$$, which are analytic, vanish on a set of positive measure. 2 An integral identity We now give a very rough outline of how to show that $$\Lambda_{A_1,q_1}\,{=}\,\Lambda_{A_2,q_2}$$ implies that $$(\text{curl}A_1,q_1)\,{=}\,(\text{curl} A_2,q_2)$$ using the Sylvester–Uhlmann strategy. The first step is to write the condition that $$\Lambda_{A_1,q_1} = \Lambda_{A_2,q_2}$$ as an integral identity. Lemma 2.1. Let $$B$$ be the unit ball in $$\mathbb{R}^n$$. Suppose that $$A_i \in L^n$$ and $$q_i \in W^{-1,n}$$ have support in $$\tfrac{1}{2} B$$. If $$\Lambda^B_{A_1, q_2} = \Lambda^B_{A_2,q_2}$$, then the integral identity   ∫[i(A1−A2)⋅(u1∇u2−u2∇u1)+(A12−A22+q1−q2)u1u2]dx=0 (3) holds for any $$u_i \in H^1(B)$$ solving $$L_{A_1,q_1}u_1 = 0$$ and $$L_{- A_2, q_2} u_2 = 0$$ in $$B$$. □ Proof. Define the bilinear form $$Q_{A,q}$$ by   QA,q(u,v)=∫B(∇u⋅∇v+iA⋅(u∇v−v∇u)+(A2+q)uv)dx. If $$u$$ and $$v$$ are functions in $$H^1(B)$$ and $$u$$ is a weak solution to the equation $$L_{A,q}u =0$$ in $$B$$, then   0=∫B(−div∇u−idiv(Au)−iA⋅∇u+A2+q)vdx=QA,q(u,v)−∫∂B∂νu⋅vdx, (4) where $$\nu$$ is the outward unit normal to $$\partial B$$. Thus we have the identity   QA,q(u,v)=⟨∂νu|∂B,v¯|∂B⟩L2(∂B). (5) Suppose we are given functions $$u_1$$ and $$u_2$$ in $$H^1(B)$$ satisfying the equations $$L_{A_1,q_1} u_1 =0$$ and $$L_{-A_2,q_2} u_2 =0$$. The assumption that $$\Lambda_{A_1,q_2} \,{=}\, \Lambda_{A_2,q_2}$$ implies that there is some $$v_2$$ in $$H^1(B)$$ such that $$L_{A_2,q_2} v_2 = 0$$ and   u1−v2|∂B=0,∂ν(u1−v2)|∂B=0. Thus, by the identity (5), we derive that   QA1,q1(u1,u2)=⟨∂νu1|∂B,u¯2|∂B⟩L2(∂B)=⟨∂νv2|∂B,u¯2|∂B⟩L2(∂B)=QA2,q2(v2,u2) On the other hand, by the definition of $$Q_{A,q}(u,v)$$, we have $$Q_{A,q}(v,u) = Q_{-A,q}(u,v)$$. Thus, using the identity (5) again, we derive   QA2,q2(v2,u2)=Q−A2,q2(u2,v2)=Q−A2,q2(u2,u1)=QA2,q2(u1,u2). We conclude that $$Q_{A_1,q_1}(u_1,u_2) - Q_{A_2,q_2}(u_1,u_2) = 0$$, which is (3). ■ To use this integral identity, we construct CGO solutions $$u_1$$ and $$u_2$$ to the equations $$L_{A_1,q_1} u_1 = 0$$ and $$L_{-A_2,q_2} u_2 = 0$$. The CGO solutions $$u_i$$ are approximately complex exponentials $${\rm e}^{x\cdot \zeta_i}$$, where the $$\zeta_i$$ are chosen such that   ζ1=τ(e1+ie2)+O(|k|)ζ2=−τ(e1+ie2)+O(|k|)ζ1+ζ2=ik for some arbitrary vectors $$e_1, e_2, k \in \mathbb{R}^3$$ satisfying   e1⊥e2⊥k|e1|=|e2|=1. Substituting the CGO solutions $$u_i \sim e^{x\cdot \zeta_i}$$ into the integral identity (3) gives   0∼−2iτ(e1+ie2)⋅∫B(A1−A2)eik⋅xdx+∫B(A12−A22+q1−q2)eik⋅xdx. Taking the limit as $$\tau \to \infty$$, we have   0=(e1+ie2)⋅A1−A2^(k) for every pair $$\{e_1, e_2\}$$ of orthonormal vectors perpendicular to $$k$$. This implies that $$\text{curl} A_1 = \text{curl} A_2$$. In particular, by Poincaré’s lemma, there is a gauge transform $$\psi$$ such that $$A_1 - A_2 = \nabla \psi$$. The Dirichlet–Neumann relation is invariant under such gauge transforms. Lemma 2.2. Suppose $$\psi$$ is a function supported in $$\tfrac{1}{2} B$$ such that $$\psi \in W^{1,3}(B)\cap L^\infty(B)$$. Then $$\Lambda_{A + \nabla \psi, q} = \Lambda_{A, q}$$. □ Proof. We have   e−iψLA,qeiψ=−(∇+iA+i∇ψ)2+q. Thus the map $$u \mapsto e^{-i \psi} u$$ is a bijection between solutions to $$L_{A,q} u = 0$$ and solutions to $$L_{A + \nabla \psi,q} u =0$$. Since $$\psi$$ is supported in $$\tfrac{1}{2} B$$, multiplication by $${\rm e}^{i\psi}$$ does not change the boundary data, so the conclusion of the lemma follows. ■ By the gauge-invariance of the Dirichlet–Neumann relation, we have $$\Lambda_{A_2,q_2} = \Lambda_{A_1,q_2}$$. Since we assumed that $$\Lambda_{A_2,q_2} = \Lambda_{A_1,q_1}$$, this implies that $$\Lambda_{A_1,q_1} = \Lambda_{A_1,q_2}$$. Now construct CGO solutions to the equations $$L_{A_1,q_1} u_1 = L_{-A_1,q_2}u_2 = 0$$. Substituting the $$u_i$$ into the integral identity (3) again gives   0∼∫B(q1−q2)eik⋅xdx, and we can conclude that $$q_1 = q_2$$. 3 A transport equation When the magnetic potential $$A$$ is nonzero, the form of the CGO solutions will depend on $$A$$. We construct solutions of the form   u=ex⋅ζ(a+ψ), (6) where $$a = e^{-i\phi}$$ for a suitable function $$\phi$$ depending on $$\zeta$$. The remainder $$\psi$$ must solve the equation   LA,q,ζψ=−Δa−2ζ⋅∇a−i(∇⋅A)a−2iA⋅∇a−2iζ⋅Aa+A2a+qa, (7) where the operator $$L_{A,q,\zeta}$$ is defined by   LA,q,ζ=e−x⋅ζLA,qex⋅ζ=−(∇+iA+ζ)2+q. In order to eliminate the terms of order $$\tau$$ on the right hand side of (7), we choose $$\phi$$ so that $$a$$ solves (roughly speaking) a transport equation of the form   ζ⋅∇a=−iζ⋅Aa. Equivalently, the function $$\phi$$ satisfies an equation of the form   ζ⋅∇ϕ=ζ⋅A. Since $$\zeta = \tau (e_1 + i e_2)$$, where $$e_1$$ and $$e_2$$ are orthonormal vectors, this a $$\bar\partial$$ equation for $$\phi$$ in the plane determined by $$e_1$$ and $$e_2$$. Given $$e=e_1 + i e_2$$, where $$e_1$$ and $$e_2$$ are orthogonal unit vectors, define   ∂¯e=e1⋅∇+ie2⋅∇. We now assume for simplicity that $$e_1$$ and $$e_2$$ are the standard basis vectors. In this case, the operator $$\bar\partial$$ is given by   ∂¯=∂1+i∂2. Let $$f$$ be a function defined on the complex plane, which we identify with $$\mathbb{R}^2$$ by writing $$z = z_1 + iz_2$$. The equation   ∂¯u=f is of Cauchy–Riemann type. If $$f$$ is smooth and compactly supported, then it has a solution given by the formula   ∂¯−1f(w)=12π∫f(w−z)zdz1dz2. The kernel $$(2 \pi z)^{-1}$$ is locally integrable, so it has good mapping properties. Lemma 3.1. If $$f: \mathbb{C} \to \mathbb{R}$$ is supported in $$B(0,1/2)$$, then   ‖⟨w⟩∂¯e−1f(w)‖L∞≲‖f‖L∞. □ Proof. Write   |∂¯e−1f(w)|≲‖f‖L∞∫χB(0,1/2)(z−w)|z|−1dz1dz2 When $$\lvert{w}\rvert \leq 1$$, we estimate the integral by   ∫B(0,3/2)|z|−1dz1dz2∼1. When $$\lvert{w}\rvert > 1$$, we have $$\lvert{z}\rvert \geq \lvert{w}\rvert/2$$ in the region of integration, so we estimate instead by   |w|−1∫χB(0,1/2)(z−w)dz1dz2∼|w|−1. ■ When we substitute CGO solutions of the form $$u_i = {\rm e}^{x \cdot \zeta_i} ({\rm e}^{i \phi_i} + \psi)$$ into the integral identity (3), the main term has the form   −i(ζ1−ζ2)⋅∫B(A1−A2)ei(ϕ1−ϕ2)eix⋅kdx. The next lemma, due to [16] says that we can remove the factor $${\rm e}^{i (\phi_1-\phi_2)}$$ from this integral and recover the Fourier transform. Lemma 3.2. Let $$e_1, e_2, k \,{\in}\, \mathbb{R}^n$$ be arbitrary vectors satisfying $$\lvert{e_1}\rvert\,{=}\,\lvert{e_2}\rvert\,{=}\,1$$ and $$e_1\cdot e_2 = e_1\cdot k= e_2\cdot k=0$$. Let $$A \in C_0^\infty(\mathbb{R}^n)$$, and let $$\phi = \bar\partial_e^{-1}(e\cdot A)$$. Then   (e1+ie2)⋅∫Ae−iϕeix⋅kdx=(e1+ie2)⋅∫Aeix⋅kdx. □ Proof. We first prove the lemma in the case $$n =2$$, where by necessity $$k=0$$. Without loss of generality, we may assume that $$e_1$$ and and $$e_2$$ are the standard basis vectors. Since   (e1+ie2)⋅Ae−iϕ=i∂¯e(e−iϕ), we may write   (e1+ie2)⋅∫Ae−iϕ(x)dx=i∫∂¯e(e−iϕ)dx. (8) By the divergence theorem, we have   ∫(∂1+i∂2)(e−iϕ)dx=limR→∞∫∂B(0,R)(ν1+iν2)e−iϕdS, (9) where $$\nu$$ is the outward unit normal on the circle $$\partial B(0,R)$$. By Lemma 3.1, we have $$\lvert{\phi}\rvert = O(1/\langle {x} \rangle)$$. Thus we have a Taylor expansion of the form   e−iϕ=1−iϕ+O(⟨x⟩−2). Substituting the Taylor series into the right hand side of (9) and applying the divergence theorem again, we find that   ∫∂B(0,R)(ν1+iν2)eiϕdS=∫∂B(0,R)(ν1+iν2)dS−i∫∂B(0,R)(ν1+iν2)ϕdS+∫∂B(0,R)O(R−2)dS=∫B(0,R)∂¯e(1)dx−i∫B(0,R)∂¯eϕdx+O(R−1) Taking the limit as $$R \to \infty$$ we obtain the identity   ∫(∂1+i∂2)(eiϕ)dx=−i∫∂¯eϕdx=−i∫e⋅Adx. Substituting this identity into (8) proves the lemma in the case $$n=2$$. To prove the general case, we assume without loss of generality that $$e_1$$ and $$e_2$$ are the first standard basis vectors. Write $$x = (z, x')$$, where $$z \in \mathbb{R}^2$$ and $$x'\in \mathbb{R}^{n-2}$$. By the two-dimensional case, we have   (e1+ie2)⋅∫A(z,x′)eiϕ(z,x′)dz=(e1+ie2)⋅∫A(z,x′)dz. Since $$k$$ is orthogonal to $$e_1$$ and $$e_2$$, the function $${\rm e}^{ik\cdot x}$$ depends only on $$x'$$. Thus we can multiply both sides by $${\rm e}^{ik\cdot x}={\rm e}^{ik'\cdot x'}$$ and integrate in $$x'$$ to obtain the general case. ■ 4 The operator $$\Delta_\zeta$$ In order to construct solutions to the equation (7) for the remainder $$\psi$$, we consider operators of the form   Δζ=e−x⋅ζΔex⋅ζ. The complex vector $$\zeta \in \mathbb{C}^3$$ is given by   ζ=τ(e1+iη), where $$\tau > 0$$, $$\lvert{e_1}\rvert = 1$$, $$\lvert{\eta}\rvert \leq 1$$ and $$\eta \perp e_1$$. The symbol of $$\Delta_\zeta$$ is   pζ(ξ)=(iξ+ζ)2=−(ξ+τη)2+2iτe1⋅ξ+τ2. The characteristic set $$\Sigma_\zeta$$ is the intersection of the plane perpendicular to $$e_1$$ and a sphere centered at $$-\tau \eta$$.   Σζ={ξ:ξ⋅e1=0,|ξ+τη|=τ}. We will refer to the distance from this set as the modulation. The symbol $$p_\zeta$$ is elliptic at high modulation and vanishes simply on $$\Sigma_\zeta$$, as the reader can easily check (or see [23]).   |pζ|∼{τd(ξ,Σζ) when d(ξ,Σζ)≤τ/8τ2+|ξ|2 when d(ξ,Σζ)≥τ/8. (10) 5 Dyadic projections If $$m$$ is a smooth function on $$\mathbb{R}^n$$, then $$m(D)$$ will denote the Fourier multiplier with symbol $$m(\xi)$$. Let $$\chi \in C^\infty_0([0,1])$$ be a smooth function such that $$\chi = 1$$ on $$[0,3/4]$$. For each dyadic integer $$\lambda = 2^k$$, define the Littlewood–Paley projection $$P_{\leq \lambda}$$ on to frequencies of magnitude $$\lvert{\xi}\rvert \leq \lambda$$ by $$P_{\leq \lambda} = \chi(\lvert{D}\rvert/\lambda)$$. Similarly, define the projection $$P_{>\lambda}$$ on to frequencies of magnitude $$\lvert{\xi}\rvert \gtrsim \lambda$$ by $$P_{>\lambda} = I-P_{\leq \lambda}$$, and define the projection $$P_{\lambda}$$ on to frequencies of magnitude $$\lvert{\xi}\rvert \sim \lambda$$ by $$P_\lambda = P_{\leq \lambda} - P_{\leq \lambda/2}$$. Note that $$I = \sum_{\lambda} P_\lambda$$. Thus we can decompose a function $$f$$ into a sum of dyadic pieces $$f_\lambda = P_\lambda f$$. We can use the Littlewood–Paley decomposition to characterize the Besov spaces $$B^s_{p,q}$$. Given $$s \in \mathbb{R}$$, $$p \in (1,\infty)$$, and $$q \in [1,\infty]$$, the Besov space $$B^s_{p,q}$$ is characterized by the norm   ‖u‖Bp,qs=‖u≤1‖p+(∑λ>1(λs‖uλ‖p)q)1/q. For any integer $$k$$, the Littlewood–Paley square function estimate implies that $$B^k_{p,2} \subset W^{k,p} \subset B^k_{p,p}$$ when $$p\geq 2$$ and that $$B^k_{p,p} \subset W^{k,p} \subset B^k_{p,2}$$ when $$p \leq 2$$. When $$s$$ is not an integer, the Sobolev space $$W^{s,p}$$ is usually defined in such a way that $$W^{s,p} = B^s_{p,p}$$. We will make frequent use of Nikol’skii’s inequality [39] (which is usually referred to as Bernstein’s inequality for some reason). If $$\lambda$$ is any dyadic integer and $$q \geq p$$, then   ‖f≤λ‖q≲λn(1/p−1/q)‖f‖p. (11) Given a pair $$\{e_1,e_2\}$$ of orthonormal vectors, we set $$e = e_1 + i e_2$$ and define partial Littlewood–Paley projections   P≤λe1=χ(|D⋅e1|/λ)P≤λe=χ(|D⋅e|/λ). We define $$P^{e_1}_\lambda$$ and $$P^e_{\lambda}$$ in a similar way. Next, we define projections $$Q_\nu^\zeta$$ to regions where $$d(\xi,\Sigma_\zeta) \sim \nu$$. Let $$\zeta$$ be a complex vector of the form $$\zeta = \tau(e_1 + i \eta)$$. We first define the projection $$C_{\leq \nu}^\zeta$$ by   C≤νζ=χ((|D⊥+τη|−τ)/ν), where $$\xi^\perp = \xi - (\xi\cdot e_1) e_1$$. We then define the projection $$Q_{\leq \nu}^\zeta$$ by   Q≤νζ=P≤νe1C≤νζ. Finally, we define the projection $$Q_\nu^\zeta$$ by $$Q_\nu^\zeta = Q_{\leq \nu}^\zeta - Q_{\leq \nu/2}^\zeta$$ as before. Similarly, we define $$Q_{>\nu} = 1- Q_{\leq \nu}$$. When the choice of $$\zeta$$ is clear from context, we will suppress the dependence of the $$Q$$ projections on $$\zeta$$. Similarly, we will write $$P^1$$ instead of $$P^{e_1}$$ or $$P^{Ue_1}$$. Define projections $$Q_l$$ and $$Q_h$$ on to low and high modulation by   Qlζ=Q≤τ/8ζQhζ=Q>τ/8ζ. Note that the projection $$Q_h^\zeta$$ projects to the region where $$\Delta_\zeta$$ is elliptic. 6 The $$X_\zeta^{b}$$ spaces Given $$b \in (-1,1)$$, define the homogeneous $$\lVert{\cdot}\rVert_{\dot X^{b}_\zeta}$$ norm by   ‖u‖X˙ζb=‖|Δζ|bu‖2, and define the inhomogeneous $$\lVert{\cdot}\rVert_{X^b_\zeta}$$ norm by   ‖u‖Xζb=‖(|Δζ|+τ)bu‖2. By the symbol estimates (10), we have the low-modulation $$L^2$$ estimate   ‖Qμu‖X˙ζb∼(μτ)b‖Qμu‖2, (12) which holds for $$\mu \leq \tau/8$$, and the high-modulation $$L^2$$ estimate   ‖Qhu‖Xζb∼‖u‖Hτb/2, (13) where the semiclassical $$\lVert{\cdot}\rVert_{H^s_\tau}$$ norm is defined by   ‖u‖Hτs=‖(|D|+τ)su‖2. The $$X^b_\zeta$$ spaces behave well under localization, as we see from the following lemma. Lemma 6.1. ([23 Lemma 2.2]). If $$\phi$$ is a Schwartz function, then   ‖ϕu‖Xζ1/2≲ϕ‖u‖X˙ζ1/2 (14)  ‖ϕu‖X˙ζ−1/2≲ϕ‖u‖Xζ−1/2. (15) □ Let $$\phi$$ be a fixed Schwartz function which is identically equal to one on the unit ball. Then   τ1/2‖u‖L2(B(0,1))≤τ1/2‖ϕu‖L2≤‖ϕu‖Xζ1/2≲‖u‖X˙ζ1/2. Replacing $$u$$ with $$u(x/R)$$, we obtain the Agmon–Hörmander-type estimate   τ1/2R−1/2‖u‖L2(B(0,R))≲‖u‖X˙ζ1/2. (16) Using the estimate (16), it is not hard to show that the space $$\dot X^{1/2}_\zeta$$ is a Banach space and embeds continuously into $$\tau^{-1/2} L^2(\mathbb{R}^3, \langle x \rangle^{-1-\delta}\,dx)$$ for any $$\delta>0$$. The next lemma gives Strichartz-type estimates for the $$X^b_\zeta$$ spaces. Lemma 6.2. ([24 Proposition 6.3]). Suppose $$\nu\leq \tau/8$$. Then for any $$f \in \dot X^{1/2}_\zeta(\mathbb{R}^3)$$, we have   ‖Qνf‖6≲(ν/τ)1/3‖f‖X˙ζ1/2. (17)  ‖f‖6≲‖f‖X˙ζ1/2. (18) □ We also have the dual estimates   ‖Qνf‖Xζ−1/2≲(ν/τ)1/3‖f‖6/5 (19)  ‖f‖Xζ−1/2≲‖f‖6/5. (20) 7 Averaging estimates We will need to average various norms with respect to parameters $$(\tau, U)$$, which will be chosen from the set $$[2,\infty) \times O(3)$$. In order to distinguish this averaging from integration over physical space, we will use probabilistic notation. Let $$(X,\sigma)$$ be a finite measure space such that $$\sigma(X)>0$$. Let $$Z$$ be an integrable function on $$X$$. We write the average of $$Z$$ over $$X$$ as   E[Z∣X]=σ(X)−1∫XZdσ. Similarly, for a measurable subset $$Y$$ of $$X$$, we write   P[Y∣X]=σ(Y∩X)σ(X). Define the $$L^p$$ average of $$Z$$ over $$X$$ by   Ep[Z∣X]=‖Z‖Lp(X,dσ)‖1‖Lp(X,dσ). Unless otherwise specified, the set $$X$$ will be the orthogonal group $$O(3)$$, and $$\sigma$$ will be normalized Haar measure on $$O(3)$$. Given a measurable function $$Z$$ on $$[\tau_*,2\tau_*] \times O(3)$$, define   Eτ∗p[Z]=Ep[Z∣[τ∗,2τ∗]×O(3)], where the average here is taken with respect to the measure $$m$$ on $$[2, \infty) \times O(3)$$ given by   dm(τ,U)=(τlog⁡τ)−1dτdσ(U). For a positive integer $$K$$, define   E~Kp[Z]=Ep[Z∣[2K,2K2]×O(3)] and   P~K[Y]=P[Y∣[2K,2K2]×O(3)]. The quantity $$m([2^K,2^{K^2}] \times O(3))$$ is given by   ∫2K2K2(τlog⁡τ)−1dτdσ(U)∼log⁡K. (21) On each dyadic interval $$[\tau^*,2\tau^*]$$, the weight $$(\tau \log \tau)^{-1}$$ is approximately constant. Thus we can estimate $$\widetilde{\mathbb{E}}_{k}^{p}[Z]$$ by   E~Kp[Z]p ~(log⁡K)−1∑2K≤τ∗<2K2(log⁡τ∗)−1Eτ∗p[Z]p. We will need the following property of the Haar measure: if $$f$$ is an integrable function on $$S^{2}$$, then for any fixed $$\theta\in S^{2}$$, we have the identity   E[f(U⋅ θ)∣O(3)]=E[f(ω)∣S2]. (22) We will use the averaging in $$\tau$$ to take advantage of the extra decay in expressions of the form $$(\lambda /\tau)^\alpha \lVert{P_\lambda f}\rVert_p$$, where $$\lambda$$ is a dyadic integer less than or comparable to $$\tau$$. Namely, if $$\alpha > 0$$ and $$p \in [2,\infty)$$, then we have frequency convolution estimate   E~Kp[∑λ≲τ(λ/τ)α(log⁡τ)1/p‖Pλf‖p]≲(log⁡K)−1/p‖f‖p. (23) To see this, we recall the normalization (21) and use Young’s inequality, which gives   ‖∑λ≲τ(λ/τ)α‖Pλf‖p‖Lp([2K,2K2],τ−1dτ)≲(∑λ‖Pλf‖pp)1/p. That is, the right-hand side has the form $$\lVert{T(\{a_\lambda\})}\rVert_{L^p}$$, where $$T: l^p \to L^p([2^K,2^{K^2}],\tau^{-1}\,{\rm d}\tau)$$ is given by $$T(\{a_\lambda\})(\tau) = \sum_{\lambda} K(\lambda,\tau) a_\lambda$$ and $$\sup_\lambda \lVert{K(\lambda,\tau)}\rVert_{L^1([2^K,2^{K^2}],\tau^{-1}\,d\tau)} + \sup_\tau \lVert{K(\lambda,\tau)}\rVert_{l^1_\lambda}$$ is bounded. By the Besov embedding $$L^p \subset B^0_{p,p}$$, the right hand side is bounded by $$\lVert{f}\rVert_p$$. Lemma 7.1. Let $$p \in [2,\infty]$$ and let $$1/p'=1-1/p$$. Let $$\mu,\nu,\lambda$$ be dyadic integers, such that $$\mu,\nu\lesssim \lambda$$. Then   Ep[⟨λ/ν⟩1/p⟨λ/μ⟩1/p‖P≤μUe1P≤νUe2Pλu‖p]≲‖u‖p (24)  Ep′[⟨λ/ν⟩1/p⟨λ/μ⟩1/p‖P≤μUe1P≤νUe2Pλu‖p′]≲‖u‖p′. (25) As usual, the averages are taken over $$U$$ in $$O(3)$$. If $$p \in [2,4]$$, then we also have   Eτ∗p[(1+log+⁡(τ/ν))5(1/p−1/2)⟨λ/ν⟩3/p−1/2‖Q≤ντ(Ue1+iUe2)Pλu‖p]≲‖u‖p (26) □ Proof. When $$p=2$$, all of these estimates follow from Plancherel’s theorem and Fubini. To prove the first estimate (24) when $$p=2$$, we write   E2[‖P≤νUe1P≤μUe2Pλu‖2]2∼∫RnE|ϕ(ξλ)χ(ξ⋅(Ue1)μ,ξ⋅(Ue2)ν)|2|u^(ξ)|2dξ Here $$\phi$$ is supported on an annulus, and $$\chi$$ is supported on a square. Since $$U$$ is orthogonal, we have $$\xi \cdot (Ue_i) = (U^{-1} \xi) \cdot e_i$$. Thus we can compute the last integral using the identity (22):   E|ϕ(ξλ)χ((U−1ξ)⋅e1μ,(U−1ξ)⋅e2ν)|2≲sup|ξ|∼λE[|χ(|ξ|ω⋅e1μ,|ξ|ω⋅e2ν)|2|ω∈S2] The quantity on the right is bounded by the area of the intersection of the unit sphere with a rectangle centered at the origin of size proportional to $$(\mu/\lambda) \times (\nu/\lambda)\times 1$$. Since the area of such a region is bounded by $$\langle{\lambda/\mu}\rangle^{-1} \langle{\lambda/\nu}\rangle^{-1}$$, we have   E2[‖P≤νUe1P≤μUe2Pλu‖2]2≲⟨λ/μ⟩−1⟨λ/ν⟩−1‖u‖22. The $$p=2$$ case of the last estimate (26) is proven in the same way. Since $$\lvert {p_\zeta(\xi)} \rvert \lesssim \tau \nu$$ on the Fourier support of $$Q_\nu$$, it suffices to show that   Eτ∗[Z(τ,U)]≲⟨λ/ν⟩−1, (27) where   Z(τ,U)=sup|ξ|∼λ|χ(|−|ξ|2−2τξ⋅(Ue2)+2τ|ξ⋅(Ue1)|τν)|2 and $$\chi$$ is compactly supported. Using the identity (22) again yields   Eτ∗[Z]≲sup|ξ|∼λ1τ∗∫τ∗2τ∗∫S2|χ(|−|ξ|2−2τ|ξ|ω⋅e2|+2τ||ξ|ω⋅e1τν)|2dS(ω)dτ. (28) View $$(\tau,\omega)$$ as polar coordinates on $$\mathbb{R}^3$$, and change variables to $$u = \tau\omega$$. In the annular region $$\{\lvert u \rvert\in [\tau_*,2\tau_*]\}$$, the volume element $$du$$ is bounded below by $$\tau_*^{2} \, {\rm d}S(\omega) \,{\rm d}\tau$$. Thus the integral on the right hand side of (28) is bounded by   1τ∗3∫|u|∈[τ∗,2τ∗]|χ(|ξ|(|−|ξ|−2u⋅e2|+2|u⋅e1|)τν)|2du. The integrand is supported on a rectangle of size proportional to $$\tau(\langle{\lambda/\nu}\rangle^{-1} \times \langle{\lambda/\nu}\rangle^{-1} \times 1)$$. So the integral is bounded by the quantity $$\langle{\lambda/\nu}\rangle^{-2}$$, which establishes (27). This shows that   Eτ∗2[‖Q≤νPλu‖2]2≲⟨λ/ν⟩2‖u‖22. To prove the $$p \neq 2$$ case of the first two estimates (24) and (25), we define an operator $$T$$ by   Tu(U,x):=P≤μUe1P≤νUe2Pλu(x). We have shown that $$T$$ satisfies the $$L^2$$ bound   ‖T‖L2(R3)→L2(O(3)×R3)≲⟨λ/ν⟩−1/2⟨λ/μ⟩−1/2. On the other hand, since the Littlewood–Paley projections $$P^{Ue_i}_{\leq \mu}$$ and $$P_\lambda$$ are all bounded on every $$L^p$$ space, the operator $$T$$ also satisfies the bounds   ‖T‖L∞(R3)→L∞(O(3)×R3)+‖T‖L1(R3)→L1(O(3)×R3)≲1. By interpolation, we obtain the $$L^p$$ bounds (24) and (25). To prove the $$p\neq 2$$ case of the last estimate (26) for $$Q_{\leq \nu}^{\zeta(\tau,U)}$$, we interpolate with an $$L^4$$ bound. The operator $$Q_{\leq \nu}$$ factors as   Q≤νζ(τ,U)=C≤νζ(τ,U)P≤νUe1, where the operator $$C_{\leq \nu}^{\zeta(\tau,U)}$$, defined by   C≤νζ(τ,U)=χ((|D⊥+τe2|−τ)/ν), localizes the vector $$\xi^\perp = (0,\xi \cdot U e_2, \xi\cdot Ue_3)$$ to a neighborhood of a circle of radius $$\tau$$ and center $$(\xi\cdot (Ue_1), \tau e_2)$$. The Carleson–Sjölin theorem ([11]) implies that $$C^{\zeta(\tau,U)}_{\leq \nu}$$ satisfies the $$L^4$$ bound   ‖C≤νζ(τ,U)‖L4(R3)→L4(R3)≲(1+log+⁡(τ/ν))5/4. (29) This estimate (modulo rescaling and modulation) is explicit in [14]. Thus, by combining the $$L^4$$ bound (29) with the case $$p=4$$ of the bound (24) that we have already established, we obtain the $$L^4$$ bound   Eτ∗4[‖Q≤νζ(τ,U)u‖4]≲(1+log+⁡(τ/ν))5/4Eτ∗4[‖P≲νUe1u‖4]≲(1+log+⁡(τ/ν))5/4⟨λ/ν⟩−1/4‖u‖4. Interpolating this $$L^4$$ estimate with the $$L^2$$ estimate we have already established, we obtain the $$L^p$$ estimate (26). ■ 8 Estimates for the amplitude To analyze the behavior of $$\bar\partial^{-1}$$, we introduce an auxiliary function $$\eta$$ to use as a mollifier. Let $$\eta: \mathbb{R}^2 \to \mathbb{R}$$ be a smooth compactly supported bump function, such that $$\int_{\mathbb{R}^2} \eta = 1$$ and   ∫z1α1z2α2η(z)dz1dz2=0, (30) for every pair $$(\alpha_1,\alpha_2)$$ of nonnegative integers such that $$1 \leq \alpha_1 + \alpha_2 \leq 2M$$, where $$M$$ is some large number to be determined later. The vanishing moment condition (30) ensures that the Fourier transform $$\hat \eta$$ satisfies   η^(ξ)=1+O(|ξ|2M+1). (31) Define the operator $$\tilde P$$ (acting on functions on $$\mathbb{R}^2$$) by   P~u=η^(D)u, Let $$\chi_\nu$$ be the symbol of the Littlewood–Paley projection $$P_{\nu}$$. Since $$\hat \eta$$ is Schwartz and $$\chi_\nu$$ is supported in the set $$\{\xi: \lvert {\xi} \rvert \lesssim \nu\}$$, we have   ⟨ξ⟩l|∇ξk(χνη^)(ξ)|≲k,l,Nν−N for all non-negative integers $$k,l , N$$ and uniformly in $$\nu \geq 1$$. This implies that the integral kernel $$K_\nu$$ of $$\tilde P P_\nu$$ is Schwartz, and more precisely that $$ \lvert {K_\nu(x)} \rvert \lesssim_{N} \nu^{-N}\langle {x} \rangle^{-10}$$. Thus for any $$p \in [1,\infty]$$ and $$N > 0$$, the operator $$\tilde P$$ is almost orthogonal to $$\{P_\nu\}_{\nu \geq 1}$$, in the sense that   ‖P~Pν‖Lp(R2)→Lp(R2)≲Nν−N (32) for all $$N> 0$$, and uniformly in $$\nu \geq 1$$. On the other hand, the vanishing property (31) implies that   ⟨ξ⟩l|∇ξk(χν(1−η^))(ξ)|≲k,lν2M+1−k for all non-negative integers $$k, l\leq M$$ and uniformly in $$\nu \leq 1$$. It follows that for $$M$$ sufficiently large, the operator $$(1-\tilde P)$$ is almost orthogonal to $$\{P_\nu\}_{\nu \leq 1}$$, in the sense that   ‖(1−P~)Pν‖Lp(R2)→Lp(R2)≲νM (33) for all $$p \in [1,\infty]$$ and uniformly in $$\nu \leq 1$$. It is easy to see that the same estimates hold on $$L^p(\mathbb{R}^3)$$ if $$\tilde P$$ and $$ P_\nu$$ are replaced by $$\tilde P^e = \hat \eta(D \cdot e)$$ and $$P_\nu^e = \chi_\nu(D\cdot e)$$. We apply this machinery to show that the behavior of $$\bar\partial$$ near its the characteristic set can be ignored if everything is localized. Lemma 8.1. Let $$u$$ be a function on $$\mathbb{R}^2$$ whose support lies in the unit ball. Let $$E\subset \mathbb{R}^2$$ be a set of finite measure. Then for $$p \in [1,\infty]$$ we have   ‖∂¯−1u‖Lp(E)≲(1+|E|1/p)(‖P≤1u‖p+‖∂¯−1P>1u‖p). (34) □ Proof. Decompose $$u$$ as $$u = \tilde P u + (1-\tilde P) u$$. By Hölder’s inequality, we have $$\lVert{\bar\partial^{-1} \tilde P u}\rVert_{L^p(E)} \leq \lvert {E} \rvert^{1/p} \lVert{\bar\partial^{-1} \tilde Pu }\rVert_\infty$$. By Lemma 3.1 and Nikol’skii’s inequality (11)   ‖∂¯−1P~u‖∞≲‖P~u‖∞≲‖P~P≤1u‖∞+∑ν>1‖P~Pνu‖∞≲‖P≤1u‖p+∑ν>1ν2/p‖P~Pνu‖p. By the almost orthogonality bound (32), we have   ∑ν>1ν2/p‖P~Pνu‖p≲‖P≤1u‖p+∑ν>1ν2/p−N‖Pνu‖p≲‖P≤1u‖p+∑ν>1ν2/p+1−N‖Pν∂¯−1u‖p≲‖P≤1u‖p+‖∂¯−1P>1u‖p. For $$(1-\tilde P)u$$ we use the almost orthogonality bound (33) and the fact that $$(1-\tilde P)$$ is bounded on $$L^p$$ for any $$p$$. Thus   ‖∂¯−1(1−P~)u‖p≲∑ν≤1ν−1‖(1−P~)Pνu‖p+‖∂¯−1P>1u‖p≲∑ν≤1νM−1‖P≤1u‖p+‖∂¯−1P>1u‖p≲‖P≤1u‖p+‖∂¯−1P>1u‖p. ■ Using the localization estimate (34), we show that $$\bar\partial_{Ue}^{-1} \nabla f$$ is bounded on average in $$L^2(B)$$ with a slight loss of regularity. Lemma 8.2. Let $$f \in H^s(\mathbb{R}^3)$$, where $$s>0$$, and suppose that $$\text{supp} f \subset B$$, where $$B = B(0,1)$$. Then   E2[‖∂¯Ue−1∇f‖L2(B)]≲s‖f‖Hs. □ Proof. First we apply the localization estimate (34), to obtain   ‖∂¯Ue−1∇f‖L2(B)≲‖P≤1Ue∇f‖L2+‖P>1Ue∂¯Ue−1∇f‖L2. We bound both terms using the averaging estimate (24).   E2[‖P≤1Ue∇f‖L2]≲‖∇f≤1‖2+∑λ>1λE2[‖P≤1Uefλ‖2]≲‖f‖2+∑λ>1‖fλ‖2≲‖f‖Hs. Here $$f_{\leq 1}$$ and $$f_\lambda$$ are defined using the Littlewood–Paley projections $$P_\lambda$$ on $$\mathbb{R}^3$$. Similarly, since $$P^{Ue}_\nu f_\lambda = 0$$ unless $$\nu \lesssim \lambda$$, we have   E2[‖P>1Ue∂¯Ue−1∇f‖L2]≲∑1≤ν≲λ(λ/ν)E2[‖PνUefλ‖2]≲∑λ>1(log⁡λ)‖fλ‖2≲‖f‖Hs. ■ We now show that the $$\bar\partial^{-1}_{U e}$$ operator takes compactly supported functions in the Besov space $$B^0_{3,1}(\mathbb{R}^3)$$ to bounded functions. If the $$\bar\partial^{-1}_{Ue}$$ operator was replaced by $$\lvert {D} \rvert^{-1}$$, then this property would hold without any averaging. Lemma 8.3. Let $$f \in B^0_{3,1}(\mathbb{R}^3)$$, and suppose that $$\text{supp} f \subset B(0,1)$$. Then   E3[‖∂¯Ue−1f‖∞]≲‖f‖B3,10. □ Proof. First, we show that for such $$f$$, we have the estimate   ‖∂¯Ue−1f‖∞≲Z(U), where   Z(U)=‖f‖3+∑1≤ν,λ(λ/ν)1/3‖P≤νUePλf‖3. To this end, we apply the localization estimate (34), which gives   ‖∂¯Ue−1f‖∞≲‖P≤1Uef‖∞+‖∂¯Ue−1P>1Uef‖∞. Next, we decompose $$f$$ into Littlewood-Paley pieces and apply Nikol’skii’s inequality (11). We estimate $$P^{Ue}_{\leq 1} f$$ by   ‖P≤1Uef‖∞≲‖P≤1Uef≤1‖∞+∑ν,λ>1‖P≤1Uefλ‖∞≲‖f‖3+∑ν,λ>1λ1/3‖P≤1Uefλ‖3.≤Z(U). We estimate $$\bar\partial^{-1}_{Ue}P^{Ue}_{>1} f$$ in the same way. Note that $$P_\nu^{Ue} f_\lambda$$ vanishes unless $$\nu \lesssim \lambda$$, so   ‖∂¯Ue−1P>1Uef‖∞≲∑1<ν≲λν−1‖PνUefλ‖∞≲∑1<ν≲λν−1/3λ1/3‖PνUefλ‖3≲Z(U). Finally, we show that $$Z(U)$$ is bounded on average. This follows from the averaging estimate (24), which gives   E3[Z(U)]≲‖f‖3+∑1≤ν≤λ(ν/λ)1/3‖Pλf‖+∑1≤λ≤ν(λ/ν)1/3‖Pλ‖3≲‖f‖3+∑λ≥1‖Pλf‖3≲‖f‖B3,10. ■ To state the next lemma, we introduce the mixed-norm notation   ‖f‖Lx1p1Lx2p2Lx3p3=‖‖‖f(x1,x2,x3)‖Lx3p3‖Lx2p2‖Lx1p1. Given an orthonormal frame $$\{e_1,e_2,e_3\}$$, we will also use the notation   ‖f‖Le1p1Le2p2Le3p3=‖f(x)‖Ly1p1Ly2p2Ly3p3, where the $$L^p$$ norms on the right hand side are taken with respect to the coordinates $$y_i = x \cdot e_i$$. Sometimes we will write $$L^p_e$$ for $$L^p_{e_1} L^p_{e_2}$$, where $$e = e_1 + i e_2$$. We will also omit to specify all of the directions $$e_i$$ when they can be inferred from context. Observe that the norms $$L^\infty L^1_{e}$$ and $$L^{3/2}$$ scale identically under the isotropic dilations $$x \mapsto \lambda x$$, where $$\lambda$$ is a positive real number. Of course, there cannot possibly be a straightforward relationship between these norms, since they scale differently with respect to the anisotropic dilations $$x_e + x_{e_3} \mapsto \lambda x_e + \mu x_{e_3}$$, where $$x_e$$ and $$x_{e_3}$$ are the projections of $$x$$ on to the $$e$$ and $$e_3$$ directions, respectively. The next lemma, due to Falconer [18], states that we can control one norm with respect to the other (with an $$\epsilon$$ loss of integrability) if we average over all of the different frames $$\{e_1,e_2,e_3\}$$. Since we will apply it to $$A$$ in the subcritical space $$W^{s,3}$$, this will suffice for our purposes. We will not use Lemma 8.4 as stated. Instead we will use an easy consequence: if $$q$$ lies in the range $$(3,3/(1-s))$$, then   Eq[‖A‖L∞LUe2]≲‖A‖q≲‖A‖Ws,3. (35) We prove the above estimate by applying Lemma 8.4 to the function $$A^2$$, which lies in $$L^{q/2}$$. Lemma 8.4. Let $$f \in L^{p}(\mathbb{R}^3)$$, where $$p> 3/2$$. Assume $$f$$ is supported in a ball $$B(0,1)$$. Then   Ep[‖f‖L∞LUe1]≲‖f‖p. □ Proof. Let $$g = \lvert {f} \rvert$$, and let $$\eta$$ be a mollifier as defined above. Since $$g$$ is non-negative, we have   ‖g‖Lx3∞Lx1,x21=esssupx3∫g(x1,x2,x3)dx1dx2=esssupx3∫∫η((y1−x1)+i(y2−x2))dy1dy2g(x1,x2,x3)dx1dx2=esssupx3∫η∗g(y1,y2,x3)dy1dy2 Thus we can replace $$g$$ by $$\tilde P g$$, since   ‖g‖Lx3∞Lx1,x21≤‖P~g‖Lx3∞Lx1,x21. More generally, for $$U \in O(3)$$, we have   ‖g‖L∞LUe1≤‖P~Ueg‖L∞LUe1. Now, since $$\tilde P^{Ue} g$$ is supported in the ball $$B(0,2)$$, Hölder’s inequality implies that   ‖P~Ueg‖L∞LUe1≲‖P~Ueg‖L∞LUep. Decompose $$g$$ as $$g = \sum_{\nu,\lambda\geq 1} P^{Ue}_\nu P_\lambda g$$. By abuse of notation, we redefine $$P_1$$ and $$P^{Ue}_1$$ as $$P_1 = P_{\leq 1}$$ and $$P^{Ue}_1 = P^{Ue}_{\leq 1}$$. For each of these pieces, we apply Nikol’skii’s inequality (11) in the $$U e_3$$ direction, which gives   ‖P~UePνUePλg‖L∞LUep≲λ1/p‖P~UePνUePλg‖p By the almost orthogonality bound (32), this implies that   ∑ν,λ≥1‖P~UePνUePλg‖L∞LUep≲∑ν,λ≥1λ1/pν−N‖PνUePλg‖p Averaging over $$O(3)$$ using the averaging bound (25), we obtain   ∑ν,λ≥1λ1/pν−NE3[‖P~UePνUePλg‖L∞LUep]≲∑ν,λ≥1λ1/p⟨λ/ν⟩2/p−2ν−N‖Pλg‖p≲∑λ≥1(λ1/p−N+λ3/p−2)‖g‖p≲‖g‖p. ■ We are now ready to prove estimates in the space $$X^{-1/2}_\zeta$$ for some dangerous terms that will appear as inhomogeneous terms in the equation (7), which we will use to construct the remainder term $$\psi$$. The next lemma is fairly straightforward to prove if $$a = 1$$; in that case it follows from the fact that the $$X^{-1/2}_\zeta$$ norm is controlled, on average, by the $$W^{-1,2}$$ norm. Since we control the $$W^{-1,3}$$ norm, there is some slack here. When $$a$$ is nontrivial, we will have to work harder, but this extra slack will help us get the required estimates. Lemma 8.5. Let $$q \in W^{-1,3}(\mathbb{R}^3)\cap W^{-1,2} (\mathbb{R}^3)$$, and suppose that for each $$(\tau, U)$$ in $$\mathbb{R}_+ \times O(3)$$ we are given a function $$a_{\tau,U}$$, such that   M=supτ,U(‖a‖∞+τ−1‖∇a‖∞+τ−2‖∇2a‖∞+‖∇a‖2)<∞. Then   E~K2‖a⋅q‖XτU(e1+ie2)−1/2 ≲M(log⁡K)−1/3(‖q‖W−1,2+‖q‖W−1,3)+‖q>2K‖W−1,2. (36) □ Proof. Let $$\zeta = \tau U(e_1+ie_2)$$. In what follows, we will use $$\lVert{\cdot}\rVert$$ to denote the $$X^{-1/2}_\zeta$$ norm. Since we are working with homogeneous norms, it is convenient to redefine all of our dyadic projections by $$P_1 = P_{\leq 1}$$, $$Q_1 = Q_{\leq 1}$$ and so on. At high modulation, we use the high-modulation estimate (13)   ‖Qh(aq)‖≲‖aq‖Hτ−1. We estimate this using the definition of $$H^{-1}_\tau$$. For a test function $$u$$, we have   |(aq,u)|=|(q,a¯u)|≲∑1≤λ≤τ|(qλ,a¯u)|+|(q>τ,a¯u)|≲∑1≤λ≤τ(λ/τ)‖qλ‖W−1,2‖a‖∞τ‖u‖2+‖q>τ‖W−1,2‖au‖H1≲M(∑1≤λ≤τ(λ/τ)‖qλ‖W−1,2+‖q>τ‖W−1,2)‖u‖Hτ1. Thus by duality, we have   ‖aq‖Hτ−1≲M∑1≤λ≤τ(λ/τ)‖qλ‖W−1,2+M‖q>τ‖W−1,2. Applying the frequency convolution estimate (23), we have   E~K2[‖aq‖Hτ−1] ≲M(log⁡K)−1/2‖q‖W−1,2+M‖q>2K‖W−1,2. We decompose the low-modulation part as   Ql(a⋅q)=LL+HH, (37) where the low–low part is given by   LL=Ql(a≲τq≲τ) and the high–high part is given by   HH=∑λ1,λ2≫τQl(aλ1qλ2)=∑λ≫τQl(aλq∼λ). For each $$\lambda$$, the expression $$q_{\sim \lambda}$$ denotes a sum of Littlewood–Paley projections of $$q$$ with frequencies comparable to $$\lambda$$. Here we use that fact that if the ratio between $$\lambda_1$$ and $$\lambda_2$$ is very large or very small, then $$a_{\lambda_1} q_{\lambda_2}$$ has Fourier support in $$\{\xi: \lvert {\xi} \rvert \sim \max\{\lambda_1,\lambda_2\} \gg \tau\}$$. We further decompose the low-low part as   LL=I+II+III+IV+V, where   I=∑1≤λ≲τ∑μ≥A(λ,τ)Qμ(a≲τqλ)II=∑1≤λ≲τ∑μ<A(λ,τ)Qμ(a≤μqλ)III=∑1≤λ≲τ∑μ<A(λ,τ)Qμ(a≥λqλ)IV=∑1≤λ≲τ∑μ<A(λ,τ)Qμ(a(μ,B(μ,λ,τ))qλ)V=∑1≤λ≲τ∑μ<A(λ,τ)Qμ(a[B(μ,λ,τ),λ)qλ). The cut-offs $$A(\lambda,\tau)$$ and $$B(\mu,\lambda,\tau)$$ will be chosen later. We estimate $$\lVert{I}\rVert_{X^{-1/2}_\zeta}$$ by the $$L^2$$ estimate (12).   ‖I‖≲∑1≤λ≲τ∑μ≥A(λ,τ)(μτ)−1/2λ‖a‖∞‖qλ‖W−1,2≲‖a‖∞∑λ≲τ(λ2/τ)1/2A(λ,τ)−1/2‖qλ‖W−1,2. Taking $$A(\lambda,\tau) = \lambda^{2-2\epsilon} \tau^{-1+2\epsilon}$$, we apply the frequency convolution estimate (23) again to obtain   ‖I‖≲‖a‖∞∑λ≲τ(λ/τ)ϵ‖qλ‖W−1,2E~K2[‖I‖]≲(log⁡K)−1/2‖a‖∞‖q‖W−1,2. For $$\lVert{II\rVert}$$ multiplication by $$a_{\lesssim \mu}$$ shifts the Fourier support by at most $$\mu$$. Thus we have $$Q_\mu(a_{\leq \mu}q_{>\mu}) = Q_\mu(a_{\leq \mu} Q_{\lesssim \mu}q_{>\mu})$$. By the $$L^2$$ estimate (12) and the averaging estimate (26), we have   Eτ∗2[‖II‖]≲∑μ≲A(λ,τ∗)(μτ∗)−1/2‖a‖∞Eτ∗2[‖Q≲μqλ‖2]≲M∑λ≲τ∗∑μ≲A(λ,τ∗)(μ/τ∗)1/2‖qλ‖W−1,2≲M∑λ≲τ∗(λ/τ∗)1−ϵ‖qλ‖W−1,2E~K2[‖II‖]≲M(log⁡K)−1/2‖q‖W−1,2. For $$\lVert{III}\rVert$$, we use the Strichartz estimate (19):   ‖III‖≲∑λ≲τ∑μ<A(λ,τ)∑ν≥λ(μ/τ)1/3‖aνqλ‖6/5≲∑λ≲τ∑μ<A(λ,τ)∑ν≥λ(μ/τ)1/3(λ/ν)‖∇aν‖2‖qλ‖W−1,3≲∑λ≲τ(λ/τ)(2−2ϵ)/3‖∇a‖2‖qλ‖W−1,3E~K3[‖III‖]≲M(log⁡K)−1/3‖q‖W−1,3. For the terms in $$\lVert {IV_\mu} \rVert$$ we can use the identity $$Q_\mu(a_{(\mu,B)} q_\lambda) = Q_\mu(a_{(\mu,B)} Q_{\lesssim B} q_\lambda)$$. Thus by the $$L^2$$ estimate (12) and the averaging estimate (26) we have, with $$B = \min{\{\lambda, \mu^{1/2}\tau^{1/2-2\epsilon} \lambda^{2\epsilon}\}}$$,   ‖Qμ(a(μ,B)qλ)‖≲(μτ)−1/2‖a‖∞‖Q≲Bqλ‖2Eτ∗2[‖Qμ(a(μ,B)qλ)‖]≲M(λ/τ∗)2ϵ‖qλ‖W−1,2. Summing over $$\mu$$, we obtain   Eτ∗2[‖IV‖]≲ M(log⁡τ∗)1/2∑λ≲τ∗(λ/τ∗)2ϵ‖qλ‖W−1,2E~K2[‖IV‖]≲M(log⁡K)−1/2‖q‖W−1,2. For $$\lVert {V_\mu} \rVert$$, we use the identity $$Q_\mu(a_\nu q_\lambda) = Q_\mu(a_\nu Q_{\lesssim \nu} q_\lambda)$$. Using the Strichartz estimate (19) and the averaging estimate (26), we obtain   ‖Qμ(aνqλ)‖≲(μ/τ)1/3‖aνQ≲ν‖qλ6/5≲(μ/τ)1/3‖aν‖2‖Q≲νqλ‖3≲(μ/τ)1/3ν−1‖∇aν‖2‖Q≲ν‖qλ3Eτ∗3[‖Qμ(aνqλ)‖]≲(μ/τ∗)1/3(λ/ν)1/2(τ∗/ν)2ϵM‖qλ‖W−1,3. Summing over $$\nu$$, we have   Eτ∗3[‖Qμ(a[B,λ)qλ)‖]≲(μ/τ∗)1/3(λ/B)1/2(τ∗/B)2ϵM‖qλ‖W−1,3≲μ1/12−ϵτ∗−7/12+2ϵ+4ϵ2λ1/2−ϵ−4ϵ2M‖qλ‖W−1,3≲(μ/λ)1/12−ϵ(λ/τ∗)7/12−2ϵ−4ϵ2M‖qλ‖W−1,3. Summing over $$\mu$$ and applying the frequency convolution estimate (23), this gives   Eτ∗3[‖V‖]≲ ∑λ≲τ∗(λ/τ∗)αM‖qλ‖W−1,3E~K3[‖V‖]≲M(log⁡K)−1/3‖q‖W−1,3. Finally, we estimate the high-high terms. When the modulation is sufficiently small, we use the Strichartz estimate (19)   ‖Q≤C(aλq∼λ)‖≲(C/τ)1/3λ‖aλ‖2‖q∼λ‖W−1,3≲(C/τ)1/3M‖q‖W−1,3. (38) When the modulation is large, we use the $$L^2$$ estimate (12) and then estimate $$\lVert{A}\rVert_6$$ by interpolation.   ‖Q>C(aλq∼λ)‖≲(Cτ)−1/2λ‖aλ‖6‖q∼λ‖W−1,3≲(Cτ)−1/2λ‖aλ‖∞2/3‖aλ‖21/3‖q∼λ‖W−1,3≲(Cτ)−1/2λ2/3(τ/λ)4/3M‖q‖W−1,3. (39) Here we use that $$\lVert {\nabla^2 a} \rVert_\infty \lesssim \tau^2 M$$. Let $$C = \tau \lambda^{-\epsilon}$$. Summing the inequalities (38) and (39) over $$\lambda \gtrsim \tau$$, we obtain   ‖HH‖≲ ∑λ≳τ(λ−ϵ/3+τ1/3λ−2/3+ϵ)M‖q‖W−1,3E~K[‖HH‖]≲2−ϵK/3M‖q‖W−1,3. ■ In the next lemma, we make use of the relationship between the operator $$\bar\partial_e$$ and the operator $$\Delta_\zeta$$. Lemma 8.6. Fix $$s>1$$. Let $$B = B(0,1)$$. Let $$A$$ be a smooth function supported in $$\frac{1}{2} B$$, and let $$\chi$$ be a cut-off supported in $$B$$ such that $$\chi = 1$$ on $$\tfrac{1}{2} B$$. Let $$a = \exp(\bar\partial_e^{-1} A)$$. Then   ‖Δ(χa)‖Xζ−1/2≲s(1+‖∇∂¯e−1A‖L2(B)+e‖∂¯e−1A‖∞+‖A‖L∞Le2+‖⟨∇1⟩−1/2+s⟨∇2⟩−1/2+s∇A‖2+‖⟨∇1,2⟩−1+s∇A‖2)4. □ Proof. As in the previous lemma, we redefine $$P_1$$ as $$P_{\leq 1}$$ and so on. At high modulation we use the high-modulation estimate (13).   ‖QhΔ(χa)‖X−1/2≲‖χa‖H1≲‖a‖H1(B). It remains to consider the low-modulation part of $$\chi a$$. By the $$L^2$$ estimate (12),   ‖QlΔ(χa)‖X−1/22≲∑1≤μ≤τ/8∑λ≲τ(μτ)−1‖QμPλeΔ(χa)‖22. Now we observe that at low modulation, the symbol bounds (10) give   |ξ|2=2iζ⋅ξ−pζ(ξ)≲τ|ξ⋅e|+τd(ξ,Σ). Thus, when $$\lambda \leq \mu$$, the symbol of $$Q_\mu P_\lambda^e \nabla$$ is bounded by $$(\mu\tau)^{1/2}$$. It follows that   ∑1≤μ≤τ/8∑λ≤μ(μτ)−1‖QμPλeΔ(χa)‖22≲∑1≤μ≤τ/8∑λ≤μ‖QμPλe∇(χa)‖22≲‖∇(χa)‖22≲‖A‖H1(B)2. It remains to control the terms where $$\lambda > \mu$$. In this case the symbol of $$Q_\mu P_\lambda^e\nabla$$ is bounded by $$(\lambda\tau)^{1/2}$$, so we have   ∑1≤μ≤min{λ,τ/8}(μτ)−1/2‖QμPλeΔ(χa)‖2≲∑1≤μ≤min{λ,τ/8}μ−1/2λ−1/2‖QμPλe∇∂¯e(χa)‖2. (40) Since the commutator $$[\nabla \bar\partial_e,\chi]$$ satisfies the bound   ‖[∇∂¯e,χ]a‖2≲‖a‖H1(B), we may replace $$\nabla \bar\partial_e(\chi a)$$ with $$\chi \nabla\bar\partial_e a$$ on the right-hand side of (40). Now we use the definition of $$a$$ to write   ∇∂¯ea=∇(Aa)=∇Aa+A∇a. For $$A \nabla a$$, we apply Nikol’skii’s inquality (11) in the $$e_1$$ and $$e_2$$ directions and use the identity $$\nabla a = \nabla \bar\partial_e^{-1} A \cdot a$$.   μ−1/2λ−1/2‖QμPλe(χA∇a)‖2≲(μλ)−s/2‖QμPλe(χ⋅A∇a)‖L2Le1/(1−s/2)≲(μλ)−s/2‖A‖L∞Le2‖χ∇a‖L2Le2/(1−s)≲(μλ)−s/2‖A‖L∞Le2‖a‖∞‖χ∇∂¯e−1A‖L2Le2/(1−s). Since $$s>0$$, we can sum the right-hand side over $$\mu$$ and $$\lambda$$ as long as the last factor is bounded. To check this, we use the localization estimate (34) and Sobolev embedding.   ‖χ∇∂¯e−1A‖L2Le2/(1−s)≲‖∇P≤1eA‖L2Le2/(1−s)+‖∇∂¯e−1P>1eA‖L2Le2/(1−s)≲‖∇P≤1eA‖2+‖∇⟨∇1,2⟩s∂¯e−1P>1eA‖2≲‖⟨∇1,2⟩s−1∇A‖2. For $$(\chi a)\nabla A $$, we decompose using the Littlewood–Paley dichotomy, as we did with (37):   Pλe((χa)∇A)=∑κ≲λPλe(Pκe(χa)⋅P≲λe∇A)+Pλe(∑η≫λPηe(χa)⋅P∼ηe∇A). For the low-low terms, we have two cases. When $$\kappa \,{\leq}\, \mu$$, we use the identity $$Q_\mu (P_{\leq \mu}^e f\cdot g) = Q_\mu (P_{\leq \mu}^e f \cdot P^{e_1}_{\lesssim \mu} g)$$. Thus we have   μ−1/2λ−1/2‖QμPλe(P≤μe(χa)⋅P≲λeP≲μe1∇A)‖2≲‖a‖∞(μλ)−1/2‖P≲λe2P≲μe1∇A‖2. Summing over $$\mu$$ and $$\lambda$$, we obtain   ∑1≤μ,λ≤τ/8⋯≲‖a‖∞‖⟨∇1⟩−1/2+s⟨∇2⟩−1/2+s∇A‖2. When $$\kappa > \mu$$, we have instead $$Q_\mu(P_\kappa^e f \cdot g) = Q_\mu(P_\kappa^e f \cdot P^{e_1}_{\lesssim \kappa} g)$$. Then   μ−1/2λ−1/2‖Qμ(Pκe(χa)P≲λeP≲κe1∇A)‖2≲λ−1/2‖Pκe(χa)P≲λeP≲κe1∇A)‖Le3,e22Le11≲λ−1/2‖Pκe(χa)‖Le3∞Le2∞Le12‖P≲λe2P≲κe1∇A‖L2≲λ−1/2κ−1/2‖Pκe∂¯e(χa)‖L∞Le2‖P≲λe2P≲κe1∇A‖L2. Summing over $$\kappa$$, $$\mu$$, and $$\lambda$$, we obtain   ∑1≤μ<κ≤λ≤τ/8⋯≲‖a‖∞(1+‖A‖L∞Le2)‖⟨∇1⟩−1/2+s⟨∇2⟩−1/2+s∇A‖L2. For the high–high terms, we use Nikol’skii’s inequality and then transfer the $$\bar\partial^{-1}_e$$ from $$a$$ to $$A$$:   μ−1/2λ−1/2‖QμPλe(Pηe(χa)⋅P∼ηe∇A)‖2≲‖Pηe(χa)⋅P∼ηe∇A‖L2Le1≲‖Pηe(χa)‖L∞Le2‖P∼ηe∇A‖2≲η−1‖Pηe∂¯(χa)‖L∞Le2‖P∼ηe∇A‖2≲η−s‖a‖∞(1+‖A‖L∞Le2)‖⟨∇1,2⟩−1+s∇A‖L2. The sum of the right hand side over $$\eta \geq \lambda \geq \mu\geq 1$$ is bounded, and the proof is complete. ■ 9 Solvability of $$L_{A,q,\zeta}$$ Now we show that on average, the terms in $$L_{A,q,\zeta} + \Delta_\zeta$$ are all perturbative. Here we note an important difference between the estimate for $$q$$ and the estimate for $$A$$: as the parameter $$K$$ gets large, the right-hand side of the estimate (43) for $$q$$ goes to zero. However, this does not hold for the estimate (41) for $$A$$, and for this reason we can only handle the case where $$A$$ is small. Lemma 9.1. Let $$e_1$$ be a fixed unit vector in $$\mathbb{R}^3$$, and let $$\eta$$ be a vector in $$\mathbb{R}^3$$ such that $$\lvert {\eta} \rvert\leq 1$$. Define the operator norm $$\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert{\cdot}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert_{\tau,U }$$ by $$\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert{T}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert = \lVert{\cdot}\rVert_{X^{1/2}_{\zeta(\tau,U)} \to X_{\zeta(\tau,U)}^{-1/2}}$$, where $$\zeta(\tau,U) = \tau U(e_1 + i \eta)$$. Suppose $$A \in L^3(\mathbb{R}^3)$$. For every dyadic integer $$\lambda$$ such that $$1\leq \lambda \leq 100\tau$$ we have   E3[τ|||Aλ|||τ,U+|||Aλ⋅∇|||τ,U]≲min{⟨log+⁡λ⟩,⟨log+⁡τ/λ}⟩1/3‖Aλ‖L3. (41) On the other hand, we have the high-frequency estimate   |||∇A>100τ|||τ,U+|||A>100τ⋅∇|||τ,U≲‖A>100τ‖L3. (42) Finally, for $$q \in W^{-1,3}$$ we have   E~K3[‖|q|‖τ,U]≲(log⁡K)−1/3‖q‖W−1,3+‖q≥2K‖W−1,3. (43) □ Proof. It is convenient to use a bilinear characterization of the $$\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert{\cdot}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert$$ norm   |||A|||=sup{|⟨Au,v⟩|:‖u‖Xζ1/2=‖v‖Xζ1/2=1}. Decompose $$u$$ and $$v$$ into low and high modulation parts:   ⟨Au,v⟩=⟨AQhu,v⟩+⟨AQlu,Qhv⟩+⟨AQlu,Qlv⟩. The terms with $$Q_h$$ can be estimated by the high-modulation estimate (13) and the Strichartz estimate (18). For example,   τ|⟨AQhu,v⟩|≲τ‖A‖3‖Qhu‖2‖v‖6≲‖A‖3‖Qhu‖Xζ1/2‖v‖Xζ1/2. It remains to estimate the low modulation terms. Write   ⟨Aλ⋅Qlu,Qlv⟩=∑τ/8≥μ,ν≥1∫Aλ⋅Qμu⋅Qνv¯dx, (44) where we make the notational convention that $$Q_1 = Q_{\leq 1}$$. Note that when $$\lambda\geq 100\tau$$ these terms are all zero, so for the high-frequency estimate (42) there is nothing left to prove. Set   aμ=‖Qμu‖Xζ1/2bμ=‖Qμv‖Xζ1/2, and   Bμ,ν=τ|∫Aλ⋅Qμu⋅Qνv¯dx|. We claim that   ∑μ(aμ2+bμ2)≲1 implies that ∑μ,νBμ,ν≲Z(U), (45) where   E3[Z(U)]≲‖Aλ‖L3. By symmetry, it suffices to treat the terms where $$\mu \leq \nu$$. Since $$Q_\mu u \cdot \bar{Q_\nu v}$$ has Fourier support in the set $$\{\lvert {\xi\cdot( Ue_1)} \rvert \leq 2\nu\}$$, we have   ∫Aλ⋅Qμu⋅Qνv¯dx=∫P≤8νUe1Aλ⋅Qμu⋅Qνv¯dx. Suppose first that $$\lambda^2 > \mu\tau$$. In this case we use Hölder’s inequality and estimate $$Q_\mu u$$ by using the Strichartz estimate (18) and the $$L^2$$ estimate (12):   Bμ,ν≲τ‖P≤8νUe1Aλ‖3‖Qμu‖6‖Qνv‖2≲‖P≤8νUe1Aλ‖3τ(μ/τ)1/3(ντ)−1/2aμbν. By Young’s inequality, we have   ∑μ≤ν(μ/ν)1/12aμbν≲1, so that the sum over $$\lambda^2 > \mu\tau$$ is bounded by   ∑μ≤νλ2>μτBμ,ν≲supν≥μλ2>μτ(λ/ν)1/3(μ/ν)1/12(μτ/λ2)1/6‖P≤8νUe1Aλ‖3≲Z1(U)+Z2(U)+‖Aλ‖3, where $$Z_1(U)$$ and $$Z_2(U)$$ are given by   Z1(U)3=∑max{1,λ2/τ}≤ν≤λ(λ/ν)(λ2/ντ)1/12‖P≤8νUe1Aλ‖33Z2(U)3=∑ν≤λ2/τ(λ/ν)(ντ/λ2)1/2‖P≤8νUe1Aλ‖33. Now we check that $$Z_1$$ and $$Z_2$$ are bounded on average by applying the averaging estimate (24).   E3[Z1(U)]3≲∑max{1,λ2/τ}≤ν≤λ(λ2/ντ)1/12‖Aλ‖33≲‖Aλ‖33E3[Z2(U)]3≲∑ν≤λ2/τ(ντ/λ)1/2/λ‖Aλ‖33≲‖Aλ‖33. Next, we treat the case $$\lambda \leq (\mu\tau)^{1/2}$$. Note that   ⟨Au,v⟩=⟨Ae−iv⋅xu,e−iv⋅xv⟩, and that the $$X^{1/2}_\zeta$$ spaces have the modulation invariance   ‖e−iv⋅xu‖Xζ1/2∼‖u‖Xζ+iv1/2. Thus we may as well assume that $$\eta$$ is zero. We subdivide the set   El={ξ:d(ξ,Σζ)≤τ/8} into $$M=\left \lfloor {(\tau/\mu)^{1/2}} \right \rfloor$$ sectors $$S_k$$, defined for $$k=0,\dotsc,M-1$$ by   Sk=El∩{(ξ1,rcos⁡θ,rsin⁡θ):θ∈(2π/M)[k,k+1),r∈R+}. Here we recall that Let $$R_k$$ be Fourier projection on to $$S_k$$. The distance between two points in $$E_l$$ is bounded below by $$\tau \theta$$, where $$\theta$$ is the angular separation between the points. Thus for any two sectors $$S_j$$ and $$S_k$$, we have   d(Sj,Sk)≳(μτ)1/2dM(j,k), where $$d_M(j,k) = \min \{\lvert {j-k} \rvert,M-\lvert {j-k} \rvert\}$$. Since $$A_\lambda \cdot R_k f$$ has Fourier support in the set $$\{S_k + B(0, 2\lambda)\}$$, we find that the inner product $$\langle{A_\lambda \cdot R_k f, R_j g}\rangle$$ vanishes unless $$\lvert {d_M(j,k)} \rvert \leq C$$, so that   Bμ,ν≲τ∑dM(j,k)≤C|⟨P≤8νUe1Aλ⋅RkQμu,RjQνv⟩|. The Fourier support of $$R_k Q_\mu u$$ is contained in a rectangle of size proportional to $$\mu^{1/2}\tau^{1/2}\times \mu \times \mu$$. Thus, applying Hölder and Nikol’skii’s inequality (11) in each direction separately, we obtain   ‖P≤8ν1Aλ⋅RkQμu‖L2≲‖P≤8νUe1Aλ‖L∞L3L3‖RkQμu‖L2L6L6≲λ1/3μ2/3‖P≤8νUe1Aλ‖L3‖RkQμu‖L2. Now apply Cauchy–Schwarz to the sum over $$j$$ and $$k$$. This gives   Bμ,ν≲λ1/3μ2/3‖P≤8νUe1Aλ‖3(∑k‖RkQμu‖22)1/2(∑k‖RkQνv‖22)1/2≲μ2/3λ1/3‖P≤8νUe1Aλ‖L3‖Qμu‖L2‖Qνv‖L2≲(μ/ν)1/6(λ/ν)1/3aμbν‖P≤8νUe1Aλ‖L3. Thus   ∑λ2≤μτμ≤νBμ,ν≲supν≥max{λ2/τ,1}(λ/ν)1/3‖P≤8νUe1Aλ‖3≲Z3+‖Aλ‖3, where $$Z_3$$ is given by   Z3(U)3=∑max{λ2/τ,1}≤ν≤λ(λ/ν)1/3‖P≤8νUe1Aλ‖3. Applying the averaging estimate (24), we have   E3[Z3(U)]3≲∑max{λ2/τ,1}≤ν≤λ‖Aλ‖3≲min{⟨log+⁡λ⟩,⟨log+⁡τ/λ⟩}1/3‖Aλ‖3. This proves the claim (45), which shows that   E3[τ|||Aλ|||τ,U]≲min{⟨log+⁡λ⟩,⟨log+⁡τ/λ⟩}1/3‖Aλ‖3. The estimate for $$\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert{A\cdot \nabla}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert$$ now follows, since   ‖A⋅∇u‖Xζ−1/2≲|||A|||‖∇Qlu‖Xζ1/2+‖A⋅∇Qhu‖Xζ−1/2. Since $$Q_l$$ localizes to frequencies $$\lvert \xi \rvert \lesssim \tau$$, we can estimate the first term using the bound   ‖∇Qlu‖Xζ1/2≲τ‖u‖Xζ1/2. For second term, we apply the Strichartz estimate (20) and the high-modulation estimate (13)   ‖A⋅∇Qhu‖Xζ−1/2≲‖A⋅∇Qhu‖L6/5≲‖A‖3‖∇Qhu‖L2≲‖A‖3‖u‖Xζ1/2. Thus we have   E3[|||Aλ⋅∇|||]≲E3[τ|||Aλ|||]+‖Aλ‖3≲⟨log+⁡τ/λ⟩1/3‖Aλ‖3. Finally we derive the estimate (43) for $$\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert{q}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert$$. By Nikol’skii’s inequality, we can control $$\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert{q_{\leq 1}}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert$$ by   |||q<1|||≲τ−1‖q≤1‖∞≲τ−1‖q‖W−1,3. On the other hand, the estimate (41) for $$\tau \left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert{A_\lambda}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert$$ applied to $$q_\lambda$$ gives   E3[|||q[1,100τ]|||]≲∑1≤λ≤100τ(λ/τ)⟨log+⁡τ/λ⟩1/3‖qλ‖W−1,3≲∑1≤λ≤100τ(λ/τ)1−ϵ‖qλ‖W−1,3. Thus the frequency convolution estimate (23) gives   E˜K3[‖|q[1,100τ]|‖]≲(logK)−1/3‖q‖W−1,3. Together with the high-frequency estimate (42), this gives the desired bound (43) for $$\left\vert\kern-0.25ex\left\vert\kern-0.25ex\left\vert{q}\right\vert\kern-0.25ex\right\vert\kern-0.25ex\right\vert$$. ■ 10 Proof of the main theorem We will need the following Poincaré lemma: Lemma 10.1. Suppose that $$A \in L^3(\mathbb{R}^3)$$ is compactly supported, and that   curlA=0. Then there exists $$\psi \in W^{1,3}(\mathbb{R}^3)$$ supported in $$\text{supp} A$$ such that   ∇ψ=A. □ Proof. If $$\psi$$ exists then   Δψ=div(∇ψ)=div(A). Thus we set   ψ(x)=c∫∑i∂iK(y)Ai(x−y)dy, where $$K(y) \sim \lvert{y}\rvert^{-1}$$ is the fundamental solution of the Laplacian. Since $$\lvert{\partial_i K}\rvert \lesssim \lvert{y}\rvert^{-2}$$, the kernel is locally integrable. By Young’s inequality and the fact that $$A$$ is compactly supported, we have $$\psi \in L^3_{{\mathrm{loc}}}$$. Furthermore, the function $$\psi$$ is smooth away from $$\text{supp} A$$ and decays to zero at infinity. The vector Laplacian is given by   Δ=grad∘div−curl∘curl. Since $$\text{curl} A$$ and $$\text{curl}\nabla \psi$$ are both zero, we have   Δ(A−∇ψ)=∇(divA−Δψ)=0. Since $$A - \nabla \psi$$ vanishes at infinity, this implies, by the maximum principle, that $$\nabla \psi = A$$. In particular, $$\nabla \psi = 0$$ away from $$\text{supp} A$$, and since $$\psi$$ decays at infinity we conclude that $$\psi = 0$$ away from $$\text{supp} A$$. ■ Proof. (of Theorem 1.1) Let $$B = B(0,1)$$. We construct solutions in $$H^1(B)$$ to the Schrödinger equation $$L_{A,q} u = 0$$ of the form   u=ex⋅ζ~(a+ψ). (46) Let $$\chi,\tilde \chi \in C^\infty_0(B)$$ be cutoff functions satisfying $$\chi = \tilde \chi = 1$$ in $$\tfrac{1}{2} B$$ and $$\chi \tilde \chi = \tilde \chi$$. We construct $$\psi$$ by solving the equation   LA,q,ζ~ψ=χ~(F+G), (47) where   LA,q,ζ~=−Δζ~−2iζ~⋅A+D⋅A+2A⋅D+q, (48)  F=−Δ(χa)+(D⋅A)a+2A⋅Da+A2a+qa, (49) and   G=−2ζ~⋅∇a−2iζ~⋅Aa. (50) In order to eliminate the terms of order $$\tau$$ in $$G$$, we let $$a = {\rm e}^{-i\phi_\zeta}$$, where   ϕζ=∂¯ζ−1(χζ⋅A≤100τ), (51) where $$\zeta \in \mathbb{C}^n$$ will be chosen such that $$\lvert{\zeta - \tilde \zeta}\rvert = O(1)$$. With this choice of $$\phi$$, the function $$G$$ satisfies   χ~G=χ~(−2iζ⋅A>100τa−(ζ~−ζ)⋅(2∇a+2iAa)). We choose the parameters $$\zeta_i,\tilde \zeta_i$$ as follows: Fix a radius $$r\in [1,2]$$ and an orthonormal frame $$\{e_1,e_2, e_3\}$$, and define   ζ1(τ,U)=τU(e1+ie2)ζ2(τ,U)=−ζ1(τ,U)ζ~1(τ,U):=τUe1+i(r2Ue3+τ2−r2Ue2)ζ~2(τ,U):=−τUe1+i(r2Ue3−τ2−r2Ue2). Note that $$\lvert{\zeta_i - \tilde \zeta_i}\rvert \lesssim 1$$. In particular, the spaces $$X^b_{\zeta_i}$$ and $$X^b_{\tilde \zeta_i}$$ have equivalent norms. Let $$\tilde A_1 = A_1$$ and $$\tilde A_2 = -A_2$$, define $$F_i,G_i,\phi_{i,\zeta}$$ as in (49), (50), (51) by replacing $$A$$ with $$\tilde A_i$$. Let   M=∑i(‖Ai‖Ws,3+‖qi‖W−1,3)Z0=∑i(∑k=0,1,2τ−k‖∇kϕi,ζi‖∞+‖∇ϕi,ζi‖L2(B)+‖Ai‖L∞Le2+‖⟨∇1⟩−1/2+s⟨∇2⟩−1/2+s∇Ai‖2+‖⟨∇1,2⟩−1+s∇Ai‖2)Z1=∑i,l,m(τ‖Ai‖Xζj1/2→Xζj−1/2+‖Ai∇‖Xζj1/2→Xζj−1/2+‖ql,m‖Xζj1/2→Xζj−1/2)Z2=∑i,j,l,m‖χaiqj,l‖Xζm−1/2. Here the $$q_{j,l}$$ are all the terms that are bounded in $$W^{-1,3}$$, namely   qj,1=Aj2qj,2=qjqj,3=τP>100τAjqj,4=∇Aj. By the localization estimate (14) and the fact that $$\lvert{\zeta - \tilde \zeta}\rvert \lesssim 1$$, we have   ‖LA~i,q,ζ~i+Δζ~i‖X˙ζ~i1/2→X˙ζ~i−1/2≲Z1. If $$Z_1$$ is sufficiently small, then by the contraction mapping principle there are $$u_i = e^{x\cdot \tilde \zeta_i} (e^{-i\phi_i} + \psi_i)$$ solving $$L_{\tilde A_i,q_i} u_i = 0$$ in $$B$$ such that   ‖ψi‖X˙ζ~i1/2≲‖χ~(Fi+Gi)‖Xζi−1/2. By Lemma 8.6, the Strichartz estimate (20) and Hölder’s inequality, we have   ‖Fi‖Xζi−1/2≲(1+Z0)4+Z2+‖Ai‖3‖∇ai‖L2(B)≲(1+Z0)4+Z2+MeZ0Z0. On the other hand we can simply estimate $$\tilde \chi G$$ by   ‖χ~Gi‖Xζi−1/2≲Z2+eZ0(Z0+M).  Thus we have   ‖ψi‖X˙ζ~i1/2≲g(M,Z0,Z2), where $$g$$ is continuous. Now we apply the integral identity (3) to the solutions $$u_1$$ and $$u_2$$ to obtain   0=∫[i(A1−A2)⋅(u1∇u2−u2∇u1)+(A12−A22+q1−q2)u1u2]dx=I+II+III+IV, where   I=i(ζ2−ζ1)∫χ(A1−A2)≤100τe−i(ϕ1+ϕ2)eik⋅xdxII=∫[(ζ−ζ~)χA≤100τ+χζ~A>100τ+A∇ϕ+χq]e−iϕeik⋅xdxIII=∫[ζ~Aaψ+A∇aψ+Aa∇ψ+χqaψ]eik⋅xdxIV=∫qψ1ψ2eik⋅xdx. Here $$k = rUe_3$$. The error terms $$II$$–$$IV$$ are in schematic form. For example, the notation $$\int \chi qa \psi {\rm e}^{ik\cdot x}$$ represents a linear combination of terms $$\int\chi q_{l,m} a_i \psi_j {\rm e}^{ik \cdot x}$$. The first expression $$I$$ contains the main term. We remove the exponential factor $${\rm e}^{-i(\phi_1+\phi_2)}$$ using Lemma 3.2. We use the Littlewood–Paley commutator estimate $$\lVert{[\chi,P_{\leq 100\tau}]}\rVert_{L^1\to L^1} \lesssim \tau^{-1}$$ to control some of the errors.   I=i(ζ2−ζ1)⋅(χ(A1−A2)≤100τ)∧(k)=i(ζ2−ζ1)⋅(P≤100τ(A1−A2)+[χ,P≤100τ](A1−A2))∧(k)=i(ζ2−ζ1)⋅(A1−A2)∧(k)+O(‖A1−A2‖1). Next we estimate the terms in $$II$$.   |II|≲‖aA‖L1+τ‖A>100τ‖H−1‖χa‖H1+‖A‖2‖χa∇ϕ‖2+‖q‖H−1‖χa‖H1≲eZ0Z0M. For the terms in $$III$$ we use the the duality between $$\dot X^{1/2}_\zeta$$ and $$\dot X^{-1/2}_\zeta$$, the localization estimate (15), the trivial estimate $$\lVert{u}\rVert_{X^{-1/2}_\zeta} \lesssim \tau^{-1/2} \lVert{u}\rVert_2$$, the Strichartz estimate (20), and the fact that multiplication by $$e^{ik\cdot x}$$ is bounded in $$X^{-1/2}_\zeta$$:   |III|≲(τ‖Aa‖Xζ−1/2+‖A∇a‖Xζ−1/2+‖∇Aχa‖Xζ−1/2+‖qχa‖Xζ−1/2)‖ψ‖X˙ζ1/2≲(τ1/2‖A‖2eZ0+‖A‖3‖∇a‖L2(B)+Z2)g(M,Z0,Z2)≲(τ1/2MeZ0+MeZ0Z0+Z2)g(M,Z0,Z2). For $$IV$$, we estimate by $$\lVert{q}\rVert_{X^{1/2}_\zeta\to X^{-1/2}_\zeta}\lVert{\psi_1}\rVert_{X^{1/2}_\zeta} \lVert{\psi_2}\rVert_{X^{1/2}_\zeta}$$. Thus   |IV|≲Z1g(M,Z0,Z2)2. Combining all these estimates, we obtain   |i(ζ2−ζ1)⋅(A1−A2)∧(k)|≤τ1/2f(M,Z0,Z1,Z2), (52) where $$f$$ is continuous. To conclude, we must select $$\zeta$$ such that all of the constants $$Z_i$$ are bounded uniformly in $$\zeta$$. Let $$\epsilon = \sum_i \lVert{A_i}\rVert_{W^{s,3}}$$, which we assume to be small. By Lemmas 7.1, 8.2, 8.3 and the corollary (35) of Falconer’s maximal function estimate, we have (using Hölder’s inequality to control $$\mathbb{E}$$ by $$\mathbb{E}^p$$), that   E[Z0∣SO(3)]≤Cϵ. Define the set $$V$$ by   V={U∈SO(3):Z0<2Cϵ}, By Chebyshev’s inequality, we have $$\mathbb P[V] \geq \frac{1}{2}$$. Similarly, Lemma 9.1 implies that   E~K[Z1∣V]≲∑1≤λ≲τ⟨log+λ⟩λ−s‖Aλ‖Ws,3+‖A>2K‖L3+(log⁡K)−1/3‖q‖W−1,3+‖q≥2K‖W−1,3. Since the $$q$$ terms decay to zero as $$K \to \infty$$, we have   lim supK→∞E~K[Z1∣V]≤Cϵ. Lemma 9.1 implies that   E~K[Z2∣[2K,2K2]×V]≤h(Z0,M), where $$h$$ is continuous. Thus for sufficiently large $$K$$, it follows that the inequality   (2Cϵ)−1Z1+(2h(Z0,M))−1Z2≤2 holds on a set $${{\tilde{V}}_{K}}\subset [{{2}^{K}},{{2}^{{{K}^{2}}}}]\times V$$ with $${{\widetilde{\mathbb{P}}}_{K}}[{{\tilde{V}}_{K}}]\ge \frac{1}{4}$$. By choosing $$\epsilon$$ small, we can ensure that $$Z_1$$ is sufficiently small on $$\tilde V_K$$ that we can use the contraction mapping principle to construct $$\psi_i$$ as above. Furthermore, for $$(\tau,U) \in \tilde V_K$$, the quantities $$Z_i$$ are all bounded independently of $$\tau, U, K$$. Let   J(r,U)=U(e1+ie2)⋅(A1−A2)∧(rUe3). By (52), we have $$\lvert{J(r,U)}\rvert \lesssim \tau^{-1/2}$$ for all $$(\tau,U)$$ in the set $$\tilde V_K$$. Integrating this inequality over all $$(\tau, U)$$ in $$\tilde V_K$$ and $$r$$ in $$[1,2]$$, we have   ∫SO(3)∫12∫[2K,2K2]1V~K|J(r,U)|(log⁡K)−1(τlog⁡τ)−1dτdrdσ(U)=O(e−K/2) Let   ηK=∫[2K,2K2]1V~K(log⁡K)−1(τlog⁡τ)−1dτ, and note that   ∫SO(3)∫12ηV~Kdrdσ(U) ~P~K[V~K]≥14. By the Banach–Alaoglu theorem, there is a sequence $$K_i\to \infty$$ and a function $$\eta \in L^\infty(SO(3) \times [1,2])$$ such that $$\eta_{K_i} \rightharpoonup \eta$$. Since $$\int \eta = \lim \int \eta_{K_i} \geq \tfrac{1}{4}$$, it is clear that $$\eta \neq 0$$. On the other hand,   ∫η(r,U)|J(r,U)|drdσ(U)=limi→∞∫ηKi(r,U)|J(r,U)|drdσ(U)=0. It follows that $$J(r,U)$$ vanishes on a set of positive measure. But $$A_1 - A_2$$ is a compactly supported function, which implies that $$J(r,U)$$ is analytic in $$r$$ and $$U$$. Thus we can conclude that $$J(r,U) = 0$$ in $$\mathbb{R}_+\times SO(3)$$. By replacing $$SO(3)$$ by its complement throughout the argument, we find that $$J(r,U) = 0$$ in $$\mathbb{R}_+ \times (O(3)\setminus SO(3))$$ as well. Let $$H = A_1 - A_2$$. Since $$J(r,U)$$ vanishes uniformly, we must have $$v \cdot \hat H(k)=0$$ whenever $$v \cdot k = 0$$. In particular, $$0 = (w \times k) \cdot \hat H(k) = (\text{curl} H)^\wedge(k) \cdot w$$ for any $$w,k\in \mathbb{R}^n$$, so $$\text{curl} H = 0$$. By Lemma 10.1, there is a gauge transformation $$\psi$$ such that $$A_2 = A_1 + \nabla \psi$$, which implies that $$\Lambda_{A_1,q_1} = \Lambda_{A_2,q_2} = \Lambda_{A_1,q_2}$$. We can repeat the whole argument to obtain   0=∫((q1−q2)eik⋅x+χqaψeik⋅x+χqψ1ψ2eik⋅x)dx. Since $$\tilde E_K[\lVert{\chi a q}\rVert_{X^{-1/2}_\zeta} + \lVert{q}\rVert_{X^{1/2}_\zeta \to X^{-1/2}_\zeta}\mid \tilde V_K] \to 0$$ as $$K \to \infty$$ by Lemma 9.1 and Lemma 9.1, we can repeat the arguments above to show that that $$(q_1 -q_2)^\wedge(k) = 0$$ for all $$k$$. 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