Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {L}^p$$\end{document}-extrapolation of non-local operators: Maximal regularity of elliptic integrodifferential operators with measurable coefficients

The aim of this article is to deepen the understanding of the derivation of Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {L}^p$$\end{document}-estimates of non-local operators. We review the Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {L}^p$$\end{document}-extrapolation theorem of Shen (2005) which builds on a real variable argument of Caffarelli and Peral (1998) and adapt this theorem to account for non-local weak reverse Hölder estimates. These non-local weak reverse Hölder estimates appear, for example, in the investigation of non-local elliptic integrodifferential operators. This originates from the fact that here only a non-local Caccioppoli inequality is valid, see Kuusi, Mingione, and Sire (2015). As an application, we prove resolvent estimates and maximal regularity properties in Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm {L}^p$$\end{document}-spaces of non-local elliptic integrodifferential operators.


Introduction
Non-local phenomena play a major role in many different areas in the study of partial differential equations [1,7,9,10,12,25,31,33]. In particular, in mathematical fluid mechanics non-local phenomena arise naturally due to the presence of the pressure and the imposed solenoidality of the velocity field. One prominent example of a non-local operator in the study of mathematical fluid mechanics is the Stokes operator that is given-if the underlying domain is regular enough-by the Helmholtz projection P applied to the Laplacian − .
Another prominent example is the so-called dissipative surface quasi-geostrophic equation. It is often studied as a model equation to understand nonlinear mechanisms [11,25] connected to the Navier-Stokes equations. The dissipative surface quasigeostrophic equation involves the fractional Laplacian which is given in the whole space for α ∈ (0, 1) and u ∈ C ∞ c (R d ) by Here, C d,α > 0 denotes a suitable normalization constant. For both types of operators, certain mapping properties cease to exist in irregular situations. Indeed, it is well known that the Stokes operator does not generate a strongly continuous semigroup on L p σ if p > 2 is large enough and if it is considered in a bounded Lipschitz domain [14]. Also, the Riesz transform ∇(− D ) −1/2 is not bounded in general on L p for p > 3 if it is considered in an arbitrary bounded Lipschitz domain [19]. Here, − D denotes the Dirichlet Laplacian on the underlying domain. Such phenomena do not only occur in the presence of an irregular boundary, but also in smooth geometric constellations in the presence of irregular coefficients. For example, if A = −∇ · μ∇ denotes an elliptic operator with complex L ∞ -coefficients, an example by Frehse [16,18] shows that for any p > 2d/(d − 2) there exists a complex-valued, strongly elliptic matrix μ ∈ L ∞ such that (Id +A) −1 does not map L p into itself. Here, the dimension d satisfies d ≥ 3.
Important tools to reveal for which numbers p ∈ (1, ∞) certain L p -mapping properties do still hold are so-called p-sensitive Calderón-Zygmund theorems. Amongst others, there is the L p -extrapolation theorem of Shen [27] which builds on a real variable argument of Caffarelli and Peral [8,Sec. 1]. This argument was already successfully applied to reveal properties of the Stokes operator in Lipschitz domains, Riesz transforms of elliptic operators, homogenization theory of elliptic operators, maximal regularity properties of elliptic operators, and elliptic boundary value problems [8,15,21,[27][28][29][30]34,37]. In its whole space version, this L p -extrapolation theorem reads as follows, cf. [27,Thm. 3.1]. For its formulation we denote by L(E, F) set of all bounded linear operators between two Banach spaces E and F and by ffl B dx the mean value integral over a measurable set B with 0 < |B| < ∞. Theorem 1.1. (Shen) Let T ∈ L(L 2 (R d ), L 2 (R d )) be a bounded linear operator. Assume there exist p > 2, ι 2 > ι 1 > 1, and C > 0 such that for all x 0 ∈ R d , r > 0, and all compactly supported However, these basic ingredients suffice to unveil optimal mapping properties for the resolvent operators. Unfortunately, these basic techniques cease to work to establish a weak reverse Hölder estimate suitable for Theorem 1.1 if the elliptic operator in divergence form is replaced by an operator that is non-local as, for example, the Stokes operator. Motivated by this fact, we study here as "toy operators" non-local elliptic integrodifferential operators of order 2α, where α ∈ (0, 1). These operators generalize the fractional Laplacian in the whole space given by (1.1) and are defined via the form method as follows: Define the for α ∈ (0, 1) the fractional Sobolev space W α,2 (R d ) by Having second-order elliptic operators in divergence form in mind, we generalize (1.1) the variational definition of by considering the sesquilinear form defined by where K : R d × R d → C is measurable and satisfies for some 0 < < 1 the ellipticity and boundedness conditions and for u ∈ D(A) with associated function f the image of A under u is defined as Recently, there was a brisk interest in such operators and in certain nonlinear counterparts [2,[4][5][6]20,23,24,26]. We would like to highlight the work of Kuusi, Mingione, and Sire [23] where the following non-local Caccioppoli-type inequality was proven J. Evol. Equ.
and for any smooth function 2r ), and ∇η L ∞ ≤ C/r . The authors in [23] proved this inequality under slightly different assumptions on the sesquilinear form, the most important is that they considered even nonlinear equations. In contrast to the results in [23], we consider here also complex-valued coefficients K what can be seen as a minor improvement to [23]. Additionally, we do not consider solutions to (1.5), but we prove that the same Caccioppoli-type inequality is valid for solutions to the homogeneous resolvent problem where λ ∈ S θ := {z ∈ C \ {0} : |arg(z)| < θ} for some θ ∈ (π/2, π) depending on . It is important to note that the constant C in the Caccioppoli-type inequality is uniform with respect to λ. This is proven in Proposition 3.1. This allows us by an application of Sobolev's embedding theorem to establish in Lemma 3.2 a non-local weak reverse Hölder estimate for such solutions. In particular, this enables us to apply to each of the resolvent operators T λ := λ(λ + A) −1 the following L p -extrapolation theorem for non-local operators which is proven in Sect. 2.
Suppose that there exist constants p > 2, ι > 1, and C > 0 such that for all balls B ⊂ R d and all compactly supported holds. Here, the supremum runs over all balls B containing B. Then, for each 2 < q < p there exists a constant K > 0 such that for all f ∈ L ∞ (R d ; X ) with compact support it holds In particular, if C is bounded from L q (R d ; X ) into L q (R d ; Z ), then the restriction of T onto L 2 (R d ; X ) ∩ L q (R d ; X ) extends to a bounded linear operator from L q (R d ; X ) into L q (R d ; Y ). The constant K depends only on d, p, q, ι, C, M, and N .
Notice that in the original formulation of Shen, the operator C is simply the identity. In many applications-as also in ours-the operator C is zero. We hope, however, that a theorem of this form can be applied in fluid mechanics with C being the Helmholtz projection. Our application reads as follows. (1.4) for some 0 < < 1. Let A denote the operator associated with the sesquilinear form (1.3). Then, for := π − arccos( 2 ) one has that S is contained in the resolvent set of −A. Moreover, for each θ ∈ (0, ) there exists ε > 0 such that for all numbers p satisfying and for all λ ∈ S θ the restriction of the resolvent operator (λ holds. Here, the constant ε depends on d, θ , and and the constant C depends on d, θ , , and p. It is well known that the resolvent estimate presented in Theorem 1.3 forms the basis to a rich parabolic theory of the operator A. Indeed, since > π/2 the resolvent estimate (1.7) is equivalent to the fact that the L p -realization of −A generates a bounded analytic semigroup (e −t A ) t≥0 on L p (R d ).
An important notion in the parabolic theory is the notion of maximal regularity, cf. [13,22,38]. To this end, consider the Cauchy problem . The unique mild solution to (1.8) is given by the variation of constants formula In this situation, it is well known that the closed graph theorem implies that there exists a constant C > 0 such that for all g ∈ L r (0, ∞; L p (R d )) the estimate holds. That A has indeed maximal L r -regularity is formulated in the last theorem.
and for all 1 < r < ∞ the operator A has maximal L r -regularity.
We shortly outline the structure of the paper. As mentioned above, Theorem 1.2 is proven in Sect. 2. Sect. 3 is reserved to prove the Caccioppoli-type estimate and the non-local weak reverse Hölder estimate. In the final Sect. 4, we prove Theorems 1.3 and 1.4.

An L p -extrapolation theorem for non-local operators
This section is devoted to prove a non-local version of the L p -extrapolation theorem of Shen [27,Thm. 3.1]. The proof follows Shen's original argument and is only modified slightly. However, for the convenience of the reader, we present the complete argument here. The proof carries out a good-λ argument and bases on the following version of the Calderón-Zygmund decomposition which was proven by Caffarelli and Peral [8, Lem. 1.1].

Then, there is a family of disjoint dyadic cubes {Q k } k∈N obtained by suitable selections of successive bisections of Q, such that for all
where Q * k is the dyadic parent of Q k .
Here and in the following, we denote for ι > 0 and a ball B ⊂ R d or a cube Q ⊂ R d the by ι dilated ball and cube with the same center by ιB and ιQ.
Proof. For this proof, we denote a generic constant that depends solely on d, p, q, ι, or the constant C in inequality (1.6) by C g . We abbreviate the operator norms T L(L 2 ( ;X ),L 2 ( ;Y )) and C L(L 2 (R d ;X ),L 2 (R d ;Z )) by T and C . Let x 0 ∈ R d , r > 0, and Q be a cube in R d with diam(Q) = 2r and midpoint x 0 . Let B := B(x 0 , r ). One directly verifies that Thus, without loss of generality we can assume that estimate (1.6) is valid for cubes centered in x 0 instead of balls and for some possibly different ι.
where M denotes the Hardy-Littlewood maximal operator.
Decompose R d into a dyadic grid. Then, by (2.1) we find a mesh size such that each cube Q 0 from the grid satisfies Note that the mesh size is allowed to depend on λ, δ, f , and T . If the case |Q 0 ∩ E(Aλ)| = 0 applies, do nothing. In the other case, the set defined by A := Q 0 ∩ E(Aλ) together with the cube Q 0 satisfy the assumptions of Lemma 2.1. Proceeding in that way for every cube Q 0 in the grid and enumerating all cubes obtained in this way by Lemma 2.1 by {Q k } k∈N yields a countable family of mutually disjoint cubes satisfying for all k ∈ N Note that as in Lemma 2.1, Q * k denotes the dyadic parent of Q k . Claim 1. The operator T is L q -bounded, once there are constants δ, γ > 0 such that for all λ > 0 holds.
Perform a linear transformation in the second integral and notice that the resulting integral coincides modulo a factor by a generic constant times Using that A > 1, the first term on the right-hand side can be absorbed by the left-hand side. This yieldŝ Taking λ 0 → ∞ and using . The conclusion of the theorem follows by density (note that simple functions with bounded support are dense in all L q ( ; X )-spaces by construction of the Bochner integral). Claim 2. The premise of Claim 1 follows if there are constants δ, γ > 0 such that for all dyadic parents Q * k of the family of cubes {Q k } k∈N constructed before (i)-(iii) the following statement is valid: Dealing the first term on the right-hand side by the maximality of {Q * l } l∈I together with (i) and the second term by using the monotonicity of the Lebesgue measure yields Next, use (iii) first and then the mutual disjointness of the family {Q * l } l∈I together with the assertion of Claim 2 to get Claim 3. There exist δ, γ > 0 such that To conclude this statement, we argue by contradiction. For this purpose, suppose that there exists a Q k with {x ∈ Q * k : We show that the existence of such a cube contradicts (ii). In this situation, for every cube Q that contains Q * k , we have Next, let x ∈ Q k and Q be a cube with x ∈ Q and Q ⊂ 2Q * k . Then, we find for the side length of Q that (Q ) > (Q k ). If y ∈ Q * k , 1 ≤ i ≤ d, and if x denotes the center of Q , then Consequently, we have Q * k ⊂ 5Q and thus by virtue of (2.3) we have for where M 2Q * k denotes the localized maximal operator (2.4) By means of the weak-type estimate of M 2Q * k in the first inequality below, and the boundedness of T from L 2 (R d ; X ) into L 2 (R d ; Y ) together with (2.3) in the second inequality below, we derive (2.5) Next, the continuous embedding An application of (1.6) yields Add and subtract f χ 2ιQ * k in the arguments of T and C and use Use the L 2 -boundedness of T and C to get Notice that |2ιQ * k |/|Q | ≤ ι d so that another application of (2.3) finally yields Recall that A = 1/(2δ 2/q ), that T ≤ M, and that C ≤ N . Thus, a combination of (2.5) and (2.6) yields Since p > q, we can choose δ small enough such that {M + N + √ 2} p δ p q −1 ≤ 1/(2C g ). For this fixed value of δ choose γ ≤ min{1, δ 1− 2 q /(2M 2 C g )}. Then, we obtain |E(Aλ) ∩ Q k | ≤ δ|Q k | which is a contradiction to (ii) of the Calderón-Zygmund decomposition.

Remark 2.2.
In the third part of the proof above, we performed one particular choice. Indeed, in (2.4) we needed to decompose the function T f in the cube 2Q * k as the sum of two functions. Due to the linearity of T , this decomposition was given by . Thus, the decomposition of the function T f was done by decomposing f suitably. In some situations, however, it can be helpful to decompose T f = v + w by another choice of functions that might not be induced by a decomposition of f into a sum of two functions. This happens, for example, if T is considered on a subspace of L 2 that is not stable under multiplication by characteristic functions. Such an argument was used, for example, in [36].

Verification of the non-local weak reverse Hölder estimate
The main ingredient for the verification of the non-local weak reverse Hölder estimate is the following proposition. In this proposition a non-local Caccioppoli inequality for the resolvent equation is proved. The proof follows the ideas of the proof for the special case λ = 0 and for real-valued kernel functions K (x, y) which can be found in the work of Kuusi, Mingione, and Sire [23,Thm. 3.2].
Let η ∈ C ∞ c (B(x 0 , 3r/2)) be a function that satisfies 0 ≤ η ≤ 1 and ∇η L ∞ ≤ C d /r for some constant C d > 0 depending only on d. Then, there exists a constant C > 0 depending only on d, α, , and θ such that Proof. Write B := B(x 0 , r ).
it is possible to choose v := uη 2 as a test function. Thus, by virtue of (3.1) the following identity holds After rearranging the terms, we find by (3.2 To rewrite the second term on the left-hand side of (3.3), calculate Switch the roles of x and y, perform the same calculation as in (3.4), and add the resulting identity to (3.4) to obtain (u(x) + u(y))(η(x) − η(y))(η(x) + η(y)). (3.5)

Now, plug (3.5) into (3.3) and rearrange terms to obtain
Notice that on the left-hand side, the complex number λ is multiplied by a nonnegative real number and that in all other integrals on the left-hand side, the complex-valued function K (x, y) is multiplied by nonnegative real functions. The condition (1.4) implies that arg (K (x, y)) ∈ S π − with = π − arccos( 2 ).
Apply this inequality to (3.6) together with (1.4) to obtain |u(x)||u(y)|η(y) 2 |x − y| d+2α dx dy First, we will rewrite the second term on the left-hand side of (3.7). To this end, use |z + w| 2 = |z| 2 + |w| 2 + 2 Re(zw) for z, w ∈ C and calculate Now, switch the roles of x and y, perform the same calculation as in (3.8), and add the resulting identity to (3.8) to obtain Notice that the first term on the right-hand side of (3.9) appears in the second term on the left-hand side of (3.7). Replace half of the second term on the left-hand side of (3.7) by employing (3.9) and leave the other half as it is. After rearranging the terms, one gets |u(x)||u(y)|η(y) 2 |x − y| d+2α dx dy (3.10) The first two terms on the right-hand side are estimated by using Fubini's theorem first and second, if x 0 denotes the midpoint of B, by using that for x ∈ supp(η) and y ∈ R d \ 2B one has due to |x − y| ≥ r/2 This yieldŝ For the third term on the right-hand side of (3.10), employ Young's inequality together (3.11) For the time being, the fourth term on the right-hand side of (3.10) is not estimated further. Concerning the fifth term on the right-hand side of (3.10), employ again Young's inequality to deducê (3.12) Now, absorb each of the first terms on the right-hand sides of (3.11) and (3.12) to the left-hand side of (3.10).
The following terms remain on the right-hand side: The second term is already one of the desired terms, so we analyze the first term. Notice that by symmetry and the condition ∇η Here, C d,α > 0 is a constant that depends only on d and α. Summarizing everything, there exists a constant C > 0 depending only on d, α, , and θ such that This proves the proposition.
The following lemma brings the right-hand side of Caccioppoli inequality in Proposition 3.1 into a suitable form for the non-local weak reverse Hölder estimate. Lemma 3.2. Let x 0 ∈ R d , r > 0, α ∈ (0, 1), and u ∈ L 2 loc (R d ). Then, there exists a constant C > 0 depending only on d and α such that Proof. Jensen's inequality applied to the first integral followed by Young's inequality ensures for some constant C > 0 depending only on d that Vol. 21 (2021) L p -extrapolation of non-local operators: Maximal regularity 3145 Furthermore, decomposing the second integral on the right-hand side into dyadic annuli yieldŝ |u(y)| |x 0 − y| d+2α dy. Now, apply Jensen's inequality to each of the integrals and further use that to establish for some constant C > 0 depending only on d and α Finally, Hölder's inequality for series together with α > 0 yield This readily yields the desired estimate.

Proofs of Theorems 1.3 and 1.4
Let A denote the operator associated with the sesquilinear form Define := π − arccos( 2 ). Notice that (1.4) implies that Thus, if 0 < θ < and if λ ∈ S θ , then an elementary trigonometric consideration shows that the sesquilinear form is bounded and coercive. Thus, by the Lax-Milgram lemma, we find that λ ∈ ρ(−A), the resolvent set of −A. Thus, for f ∈ L 2 (R d ) there exists a unique u ∈ D(A) with λu+ Au = f . Testing this equation by u yields by the same trigonometry consideration as above for a constant C > 0 depending only on θ and λ that Now, forgetting about the double integral on the left-hand side and dividing by u L 2 (R d ) shows that the L 2 resolvent estimate is valid. We remark that for an operator A, the property of having maximal L r -regularity on the time interval (0, ∞) is in general stronger than the property that −A generates a bounded analytic semigroup. Indeed, combining the characterization of maximal L r -regularity via the notion of R-boundedness [38,Thm. 4.2] and using the reformulation of R-boundedness via square function estimates if operators on L p -spaces are considered [22,Rem. 2.9] we arrive at the following statement. Namely, for any given 1 < r < ∞ the operator A has maximal L r -regularity if there exists θ > π/2 and a constant C > 0 such that for all n 0 ∈ N, . (4.2) Notice that C has to be uniform with respect to n 0 , Regarding the square root over the sum of squares as an Euclidean norm, this statement is equivalent to the boundedness of the following family of operator T θ in the space L(L p (R d ; 2 )) T θ := {(λ 1 (λ 1 + A) −1 , . . . , λ n 0 (λ n 0 + A) −1 , 0, . . . ) : n 0 ∈ N, (λ n ) n 0 n=1 ⊂ S θ }.
Notice that the square function estimate (4.2) is in the case p = 2 equivalent to the uniform resolvent estimate in (4.1). Thus, we already know that the family of operators T θ is bounded in L(L 2 (R d ; 2 )). Let θ ∈ (0, ). We show in the following that there exists ε > 0 such that for all p ≥ 2 that satisfy the family T θ is bounded in L(L p (R d ; 2 )). To this end, we verify that each operator in T θ fulfills the assumptions of Theorem 1.2 with uniform constants for all operators in T θ . As the knowledge of the L q -operator norm in Theorem 1.2 is known to depend only on the quantities at stake, this will imply that T θ is bounded in L(L p (R d ; 2 )). As the L 2 (R d ; 2 )-boundedness is already established, we concentrate on the non-local weak reverse Hölder estimate, i.e., estimate (1.6).
In particular, K (y, x) fulfills the ellipticity assumption (1.4) for the same constant as K did. Now, since the dual space of L p (R d ; 2 ) is L p (R d ; 2 ) and p > 2 satisfies (4.3), we conclude the statements of Theorems 1.3 and 1.4 in the case p < 2 as well.