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Comp. Appl. Math. https://doi.org/10.1007/s40314-018-0618-2 An elliptic regularity theorem for fractional partial differential operators Arran Fernandez Received: 17 February 2018 / Revised: 2 April 2018 / Accepted: 5 April 2018 © The Author(s) 2018 Abstract We present and prove a version of the elliptic regularity theorem for partial differ- ential equations involving fractional Riemann–Liouville derivatives. In this case, regularity is deﬁned in terms of Sobolev spaces H ( X ): if the forcing of a linear elliptic fractional PDE is in one Sobolev space, then the solution is in the Sobolev space of increased order corre- sponding to the order of the derivatives. We also mention a few applications and potential extensions of this result. Keywords Fractional derivatives · Elliptic partial differential equations · Regularity theorems · Sobolev spaces Mathematics Subject Classiﬁcation 26A33 · 35R11 · 35B65 · 46E35 1 Introduction In fractional calculus, the orders of differentiation and integration are extended beyond the integer domain to the real line and even the complex plane. This ﬁeld of study has a long history, having been considered by Leibniz, Riemann, and Hardy among others (Miller and Ross 1993). It also has a wide variety of applications, including in bioengineering (Glöckle and Nonnenmacher 1995; Magin 2006), chaos theory (Zaslavsky 2012), drug transport (Dok- Communicated by Vasily E.Tarasov. The original version of this article was revised due to a retrospective Open Access order. This work was supported by a research student grant from the Engineering and Physical Sciences Research Council, UK. B Arran Fernandez af454@cam.ac.uk Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK 123 A. Fernandez oumetzidis and Macheras 2009; Petráš and Magin 2011; Sopasakis et al. 2017), epidemiology (Carvalho et al. 2018), geohydrology (Atangana 2017), random walks (Zaburdaev et al. 2015), thermodynamics (Vazquez et al. 2011), and viscoelasticity (Koeller 1984). Many of these cited papers are from the last few years, indicating the importance and relevancy of fractional calculus in modern science. Fractional derivatives and integrals can be deﬁned in several different ways, not all of which agree with each other, and thus it must always be clear which deﬁnition is being used. In fact, new models of fractional calculus are being developed all the time, including just in the last few years. In this paper, however, we shall always use the classical Riemann–Liouville formula (Deﬁnitions 1 and 2) unless otherwise stated. Deﬁnition 1 (Riemann–Liouville fractional integral)Let x and ν be complex variables, and c be a constant in the extended complex plane (usually taken to be either 0 or negative real inﬁnity). For Re(ν) < 0, the νth derivative, or (−ν)th integral, of a function f is ν −ν−1 D f (x ) := (x − y) f ( y) d y, c+ Γ(−ν) provided that this expression is well deﬁned. (If c =−∞, the operator is denoted by simply ν ν D instead of D .) + c+ Since x, ν,and c are deﬁned to be in the complex plane, we must consider the issue −ν−1 of which branch to use when deﬁning the function (x − y) and which contour from c to x to use for the integration. Clearly, arg(x − y) can be ﬁxed to be always equal to arg(x − c), i.e., by taking the contour of integration to be the straight line segment [c, x ]. And the choice of range for arg(x − c) usually depends on context: the essential properties of Riemann–Liouville integrals remain unchanged whether arg(x − c) is assumedtobein [0, 2π), (−π, π ], or any other range. Here, we shall follow (Samko et al. 2002, §22) in using arg(x − c) ∈[0, 2π), because we shall usually be assuming c =−∞ and x ∈ R,inwhich case arg(x − c) = 0 is the obvious choice to make. Deﬁnition 2 (Riemann–Liouville fractional derivative)Let x,ν, c be as in Deﬁnition 1 except with Re(ν) ≥ 0. The νth derivative of a function f is ν d ν−n D f (x ) := D f (x ) , c+ c+ dx where n := Re(ν)+ 1, provided that this expression is well deﬁned. (Again, if c =−∞, ν ν the operator is denoted by simply D instead of D .) + c+ For functions f such that D f (x ) is analytic in ν, Deﬁnition 2 is the analytic continuation c+ in ν of Deﬁnition 1. This provides some motivation for why this formula should be used. When the order of differentiation and integration becomes continuous, the term differin- tegration is often used to cover both. When the order of differintegration lies in the complex plane, its real part is what deﬁnes the difference between differentiation and integration. In the case where f is holomorphic, the following deﬁnition (Deﬁnition 3) can be more useful for applications in complex analysis. It is equivalent to the Riemann–Liouville deﬁ- nition wherever both are deﬁned, as proved in (Oldham and Spanier 1974, Chapter 3). Deﬁnition 3 (Cauchy fractional differintegral)Let x and ν be complex variables, and c be a constant in the extended complex plane. For ν ∈ C\Z ,the νth derivative of a function f analytic in a neighbourhood of the line segment [c, x ] is Γ(ν + 1) ν −ν−1 D f (x ) := ( y − x ) f ( y) d y, c+ 2π i 123 An elliptic regularity theorem for fractional partial... provided that this expression is well deﬁned, where H is a ﬁnite Hankel-type contour with both ends at c and circling once counterclockwise around x. Note that, Deﬁnition 1 is the natural generalisation of the Cauchy formula for repeated real integrals (see Miller and Ross 1993, Chapter II), while Deﬁnition 3 is similarly the natural generalisation of Cauchy’s integral formula from complex analysis. Since the Riemann–Liouville fractional derivative is deﬁned using ordinary derivatives of fractional integrals, one might wonder what would happen if the order of these operations were reversed. Using fractional integrals of ordinary derivatives instead, we obtain a different deﬁnition of fractional differentiation, this one due to Caputo. Deﬁnition 4 (Caputo fractional derivative)Let x,ν, c be as in Deﬁnition 1 except with Re(ν) ≥ 0. The νth derivative of a function f is d f ν ν−n D f (x ) := D , c+ c+ dx where n := Re(ν)+ 1, provided that this expression is well deﬁned. Fractional integrals in the Caputo context are exactly Riemann–Liouville integrals, so a new deﬁnition is not needed for them. Lemma 2 below shows that the Riemann–Liouville and Caputo fractional derivatives (Deﬁnitions 2 and 4) are not equivalent in general. The constant c used in the above deﬁnitions can be thought of as a constant of integration. However, in the fractional context, it appears in the formulae for derivatives as well as those for integrals. It is almost always assumed to be either 0 or −∞. Some standard properties of integer-order differintegrals extend to the fractional case: for instance, D is still a linear operator for any ν and c. But other standard theorems c+ of calculus no longer hold in the fractional case, or hold in a more complicated way. For instance, the fractional derivative of a fractional derivative is not always a fractional derivative; composition of fractional differintegrals is governed by the equations in Lemmas 1 and 2. Lemma 1 (Composition of fractional integrals) For any x,μ,ν ∈ C with Re(μ) < 0 and any μ μ+ν function f continuous in a neighbourhood of c, the identity D D f (x ) = D f (x ) c+ c+ c+ holds provided these differintegrals exist. Proof This is a simple exercise in manipulation of double integrals, and may be found in (Podlubny 1999, Chapter 2.3.2). μ n+μ n n Lemma 2 If n ∈ N and f is a C function such that one of D D f (x ) , D f (x ), c+ c+ c+ D D f (x ) exists, then all three exist and c+ c+ −μ−k (x − c) μ n+μ μ n n (n−k) D D f (x ) = D f (x ) = D D f (x ) + f (c). c+ c+ c+ c+ c+ Γ(−μ − k + 1) k=1 Proof The ﬁrst identity follows directly from Deﬁnition 2 for Riemann–Liouville fractional derivatives. For the second, use induction on n, starting with the Re(μ) < 0 case and using integration by parts, then proving the Re(μ) ≥ 0 case by performing ordinary differentiation on the previous case. A more detailed proof can be found in (Miller and Ross 1993, Chapter III). Note that, when c is inﬁnite and f has sufﬁcient decay conditions, the series term disap- pears. In this case, the Riemann–Liouville and Caputo fractional derivatives (Deﬁnitions 2 and 4) are equivalent. This fact will be used in Lemma 9 below. 123 A. Fernandez Another deﬁnition of fractional calculus involves generalising the relationship given by the Fourier transform between differentiation and multiplication by power functions. In fact, Lemma 3 shows that this model, commonly used in applications involving partial differential equations, is equivalent to the Riemann–Liouville model with c =−∞. Similarly, Lemma 4 shows that the corresponding deﬁnition with Laplace transforms instead of Fourier is equivalent to the Riemann–Liouville model with c = 0. Lemma 3 (Fourier transforms of fractional differintegrals) If f (x ) is a function with well- deﬁned Fourier transform f (λ) and ν ∈ C is such that D f (x ) is well deﬁned, then the ν ν Fourier transform of D f (x ) is (−i λ) f (λ). Proof If Re(ν) < 0, then Deﬁnition 1 can be rewritten as a convolution: D f = f ∗ Φ −ν−1 where Φ(x ) = when x > 0, Φ(x ) = 0 otherwise. Convolutions transform to products Γ(−ν) under the Fourier transform, so the result follows. If Re(ν) ≥ 0, the result follows from the fractional integral case (proved above) and the ν ∈ N case (which is standard). Lemma 4 (Laplace transforms of fractional integrals) If f (x ) is a function with well-deﬁned Laplace transform f (λ), and ν ∈ C with Re(ν) < 0 is such that D f (x ) is well deﬁned, 0+ ν ν then the Laplace transform of D f (x ) is (−i λ) f (λ). 0+ Proof As for Lemma 3. See also Miller and Ross (1993), Chapter III. The corresponding result for Laplace transforms of fractional derivatives is more compli- cated, because of the initial value terms arising. It may be found in Miller and Ross (1993), Chapter IV. Finally, we demonstrate one way, due to Osler, in which the product rule—another basic result of classical calculus—can be extended to Riemann–Liouville fractional calculus. Lemma 5 (The fractional product rule) Let u and v be complex functions such that u(x ), v(x ), and u(x )v(x ) are all functions of the form x η(x ) with Re(λ) > −1 and η analytic on a domain R ⊂ C. Then, for any distinct x , c ∈ R and any ν ∈ C, we have ν ν−n n D u(x )v(x ) = D u(x ) D v(x ), c+ c+ c+ n=0 where all differintegrals are deﬁned using the Cauchy formula. Proof See Osler (1971). Partial differential equations (PDEs) of fractional order have also become an important area of study, with entire textbooks written about them and their applications Kilbas et al. (2006), Podlubny (1999). A huge variety of methods have been devised for solving them, including by extending known results of classical calculus: see for example Podlubny et al. (2009), Yang et al. (2015), Bin (2012), Baleanu and Fernandez (2017) among many others. Even ordinary differential equations, in a non-linear fractional scenario, still present difﬁcult problems, see e.g. Area et al. (2016). The non-locality of fractional derivatives lends them utility in many real-life problems, e.g., in control theory, dynamical systems, and elasticity theory (Baleanu and Fernandez 2018; Luchko et al. 2010; Tarasov and Aifantis 2015). The elliptic regularity theorem is an important result in the theory of partial differential equations. In its most general form, it says that for any PDE satisfying certain conditions, there 123 An elliptic regularity theorem for fractional partial... are regularity properties of the solution function which depend naturally on the regularity properties of the forcing function. This is useful in cases where the solution function cannot be constructed explicitly: more information about its essential properties is the next best thing to an analytic solution. Here, we shall focus on the version of the elliptic regularity theorem given in Theorem 1, in which the PDE must be linear and elliptic with constant coefﬁcients, and ‘regularity’ is deﬁned in terms of Sobolev spaces. Deﬁnition 5 For any real number s and any natural number n,the sth Sobolev space on R is deﬁned to be s n n 2 n H (R ) := u ∈ S (R ) :ˆ u ∈ L (R ), ||u|| < ∞ , loc where the Sobolev norm || · || is deﬁned by 1/2 2 2 ||u|| := |ˆ u(λ)| 1 +|λ| dλ . For a general domain X ⊂ R ,the sth Sobolev space on X is deﬁned to be s s n H ( X ) := u ∈ D ( X ) : uφ ∈ H (R ) for all φ ∈ D( X ) . loc Theorem 1 (Elliptic regularity theorem) Let P( D) be an elliptic partial differential operator given by a complex n-variable N th-order polynomial P applied to the differential operator n n D := −i where x is a variable in R . If X is a domain in R and u, f ∈ D ( X ) satisfy ∂ x P( D)u = f, then s s+ N f ∈ H ( X ) ⇒ u ∈ H ( X ). loc loc Proof See (Folland 1999, Chapter 9). Related, more general, results are already known from the theory of pseudodifferential operators; see e.g., (Abels 2012, Theorem 7.13) for an example of an elliptic regularity theorem in this setting. However, it is not necessary to introduce the full machinery of pseudodifferential operators—with associated stronger conditions on the forcing and solution functions—to obtain a useful analogue of Theorem 1 for fractional differential equations. The structure of this paper is as follows: In Sect. 2, the bootstrapping proof used in Folland (1999) to prove Theorem 1 is adapted, with some modiﬁcations and extra lemmas, to prove an elegant analogous result in the Riemann–Liouville fractional model. The place where most new work was needed was in the proof of Lemmas 8 and 9; the ﬁnal result is Theorem 2.In Sect. 3, we consider applications and potential extensions of our work here. 2 The main result Let x ∈ R be an n-dimensional variable, and let D denote the modiﬁed n-dimensional differential operator −iD where D is the vector operation of differentiation with respect + + to x deﬁned in Deﬁnitions 1 and 2. In other words, the differential operator D is deﬁned by α −iπα/2 α D f (x ) = e D f (x ). We use the constant of differintegration c =−∞ so that we can make use of Fourier transforms in the proof (by Lemma 3), and also so that the Riemann–Liouville and Caputo 123 A. Fernandez fractional derivatives are equal (by the discussion following Lemma 2), which is required at a certain stage in the proof. Let P be a ﬁnite linear combination of power functions, i.e., P(λ) = c λ , where α is a fractional multi-index in (R ) and the sum is ﬁnite. This deﬁnes a fractional differential operator P( D), where all powers of D are deﬁned using the Riemann–Liouville formula (Deﬁnition 2) with c =−∞. The fractional partial differential equation we shall be considering is of the form P( D)u = f. Deﬁnition 6 The order ν of the operator P( D) deﬁned above is the maximal |α| such that c
= 0. Note that, ν is not necessarily an integer, and that since P is a ﬁnite sum, there exists > 0 such that |α|≤ ν − for every α such that c
= 0and |α| <ν. Deﬁnition 7 The principal symbol of P( D) is deﬁned to be the function σ (λ) = α n c λ . The operator P( D) is said to be elliptic if σ (λ)
= 0 for all nonzero λ ∈ R . α P |α|=ν Lemma 6 If P( D) is a νth-order elliptic fractional partial differential operator as above, then there exist positive real constants C, R such that for any λ ∈ C with |λ| > R, the 2 ν/2 function P satisﬁes | P(λ)|≥ C (1 +|λ| ) . Proof First consider the non-fractional case, i.e., where P is a polynomial. Here, |σ | is a continuous positive function on the compact domain |λ|= 1, so it has a positive lower bound on this domain. In other words, |σ (λ)| 1when |λ|= 1, which implies |σ (λ)||λ| P P for all λ. By the triangle inequality, this implies | P(λ) − σ (λ)| | P(λ)| 1 − |λ| . (1) |λ| Since P(λ) − σ (λ) is a polynomial of order less than ν, the ratio term is 1when |λ| is ν 2 ν/2 sufﬁciently large. Thus, for large |λ| we have | P(λ)||λ| (1 +|λ| ) as required. The above proof relies on the continuity of the function σ (λ), which is not true in general since λ has a branch cut in the complex λ-plane when α is not an integer. But σ (λ) can be approximated arbitrarily closely by a sum of rational powers of λ, i.e., a polynomial of 1/m order around mν in λ for some large natural number m. Call this function σ ˜ (λ);the 1/m above proof shows that |˜ σ (λ)| 1when |λ |= 1, i.e., when |λ|= 1. Now by letting the exponents in σ ˜ tend to those in σ ,weﬁnd |σ (λ)| 1when |λ|= 1, as before. Again P P P this gives Eq. (1). | P(λ)−σ (λ)| Because of the ﬁnite bound mentioned in Deﬁnition 6, the ratio term is still |λ| 1 for sufﬁciently large λ, and the result follows. Lemma 7 (Existence of parametrices) If P(D) is an elliptic fractional partial differential operator as above, then it has a parametrix, i.e., E ∈ D (R ) such that P( D) E = δ + ω n n ∞ n for some ω ∈ E (R ), and the parametrix E is in S (R ) and also in C (R \{0}). Proof Fix a test function χ ∈ D(R ) which is identically 1 on the domain |λ|≤ R and identically 0 on the domain |λ| > R + 1, where R is as in Lemma 6.Let 1 − χ (λ) E (λ) := . P(λ) 123 An elliptic regularity theorem for fractional partial... This is well deﬁned because 1 − χ is zero at all zeros of P, and it is bounded by Lemma 6. By deﬁnition of P, we, therefore, have the leftmost of the following inclusions, leading to the rightmost: n n n n ˆ ˆ E ∈ E (R ) ⇒ E ∈ S (R ) ⇒ E ∈ S (R ) ⇒ E ∈ D (R ), where E is the inverse Fourier transform of E. Similarly, n n n n χ ∈ D(R ) ⇒ χ ∈ S(R ) ⇒ ω ∈ S(R ) ⇒ ω ∈ E (R ), where ω is the inverse Fourier transform of −χ. Finally, P(λ) E (λ) = 1 − χ (λ) ⇒ P( D) E = δ + ω, so E is a parametrix of P( D). On the domain |λ| > R + 1, we have |α|−|β|−ν α β α β −1 α β D (x E )(λ) = λ D E (λ) = λ D P(λ) λ α β for any multi-indices α, β. Thus, for all α, β with |β| sufﬁciently large, the function D (x E ) 1 n α β n is in L (R ), which means its inverse Fourier transform D (x E ) is in C (R ). Thus, E is ∞ n n in C (R \{0}). And the fact that E ∈ S (R ) was already established above. n t n α Lemma 8 If φ ∈ D(R ) and u ∈ H (R ) for some n ∈ N, t ∈ R,then [ D ,φ](u) ∈ t −|α|+1 n n H (R ) for any α ∈ C ,where [, ] denotes a commutator. + n Proof Note that, when α is an ordinary multi-index in (Z ) , this result is straightforwardly proved using the product rule: the operator [ D ,φ] is just an (|α|− 1)th-order differential operator. In the general case, however, we need to use inﬁnite series and some more com- plicated estimates. It may appear that Osler’s generalisation of the product rule (Lemma 5) is applicable, but analyticity is out of the question when we are dealing with test functions φ ∈ D(R ). s n The property of a function f being in a Sobolev space H (R ) depends only on the large-λ behaviour of the Fourier transform f (λ), so it will sufﬁce to prove that the Fourier transform α t −|α|+1 n of [ D ,φ](u) behaves like the Fourier transform of a function in H (R ) when |λ| has some ﬁxed lower bound. First, we rewrite the expression as follows: α α α α α ˆ ˆ [ D ,φ](u)(λ) = D (φu)(λ) − (φ D u)(λ) = λ φ(λ) ∗ˆ u(λ) − φ(λ) ∗ (λ u ˆ(λ)) α α ˆ ˆ = λ φ(μ)u ˆ(λ − μ) dμ − φ(μ)(λ − μ) u ˆ(λ − μ) dμ n n R R = I (λ) + I (λ), 1 2 where the two integral expressions I , I are deﬁned by 1 2 μ α I (λ) := λ φ(μ) 1 − 1 − u ˆ(λ − μ) dμ; |μ|≤ |λ| α α I (λ) := φ(μ) λ − (λ − μ) u ˆ(λ − μ) dμ. |μ|> |λ| We shall evaluate I and I separately and prove bounds to establish that each of them is the 1 2 t −|α|+1 n Fourier transform of a function in H (R ), which will sufﬁce to prove the lemma. 123 A. Fernandez First, α α μ m I (λ) = λ φ(μ) ( ) u ˆ(λ − μ) dμ m λ |μ|≤ |λ| m=1 α μ μ = λ φ(μ) + o u ˆ(λ − μ) dμ 1 λ λ |μ|≤ |λ| α−1 ∼ αλ μφ(μ)u ˆ(λ − μ) dμ |μ|≤ |λ| α−1 αλ φ (λ) ∗ˆ u(λ) α −1 = α D (φ u). n t n Since φ ∈ D(R ),wehave φ u ∈ H (R ). By Lemma 3, this means the above expression t −|α|+1 n is the Fourier transform of a function in H (R ), as required. Now consider I . By the Paley–Wiener–Schwartz theorem (see Hörmander 1963, Chapter − N ˆ ˆ 1), the function φ is entire and satisﬁes an inequality of the form |φ(λ)| (1 +|λ|) for N ∈ N, λ ∈ R , where the subscript means the multiplicative constant depends on N.So α α I = φ(μ) λ − (λ − μ) u ˆ(λ − μ) dμ |μ|> |λ| − N −|α| |α| |α| (1 +|μ|) |2μ| +|3μ| |ˆ u(λ − μ)| dμ |μ|> |λ| − N (1 +|μ|) |ˆ u(λ − μ)| dμ |μ|> |λ| − N (1 +|•|) ∗|uˆ| (λ). t n Since u is in H (R ) and N can be arbitrarily large, this ﬁnal expression must be the Fourier t +K n a b transform of a function in H (R ) for arbitrarily large K.And H ⊂ H for a > b,so t −|α|+1 n I is the Fourier transform of a function in H (R ), as required. Lemma 9 If f and g are functions, at least one of which is a Schwartz function, and ν ∈ C ν ν ν ν is such that D f and D g are well deﬁned, then D f ∗ g = f ∗ D g, where ∗ denotes + + + + convolution. Proof When Re(ν) < 0, writing D f = f ∗ Φ as in Lemma 3 and using the associativity of convolution gives ν ν D f ∗ g = ( f ∗ Φ) ∗ g = f ∗ (Φ ∗ g) = f ∗ (g ∗ Φ) = f ∗ D g. + + When Re(ν) ≥ 0and n is deﬁned as in Deﬁnition 3, assuming without loss of generality that g is a Schwartz function, using Deﬁnition 2 and the above result gives n n d g d g ν d ν−n ν−n ν−n D f ∗ g = D f ∗ g = D f ∗ = f ∗ D . n n n + + + + dx dx dx The ﬁnal expression on the right-hand side is a Caputo derivative and not a Riemann–Liouville derivative of g. However, since g is a Schwartz function, its Caputo and Riemann–Liouville derivatives are identical (by the discussion following Lemma 2), and the result follows. 123 An elliptic regularity theorem for fractional partial... Theorem 2 (Fractional elliptic regularity theorem) If P( D) is a νth-order elliptic fractional partial differential operator as above and X is a domain in R and u, f ∈ D ( X ) satisfy P( D)u = f, then s s+ν f ∈ H ( X ) ⇒ u ∈ H ( X ). loc loc n n Proof First assume X = R and u is compactly supported (i.e., in E (R )). By Lemma 7, P( D) has a parametrix E and (using Lemma 9) u = δ ∗ u = ( P( D) E ) ∗ u − ω ∗ u = E ∗ ( P( D)u) − ω ∗ u = E ∗ f − ω ∗ u. Since u has compact support, ω ∗ u is a Schwartz function, so it will be enough to prove s+ν n E ∗ f ∈ H (R ).If |λ| > R + 1, then by Lemma 6 and the deﬁnition of E, f (λ) 2 −ν/2 E ∗ f (λ) = (1 +|λ| ) f (λ). P(λ) s n s+ν n And f ∈ H (R ),so E ∗ f ∈ H (R ) as required. To prove the general case, we shall use a bootstrapping argument. First of all, let us note that it makes sense to deﬁne fractional derivatives of functions in D ( X ) even when X does not extend to negative inﬁnity: the integrals from −∞ to x required by Deﬁnition 1 are simply taken to be zero outside of X. In other words, the arbitrary test function φ ∈ D( X ) is extendedtoafunctiononall of R which is supported on X. s+ν n Fix φ ∈ D( X ); it will sufﬁce to prove that φu ∈ H (R ).Let ψ ,ψ ,...,ψ (where 0 1 m the value of m will be decided later) be test functions in D(R ) with supports as shown in Fig. 1, i.e., such that: supp(φ) ⊂ supp(ψ ), ψ = 1 on supp(φ); m m (2) supp(ψ ) ⊂ supp(ψ ), ψ = 1 on supp(ψ ) ∀i. i i −1 i −1 i Fig. 1 The domains involved in the bootstrapping proof of Theorem 2 123 A. Fernandez n t n Now, ψ u is in E (R ), and therefore, in H (R ) for some t ∈ R.So P( D)(ψ u) = ψ P( D)u +[ P( D), ψ ]u (where [, ] is a commutator) 1 1 1 = ψ f +[ P( D), ψ ](ψ u)(by (2)) 1 1 0 s n t −ν+1) n = (element of H (R )) + (element of H (R )) (by Lemma 8) min(s,t −ν+1) n a b ∈ H (R)(since a > b ⇒ H ⊂ H ). A n Now, the ﬁrst part of the proof shows that ψ u ∈ H (R ) where A := min(s, t − ν + 1) + 1 1 ν = min(s + ν, t + 1). By exactly the same argument, P( D)(ψ u) = ψ f +[ P( D), ψ ](ψ u) and ψ u ∈ 2 2 2 1 2 A n H (R ) where A := min(s + ν, A + 1) = min(s + ν, t + 2). 2 1 min(s+ν,t +m) n Continuing in this manner eventually yields ψ u ∈ H (R ).Now,set the s+ν n natural number m to be s + ν − t+ 1, so that ψ u ∈ H (R ), which means φu ∈ s+ν n H (R ) as required, by (2). 3 Conclusions The elliptic regularity theorem is an important result in the theory of PDEs, and its fractional counterpart should have no less signiﬁcance in the theory of fractional PDEs. Elliptic frac- tional PDEs have already been studied in papers such as Bisci and Repovš (2015), Chen et al. (2015), Caffarelli and Stinga (2016), Dipierro et al. (2017), which present various methods for analysing the solutions of certain classes of elliptic fractional PDE. The current work ﬁts in with such results by providing a quick way of establishing important regularity properties of linear elliptic fractional PDEs. As example applications of our work, we consider the following two simple corollaries. Corollary 1 Let P( D) be a fractional linear partial differential operator of the form described above. If it is elliptic, then it is also hypoelliptic. Proof Recall the deﬁnition of hypoellipticity: a partial differential operator ∂ is hypoelliptic if whenever ∂ u is a smooth function, so also is u on the same domain. If P( D) is elliptic, then using all notation as in Theorem 2,wemusthave f ∈ C ( X ) ⇒ u ∈ C ( X ), i.e., P( D) is also hypoelliptic. Corollary 2 Consider the operator Δ := ∂ with 0 <α < 1, a fractional generali- i =1 sation of the Laplacian, and a function u ∈ D ( X ) where X is a domain in R . If u is a solution to the fractional Laplace-type equation Δ u = 0, then it must necessarily be smooth. More generally, if u is the solution to a fractional Poisson-type equation Δ v = f s+α with forcing f ∈ H ( X ),thenu ∈ H ( X ). loc loc α α Proof The fractional operator ∂ is elliptic when 0 <α < 1, since then λ is in the i =1 right half complex plane for all λ ∈ R. So Theorem 2 applies and the results follow. The result proved herein is only one of many possible versions of a fractional elliptic regularity theorem. For classical PDEs, there are far more elliptic regularity theorems than Theorem 1,which covers only linear partial differential operators whose coefﬁcients are constants in C. Other versions concern linear partial differential operators with non-constant coefﬁcients, perhaps k p satisfying some C or L condition; the Sobolev conditions can also sometimes be replaced 123 An elliptic regularity theorem for fractional partial... by L conditions on the functions f and u. See, e.g., (Folland 1995, Chapter 6C) and (Evans 1998, Chapter 6.3). These other variants of the elliptic regularity theorem may well be extendable to fractional PDEs just as Theorem 1 was. Furthermore, there are more models of fractional calculus than just the Riemann–Liouville formula. Some of them cooperate with the Fourier transform almost as well as Riemann– Liouville differintegrals do, which was a necessary factor in our proofs here. Thus, with a little more work we may be able to prove results analogous to Theorem 2 for fractional PDEs deﬁned using other fractional models, which have different real-world applications from the Riemann–Liouville one. 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