INTERIOR REGULARITY OF SOLUTIONS OF NON-LOCAL EQUATIONS IN SOBOLEV AND NIKOL’SKII SPACES MATTEO COZZI 6 Abstract. WeproveinteriorH2s−εregularityforweaksolutionsoflinearellipticintegro- 1 differential equations close to thefractional s-Laplacian. The result is obtained via inter- 0 mediate estimates in Nikol’skii spaces, which are in turn carried out by means of an 2 appropriate modification of the classical translation method by Nirenberg. n a J 1. Introduction 2 1 One of the first fundamental achievements in the field of the regularity theory for weak ] solutions of second order linear elliptic differential equations is the existence of weak second P derivatives. Indeed, let Ω be an open set of Rn and u∈ H1(Ω) a weak solution of A . (1.1) −div(A(·)∇u) = f in Ω, h t where the n×n matrix A = [a ] is uniformly elliptic, with entries a ∈ C0,1(Ω), and the a ij ij loc m right-hand term f ∈ L2(Ω). Then, one gets that u∈ H2 (Ω) and, for any domain Ω′ ⊂⊂Ω, loc [ kukH2(Ω′) 6 C kukL2(Ω)+kfkL2(Ω) , 1 v for some constant C > 0 independent of u(cid:0) and f. (cid:1) 9 Such result is typically ascribed to Louis Nirenberg, who in [N55] obtained higher order 1 Sobolev regularity for general linear elliptic equations. To do so, he introduced the by now 8 classical translation method. In the setting of equation (1.1) the idea is basically to consider 2 0 the difference quotients . u(x+he )−u(x) 1 Dhu(x) := i , 0 i h 6 for i= 1,...,n and h6= 0 suitably small in modulus, and use the equation itself to recover 1 a uniform bound in h for the gradient of Dhu in L2(Ω′). A compactness argument then : i v shows that u ∈ H2 (Ω). Nice presentations of this technique are for instance contained i loc X in [E98] and [GM12]. r After this, several generalizations were achieved. For example, the translation method a has been successfully adapted to study nonlinear equations, too. Indeed, in [S77] and [D82] the authors deduced higher order regularity in both Sobolev and Besov classes for singular or degenerate operators of p-Laplacian type. See also [M03, M03b] where similar fractional estimates were obtained in a non-differentiable vectorial setting. The object of this note is the attempt of a generalization of the above discussed higher differentiability to a non-local analogue of equation (1.1), modelled upon the fractional Laplacian. 2010 Mathematics Subject Classification. 35R09, 35R11, 45K05, 35B65, 46E35. Keywordsandphrases. Integro-differentialequations,fractionalLaplacian,regularitytheory,Nirenberg’s translation method, Caccioppoli inequality,fractional Sobolev spaces, Nikol’skii spaces. It is a pleasure to thank Enrico Valdinoci for his dedication, advice and encouragement. I would also like toexpress mygratitude totheWeierstraß Institutfu¨r AngewandteAnalysis undStochastik (WIAS)of Berlin, where part of this work was carried out. 1 2 MATTEOCOZZI Given any open set Ω ⊂ Rn, we consider a solution u of the linear equation (1.2) E (u,ϕ) = hf,ϕi for any ϕ ∈ C∞(Ω), K L2(Ω) 0 where f ∈L2(Ω) and E is defined by K E (u,ϕ) := (u(x)−u(y))(ϕ(x)−ϕ(y))K(x,y)dxdy. K Rn Rn Z Z Here K is a measurable function which is comparable in the small to the kernel of the fractional Laplacian. Indeed, if we take K(x,y)= |x−y|−n−2s, with s ∈ (0,1), then (1.2) is the weak formulation of the equation (−∆)su = f in Ω, for the fractional Laplace operator of order 2s u(x)−u(y) u(x)−u(y) (−∆)su(x) = 2P.V. dy = 2 lim dy. ZRn |x−y|n+2s δ→0+ZRn\Bδ(x) |x−y|n+2s On the other hand, more general kernels are admissible as well, possibly not translation invariant. However, if the kernel is not translation invariant, we need to impose on K some sort of joint local C0,s regularity. We stress that this last hypothesis seems very natural to us. Indeed, while translation invariant kernels correspond in the local framework to the constant coefficient case, asking K to be locally H¨older continuous is a legitimate counterpart to the Lipschitz regularity assumed on the matrix A in (1.1). Integro-differential equations have been the object of a great variety of studies in recent years. A priori estimates for quite general linear equations were obtained in [BK05, S06] (H¨older estimates) and in [B09] (Schauder estimates). Other fundamental results in what concernspointwiseregularity wereachieved by CaffarelliandSilvestrein[CS09,CS11]. The two authors developed there a theory for viscosity solutions, in order to deal with general fullynonlinearequations. Theframeworkconsideredhereisinsteadthatofweak (orenergy) solutions. Thesetwo notions of solutions areof coursevery close, as it is discussedin[RS14] and [SV14], but, since we have a datum f in L2, the weak formulation (1.2) seems to us more appropriate. The literature on the regularity theory for weak solutions is indeed very rich and it is not possible to provide here an exhaustive account of the many contributions. Just to name a few, Kassmann addressed the validity of a Harnack inequality and established interiorH¨olderregularityfornon-local harmonic functions throughthelanguageofDirichlet forms (see [K07, K09, K11]). In [RS14] the authors obtained H¨older regularity up to the boundary for a Dirichlet problem driven by the fractional Laplacian. Concerning regularity resultsinSobolevspaces,H2s estimatesareprovedin[DK12]forentiretranslationinvariant equations. Also, the very recent [KMS15] provides higher differentiability/integrability in a nonlinear setting quite similar to ours. Here we show that a solution u of (1.2) has better weak (fractional) differentiability propertiesintheinteriorofΩ. Byadaptingthetranslation methodtothisnon-localsetting, we prove that 2s,2 (1.3) u∈ N (Ω). loc Notice that the symbol Nr,p(Ω), for r > 0 and 1 6 p < +∞, denotes here the so- called Nikol’skii space. Since both Nikol’skii and fractional Sobolev spaces are part of the wider class of Besov spaces, standard embedding results within this scale allow us to deduce from (1.3) that (1.4) u ∈ H2s−ε(Ω), loc for any ε > 0. INTERIOR REGULARITY IN SOBOLEV AND NIKOL’SKII SPACES 3 We do not know whether or not (1.4) is the optimal interior regularity for solutions of (1.2) in the Sobolev class. While one would arguably expect u to belong to H2s(Ω), loc there is no hope in general to extend such regularity up to the boundary, as discussed in Section 8. Finally, we stress that the exponent 2s−ε still provides Sobolev regularity for the gradient of u, when s > 1/2. We point out that, almost concurrently to the present work and independently from it, a result rather similar to (1.4) has been obtained in [BL15]. Indeed, the authors address there the problem of establishing higher Sobolev regularity for a nonlinear, superquadratic generalization ofequation (1.2). Whenrestricted tothelinear case, their resultis analogous to ours, for s 6 1/2, and slightly weaker, for s > 1/2. In the upcoming section we specify the framework in which the model is set. We give formal definitions of the notion of solution and of the class of kernels under consideration. Moreover, we introduce the various functional spaces that are necessary for these purposes. After such preliminary work, we are then in position to give the precise statements of our results. 2. Definitions and formal statements Letn ∈ Nand s∈ (0,1). Thekernel K :Rn×Rn → [0,+∞] is assumedto bemeasurable and symmetric1, that is (2.1) K(x,y) = K(y,x) for a.a. x,y ∈ Rn. We also require K to satisfy (2.2a) λ 6 |x−y|n+2sK(x,y) 6 Λ for a.a. x,y ∈Rn, |x−y|< 1, (2.2b) 0 6 |x−y|n+βK(x,y) 6 M for a.a. x,y ∈Rn, |x−y|> 1, for some constants Λ> λ > 0, β,M > 0, and (2.3) |x−y|n+2s|K(x+z,y+z)−K(x,y)| 6 Γ|z|s, for a.a. x,y,z ∈ Rn, with |x−y|,|z| < 1, and for some Γ > 0. Condition (2.2a) tells that the kernel K is controlled from above and below by that of the fractional Laplacian when x and y are close. Conversely, when |x − y| is large, the behaviour of K could be more general, as expressed by (2.2b). Under these hypotheses a great variety of kernels could be encompassed, as for instance truncated ones or having non-standard decay at infinity. Naturally, these requirements are fulfilled (with β = 2s) when K is globally comparable to thekernel of the fractional Laplacian, that is when (2.2a) holds a.e. on the whole Rn×Rn. On the other hand, (2.3) asserts that the map (x,y) 7−→ |x−y|n+2sK(x,y), is locally uniformly C0,s regular, jointly in the two variables x and y. Clearly, (2.3) is satisfied by translation invariant kernels, i.e. those in the form (2.4) K(x,y) =k(x−y), 1Westressthatthesymmetryhypothesisdoesnotreallyplaymuchofarolehere. Indeed,ifoneconsiders insteadanon-symmetrickernelK,thiscanbewrittenasthesumofitssymmetricandanti-symmetricparts K(x,y)+K(x,y) K(x,y)−K(y,x) K (x,y):= and K (x,y):= . sym 2 asym 2 But then, it is easily shown that K cancels out in (2.5), thus leading to an equation driven by the asym symmetric kernelK . Hence, we may and doassume K symmetric from theoutset. sym In this regard, we refer the interested reader to [FKV13], where a class of integro-differential equations with non-symmetrickernels are studied. 4 MATTEOCOZZI for some measurable k : Rn → [0,+∞]. But more general choices are possible, as for instance kernels of the type a(x,y) K(x,y) = , |x−y|n+2s with a ∈ C0,s(Rn ×Rn). We also stress that (2.3) may be actually weakened by requiring it to hold only inside the set Ω where the equation will be valid. In order to formulate the equation and state our main results, we introduce the following functional framework. Let s > 0, 1 6 p < +∞ and U be any open set of Rn. We indicate with Lp(U) the standard Lebesgue space and with Ws,p(U) the (fractional) Sobolev space as defined, for instance, in the monograph [DPV12]. Of course, Hs(U) := Ws,2(U). Restricting ourselves to s ∈ (0,1), we denote with X(U) the space of measurable func- tions u: Rn → R such that u| ∈ L2(U) and (x,y) 7−→ (u(x)−u(y)) K(x,y) ∈ L2(C ), U U where p C := (Rn×Rn)\((Rn\U)×(Rn\U))⊂ Rn×Rn. U Notice that, by virtue of (2.2), if u ∈ X(U) and V is a bounded open set contained in U, then u| ∈ Hs(V). In addition, X (U) is the subspace of X(U) composed by the functions V 0 which vanish a.e. outside U. We refer the reader to [SV13, Section 5] for informations on very similar spaces of functions. As it is customary, given any space F(U) of functions defined on a set U, we say that u∈ F (U) if and only if u| ∈F(V) for any domain V ⊂⊂ U. loc V Let now Ω be a fixed open set of Rn. For u ∈X(Ω) and ϕ∈ X (Ω), it is well-defined the 0 bilinear form (2.5) E (u,ϕ) := (u(x)−u(y))(ϕ(x)−ϕ(y))K(x,y)dxdy. K Rn Rn Z Z Given f ∈ L2(Ω), we say that u∈ X(Ω) is a solution of (2.6) E (u,·) = f in Ω, K if (2.7) E (u,ϕ) = hf,ϕi for any ϕ ∈X (Ω). K L2(Ω) 0 We remark that, for instance whenK is symmetricand translation invariant, i.e. as in (2.4) with k even, then (2.7) is the weak formulation of the equation L u = f in Ω, k where the operator L is defined - for u sufficiently smooth and bounded - by k L u(x) := 2P.V. (u(x)−u(y))k(x−y)dy. k Rn Z As a last step towards the first theorem, we introduce a weighted Lebesgue space which we will require the solutions to lie in. Given a measurable function w : Rn → [0,+∞), we say that u∈ L1(Rn) if and only if w u :Rn → R is measurable and kukL1(Rn) := |u(x)|w(x)dx < +∞. w Rn Z In what follows we consider weights of the form 1 (2.8) w (x) = , x0,β 1+|x−x |n+β 0 forx ∈ Rn andβ > 0as in(2.2b). We denotethecorrespondingspaces justwithL1 (Rn) 0 x0,β and we adopt the same notation for their norms. Also, we simply write L1(Rn) when x is β 0 INTERIOR REGULARITY IN SOBOLEV AND NIKOL’SKII SPACES 5 the origin. Notice that, in fact, the space L1 (Rn) does not depend on x and different x0,β 0 choices for the base point x lead to equivalent norms. Lastly, we observe that, in conse- 0 quence of the fact that w ∈ L1(Rn)∩L∞(Rn), the space L1(Rn) contains both L∞(Rn) x0,β β and L1(Rn). With all this in hand,we are now ready to state the firstand principalresultof this note. Theorem 2.1. Let s ∈ (0,1), β > 0 and Ω ⊂ Rn be an open set. Assume that K satisfies assumptions (2.1), (2.2) and (2.3). Let u∈ X(Ω)∩L1(Rn) be a solution of (2.6), with f ∈ β L2(Ω). Then, u ∈H2s−ε(Ω) for any small ε > 0 and, for any domain Ω′ ⊂⊂ Ω, loc (2.9) kuk 6 C kuk +kuk +kfk , H2s−ε(Ω′) L2(Ω) L1(Rn) L2(Ω) β (cid:16) (cid:17) for some constant C > 0 depending on n, s, β, λ, Λ, M, Γ, Ω, Ω′ and ε. The technique we adopt to prove Theorem 2.1 is basically the translation method of Nirenberg, suitably adjusted to cope with the difficulties arising in this fractional, non-local framework. However, this strategy does not immediately lead to an estimate in Sobolev spaces. In fact, it provides that the solution belongs to a slightly different functional space, which is well-studied in the literature and is often referred to as Nikol’skii space. We briefly introduce such class here below. Let U be a domain of Rn. Given k ∈ N and z ∈ Rn, let (2.10) U := {x ∈U :x+iz ∈ U for any i = 1,...,k}. kz Observe that, by definition, (2.11) U ⊆ U ⊆ U if j,k ∈ N and j 6 k. kz jz For any z ∈ Rn we also define τ u(x) := u(x+z) and z ∆ u(x) := τ u(x)−u(x), z z for any x ∈ U . Sometimes we will need to deal with increments along the diagonal for the z kernel K, as previously done in (2.3). With a slight abuse of notation, we write τ K(x,y) := K(x+z,y+z) and ∆ K(x,y) := τ K(x,y)−K(x,y). z z z We also consider increments of higher orders. For any k ∈ N we set k k ∆ku(x) := ∆ ∆k−1u(x) = (−1)k−i τ u(x), z z z i iz i=0 (cid:18) (cid:19) X for any x ∈ U , with the convention that ∆0u = u. Of course, ∆1u = ∆ u. Moreover, kz z z z j notice that by (2.11) all ∆ u, as j = 0,1,...,k, are well-defined in U . z kz Given s ∈ (0,2) and 1 6 p < +∞, the Nikol’skii space Ns,p(U) is defined as the space of functions u∈ Lp(U) such that (2.12) [u] := sup |z|−sk∆2uk < +∞. Ns,p(U) z Lp(U2z) z∈Rn\{0} The norm kuk := kuk +[u] , Ns,p(U) Lp(U) Ns,p(U) makes Ns,p(U) a Banach space. We point out that the restriction to s < 2 is assumed here only to avoid unnecessary complications in the definition of the semi-norm (2.12). By the way, the above range for s is large enough for our scopes and, thus, there is no real need to deal with more general conditions. Nevertheless, such limitation will not be considered anymore in Section 3, where a deeper look at the space Ns,p(U) will be given. Now that the definition of Nikol’skii spaces has been recalled, we may finally head to our second main result. 6 MATTEOCOZZI Theorem 2.2. Let s ∈ (0,1), β > 0 and Ω ⊂ Rn be an open set. Assume that K satisfies assumptions (2.1), (2.2) and (2.3). Let u∈ X(Ω)∩L1(Rn) be a solution of (2.6), with f ∈ β L2(Ω). Then, u ∈N2s,2(Ω) and, for any domain Ω′ ⊂⊂Ω, loc (2.13) kuk 6 C kuk +kuk +kfk , N2s,2(Ω′) L2(Ω) L1(Rn) L2(Ω) β (cid:16) (cid:17) for some constant C > 0 depending on n, s, β, λ, Λ, M, Γ, Ω and Ω′. In light of this estimate, Theorem 2.1 follows more or less immediately. To see this, it is helpful to understand Sobolev and Nikol’skii spaces in the context of Besov spaces. For s ∈ (0,2), 1 6 p < +∞ and 1 6 λ 6 +∞, the Besov space Bs,p(U) is the space of λ functions u∈ Lp(U) such that [u]Bs,p(U) < +∞, where λ 1/λ dz |z|−sk∆2uk λ if 16 λ < +∞, [u]Bs,p(U) := (cid:18)ZRn z Lp(U2z) |z|n(cid:19) λ  sup(cid:0) |z|−sk∆2zukLp(U2z(cid:1)) if λ = +∞. z∈Rn\{0} Observe that, by definition, Bs,p(U) = Ns,p(U), while the equivalence Bs,p(U) = Ws,p(U)  ∞ p is also true, though less trivial. Then, since there exist continuous embeddings (2.14) Bs,p(U) ⊂Br,p(U), ν λ as 16 λ 6 ν 6 +∞ and 1 < r < s <+∞, it follows Ns,p(U) ⊂ Wr,p(U). Consequently,uptosomeminordetailsthatwillbediscussedlaterinSection7,Theorem2.1 is a consequence of Theorem 2.2. Of course, Theorem 2.2 and inclusion (2.14) yield estimates in many other Besov spaces for the solution u of (2.6). Basically, u lies in any B2s−ε,2(Ω), with ε > 0 and 1 6 λ 6 +∞. λ,loc Wepointoutherethatthroughoutthepaperthesameletter cisusedtodenoteapositive constantwhichmaychangefromlinetolineanddependsonthevariousparametersinvolved. The rest of the paper is organized as follows. In Section 3 we review some basic material on Sobolev and Nikol’skii spaces. To keep a leaner notation, we do not approach Besov spaces in their full generality and restrict in fact to the two classes to which we are interested. Despite every assertion of this section is classical and surely well-known to the experts, we choose to include here the few results that will be used afterwards, in order to make the work as self-contained as possible. Thesubsequenttwo sections aredevoted tosomeauxiliaryresults. Section 4isconcerned with a couple of technical lemmata that deal with a discrete integration by parts formula and an estimate for the defect of two translated balls. In Section 5, on the other hand, we prove a non-local version of the classical Caccioppoli inequality. The main results are proved in Sections 6 and 7. Finally, Section 8 contains some comments on the possible optimal global regularity for the Dirichlet problem associated to (2.6). 3. Preliminaries on Sobolev and Nikol’skii spaces We collect here some general facts about fractional Sobolev spaces and Nikol’skii spaces. As said before, we avoid dealing with the wider class of Besov spaces in order not to burden the notation too much. For more complete and exhaustive presentations we refer the interested reader to the books by Triebel, [T83, T92, T06] and [T95]. We remark that the proofs displayed only make use of integration techniques, mostly inspired by [S90]. While some results can not be justified with such elementary arguments, we still provide specific references to the above mentioned books. INTERIOR REGULARITY IN SOBOLEV AND NIKOL’SKII SPACES 7 Let U ⊂ Rn be a bounded domain with C∞ boundary2. Let 1 6 p < +∞ and s > 0, with s ∈/ N. Write s = k+σ, with k ∈ N∪{0} and σ ∈(0,1). We recall that the fractional Sobolev space Ws,p(U) is defined as the set of functions Ws,p(U) := u∈ Wk,p(U) :[D u] < +∞ for any |α| = k , α Wσ,p(U) n o where, for v ∈ Lp(U), |v(x)−v(y)|p 1/p [v] := dxdy . Wσ,p(U) |x−y|n+σp (cid:18)ZU ZU (cid:19) Clearly, α indicates a multi-index, i.e. α = (α ,...,α ) with α ∈ N ∪ {0}, and |α| = 1 n i α +···+α is its modulus. Moreover, Wk,p(Ω), for k ∈ N, denotes the standard Sobolev 1 n space and, when k = 0, we understand W0,p(U) = Lp(U). The space Ws,p(U) equipped with the norm kuk := kuk + [Dαu] , Ws,p(U) Wk,p(U) Wσ,p(U) |α|=k X is a Banach space. Notice that, for v ∈ Lp(U), |v(x)−v(y)|p 1/p [v] = dxdy Wσ,p(U) |x−y|n+σp (cid:18)ZU ZU (cid:19) |v(x+z)−v(x)|p 1/p = dx dz (cid:18)ZRn(cid:18)ZUz |z|n+σp (cid:19) (cid:19) 1/p dz = |z|−σk∆ vk p . (cid:18)ZRn z Lp(Uz) |z|n(cid:19) (cid:0) (cid:1) In view of this fact, we have the following characterization for Ws,p(U). Proposition 3.1. Let 1 6 p < +∞ and s > 0. Let k,l ∈ Z be such that 0 6 k < s and l > s−k. Then, 1/p p dz (3.1) kuk + |z|k−sk∆lDαuk , Lp(U) |αX|=k(cid:18)ZRn(cid:16) z Lp(Ulz)(cid:17) |z|n(cid:19) is a Banach space norm for Ws,p(U), equivalent to k·k . Ws,p(U) A reference for these equivalences is given by Theorem 4.4.2.1 at page 323 of [T95]. Note that the result is valid even if s is an integer. Remark 3.2. In what follows, we will be mostly interested in norms with k = 0 and therefore l >s. In such cases, we stress that (3.1) may bereplaced with the restricted norm 1/p p dz (3.2) kuk + |z|−sk∆luk , Lp(U) z Lp(Ulz) |z|n (cid:18)ZBδ (cid:16) (cid:17) (cid:19) for any δ > 0, with no modifications to the space Ws,p(U). Indeed, we have k∆luk 62lkuk , z Lp(Ulz) Lp(U) 2Most of the assertions contained in this section should be also true under less restrictive hypotheses on the boundary of the set. Of course, the definitions of the spaces require no assumptions at all on the boundaryandotherresultsareextendedintheliteraturetoLipschitzsets. Unfortunately,wehavenotbeen able to find completely satisfactory references for Proposition 3.1, and its counterpart for Nikol’skii spaces, undersuch weaker assumptions. Anyway,the limitation to C∞ domains will not have any influenceon our applications. 8 MATTEOCOZZI so that p dz 1/p Hn−1(∂B ) 1/p |z|−sk∆luk 6 2l 1 δ−skuk . ZRn\Bδ (cid:16) z Lp(Ulz)(cid:17) |z|n! (cid:18) sp (cid:19) Lp(U) Consequently, the norms defined by (3.1) and (3.2) are equivalent. The second class of fractional spaces which we are interested in are the Nikol’skii spaces. For s = k+σ > 0, with k ∈ N∪{0},σ ∈ (0,1], and 1 6 p < +∞, define Ns,p(U) := u∈ Wk,p(U) : [Dαu] < +∞ for any |α| = k , Nσ,p(U) where, for v ∈ Lp(U), n o [v] := sup |z|−σk∆2vk . Nσ,p(U) z Lp(U2z) z∈Rn\{0} It can be showed that Ns,p(U) is a Banach space with respect to the norm kuk := kuk +[u] . Ns,p(U) Wk,p(U) Ns,p(U) Notice that this definition of Nikol’skii space may seem to differ from that given in Section 2. In fact, this is not the case, as Ns,p(U) can be equivalently endowed with any norm of the form (3.3) kuk + sup |z|k−sk∆lDαuk , Lp(U) z∈Rn\{0} z Lp(Ulz) |α|=k X where k,l ∈ Z are such that 0 6 k < s and l > s−k (see again Theorem 4.4.2.1 of [T95]). Remark 3.3. As for the Sobolev spaces, we will consider norms with k = 0 for the most of the time. We stress that in such cases (3.3) may be replaced with kuk + sup |z|−sk∆luk , Lp(U) z Lp(Ulz) 0<|z|<δ for any integer l > s and any δ > 0. Intheconclusive partof this section westudy themutualinclusion propertiesof Ws,p(U) and Ns,p(U). In order to do this, it will be useful to consider another family of equivalent norms. To this aim, for l ∈ N we introduce the so-called l-th modulus of smoothness of u ωl(u;η) := sup k∆luk , p z Lp(Ulz) 0<|z|<η defined for any η > 0. Then, we have Proposition 3.4. Let s > 0 and 1 6 p < +∞. Let l > s be an integer and 0 < δ 6 +∞. Then, δ p dη 1/p kuk + η−sωl(u;η) , Lp(U) p η is a Banach space norm for Ws,p(U)(cid:18),Ze0qu(cid:16)ivalent to k·(cid:17)k (cid:19) . Ws,p(U) The same statement holds true for the norms kuk + sup η−sωl(u;η), Lp(U) p 0<η<δ and the space Ns,p(U). Proof. We only deal with the Sobolev space case, the Nikol’skii one being completely anal- ogous and easier. Furthermore, we assume δ = 1. Then, an argument similar to that presented in Remark 3.2 shows that the result can be extended to any δ. For u∈ Lp(U) let 1/p p dz [u]♭ := |z|−sk∆luk , Ws,p(U) z Lp(Ulz) |z|n (cid:18)ZB1(cid:16) (cid:17) (cid:19) INTERIOR REGULARITY IN SOBOLEV AND NIKOL’SKII SPACES 9 and 1 p dη 1/p [u]♯ := η−sωl(u;η) . Ws,p(U) p η (cid:18)Z0 (cid:16) (cid:17) (cid:19) We claim that there exists a constant c > 1 such that (3.4) c−1[u]♭ 6 [u]♯ 6 c kuk +[u]♭ , Ws,p(U) Ws,p(U) Lp(U) Ws,p(U) (cid:16) (cid:17) for all u∈ Lp(U). In view of Proposition 3.1 and Remark 3.2, this concludes the proof. To check the left hand inequality of (3.4) we first observe that k∆luk 6 sup k∆luk = ωl(u;|z|), z Lp(Ulz) y Lp(Uly) p 0<|y|<|z| for any z ∈ Rn. Then, using polar coordinates, 1/p p dz [u]♭ = |z|−sk∆luk Ws,p(U) z Lp(Ulz) |z|n (cid:18)ZB1(cid:16) (cid:17) (cid:19) 1 p dη 1/p 6 Hn−1(∂B ) η−sωl(u;η) 1 p η (cid:18) Z0 (cid:16) (cid:17) (cid:19) = Hn−1(∂B )1/p[u]♯ . 1 Ws,p(U) Now wefocusonthesecondinequality. Inordertoshowitsvalidity weneedthefollowing auxiliary result. For x ∈ U, η > 0 and u∈ Lp(U), let Vl(x,η) := {z ∈ B : x+τz ∈ U, for any 0 6 τ 6 l}, η Mlu(x) := η−n |∆lu(x)|dz, η z ZVl(x,η) and define 1 p dη 1/p (3.5) [u]∗ := η−skMluk , Ws,p(U) η Lp(U) η (cid:18)Z0 (cid:16) (cid:17) (cid:19) kuk∗ := kuk +[u]∗ . Ws,p(U) Lp(U) Ws,p(U) Then, by virtue of [T06, Theorem 1.118] we infer that (3.6) [u]♯ 6 ckuk∗ , Ws,p(U) Ws,p(U) for any u∈ Lp(U). Applying the generalized Minkowski’s inequality to the right-hand side of (3.5) and ob- serving that {(x,z) ∈ U ×Rn :z ∈ Vl(x,η)} ⊆ {(x,z) ∈ U ×B : x ∈ U }, η lz we get p 1/p 1 dη [u]∗ = η−(s+n)p |∆lu(x)|dz dx Ws,p(U) z η Z0 ZU ZVl(x,η) ! ! ! (3.7) p 1/p 1 dη 6 η−(s+n)p k∆luk dz . z Lp(Ulz) η Z0 ZBη ! ! Now, Jensen’s inequality implies that p k∆luk dz 6 cηn(p−1) k∆lukp dz, ZBη z Lp(Ulz) ! ZBη z Lp(Ulz) 10 MATTEOCOZZI and hence (3.7) becomes 1/p 1 [u]∗ 6 c η−n−1−sp k∆lukp dz dη . Ws,p(U) Z0 ZBη z Lp(Ulz) ! ! We finally switch to polar coordinates to compute 1/p 1 η [u]∗ 6 c η−n−1−sp k∆lukp dHn−1(z) dρdη Ws,p(U) Z0 Z0 Z∂Bρ z Lp(Ulz) ! ! 1/p 1 1 = c k∆lukp dHn−1(z) η−n−1−spdη dρ Z0 Z∂Bρ z Lp(Ulz) !(cid:18)Zρ (cid:19) ! 1/p 1 6 c k∆lukp dHn−1(z) ρ−n−spdρ Z0 Z∂Bρ z Lp(Ulz) ! ! = c[u]♭ . Ws,p(U) By combining this formula with (3.6), we obtain the right inequality of (3.4). Thus, the proof of the proposition is complete. (cid:3) We are now in position to prove the main results of this section, concerning the relation between Sobolev and Nikol’skii spaces. First, we have Proposition 3.5. Let s > 0 and 16 p < +∞. Then, Ws,p(U) ⊆ Ns,p(U), and there exists a constant C > 0, depending on n, s and p, such that kuk 6 Ckuk , Ns,p(U) Ws,p(U) for any u∈ Lp(U). Proof. In view of Proposition 3.4 it is enough to prove that, if l ∈ Z is such that l > s, then +∞ p dη 1/p (3.8) supη−sωl(u;η) 6 c η−sωl(u;η) , p p η η>0 (cid:18)Z0 (cid:16) (cid:17) (cid:19) forsomec> 0. Butthisisinturnanimmediateconsequenceofthemonotonicity ofωl(u;·). p Indeed, ωl(u;η) > ωl(u;t), for any η > t, and so p p +∞ p dη 1/p +∞ p dη 1/p η−sωl(u;η) > η−sωl(u;t) = (sp)−1/pt−sωl(u;t). p η p η p (cid:18)Z0 (cid:16) (cid:17) (cid:19) (cid:18)Zt (cid:16) (cid:17) (cid:19) Inequality (3.8) is then obtained by taking the supremum as t > 0 on the right hand side. (cid:3) The following provides a partial converse to the above inclusion. Proposition 3.6. Let s > r > 0 and 16 p < +∞. Then, Ns,p(U) ⊆ Wr,p(U), and there exists a constant C > 0, depending on n, r, s and p, such that kuk 6 Ckuk , Wr,p(U) Ns,p(U) for any u∈ Lp(U).