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International Mathematics Research Notices
, Volume 2018 (4) – Feb 1, 2018

49 pages

/lp/ou_press/unique-determination-of-a-magnetic-schr-dinger-operator-with-unbounded-S7eT41fOiH

- Publisher
- Oxford University Press
- Copyright
- © The Author(s) 2016. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals.permission@oup.com.
- ISSN
- 1073-7928
- eISSN
- 1687-0247
- D.O.I.
- 10.1093/imrn/rnw263
- Publisher site
- See Article on Publisher Site

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)=(τlogτ)−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(τlogτ)−1dτdσ(U)∼logK. (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 ~(logK)−1∑2K≤τ∗<2K2(logτ∗)−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(logτ)1/p‖Pλf‖p]≲(logK)−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(logλ)‖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(logK)−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(logK)−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‖]≲(logK)−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(logK)−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(logK)−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(logτ∗)1/2∑λ≲τ∗(λ/τ∗)2ϵ‖qλ‖W−1,2E~K2[‖IV‖]≲M(logK)−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(logK)−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]≲(logK)−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,rcosθ,rsinθ):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+(logK)−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)|(logK)−1(τlogτ)−1dτdrdσ(U)=O(e−K/2) Let ηK=∫[2K,2K2]1V~K(logK)−1(τlogτ)−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 $$k$$. It follows that $$q_1=q_2$$. ■ Funding This work was supported by the National Science Foundation [DGE 1106400, DMS 1440140 to B.H.]. The author was an NSF Graduate Research Fellow from Fall 2012-Spring 2015 and was in residence as a Postdoctoral Fellow at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2015 semester. Acknowledgments The author would like to thank his PhD advisor Daniel Tataru for suggesting the problem, for his patient guidance and encouragement, and for many helpful suggestions. 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International Mathematics Research Notices – Oxford University Press

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