HIGHER HÖLDER REGULARITY FOR THE FRACTIONAL p LAPLACIAN IN THE SUPERQUADRATIC CASE

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1 HIGHER HÖLDER REGULARITY FOR THE FRACTIONAL LAPLACIAN IN THE SUPERQUADRATIC CASE LORENZO BRASCO ERIK LINDGREN AND ARMIN SCHIKORRA Abstract. We rove higher Hölder regularity for solutions of euations involving the fractional Lalacian of order s when and 0 < s < 1. In articular we rovide an exlicit Hölder exonent for solutions of the non-homogeneous euation with data in L and > N/s which is almost shar whenever s 1+N/. The result is new already for the homogeneous euation. Contents 1. Introduction Overview State of the art 1.3. Main results A glimse of the roof Plan of the aer 6. Preliminaries 7.1. Notation 7.. Tail saces 8.3. Embedding ineualities 9.4. Existence 1 3. Basic regularity estimates Known results Hölder regularity for non-homogeneous euations Almost C s -regularity: homogeneous case 0 5. Imroved Hölder regularity: the homogeneous case Imroved Hölder regularity: the non-homogeneous case 35 Aendix A. Pointwise ineualities 40 References 4 1. Introduction 1.1. Overview. In this aer we study the Hölder regularity of local weak solutions of the nonlinear and nonlocal ellitic euation 1.1 s u = f f L loc. Here < 0 < s < 1 and s is the fractional Lalacian of order s formally defined by s ux uy u x := lim ε 0 R ux uy N \B εx x y N+s dy. The oerator s is a nonlocal or fractional version of the well studied Lalacian oerator u = div u u 010 Mathematics Subject Classification. 35B65 35J70 35R09. Key words and hrases. Fractional Lalacian nonlocal ellitic euations Hölder regularity. 1

2 BRASCO LINDGREN AND SCHIKORRA and has in recent years attracted extensive attention. In weak form this oerator naturally arises as the first variation of the Sobolev-Slobodeckiĭ seminorm for W s R N namely of the nonlocal functional RN ux ux u dx dy. x y N+s RN We are concerned with the higher Hölder regularity of solutions of 1.1. More recisely we rove that if the right-hand side f is in L loc for > N if s N s 1 if s > N then local weak solutions see Definition.9 are locally δ Hölder continuous { 1 for any δ < min s N } 1 =: ΘN s. 1 Remark 1.1 Homogeneous euation. We observe that when = the latter reduces to { } s min 1 1. Thus this is the limit exonent we get for s harmonic functions i.e. local weak solutions of the homogeneous euation s u = 0. To the best of our knowledge this is the first result with an exlicit Hölder exonent even for the homogeneous euation. We oint out that in general the exonent 1 s N 1 is the shar one see Examles 1.5 and 1.6 below. This means that our result is almost shar whenever s 1 and shar when s = 1/ = and =. For a more detailed descrition of our results we refer the reader to Section State of the art. Euations of the tye 1.1 were first considered in [] where viscosity solutions are studied. Existence uniueness and the convergence to the Lalace euation as s goes to 1 are roved. The first ointwise regularity result for euations of this tye is to the best of our knowledge [16] where the local Hölder regularity was roved generalizing the celebrated ideas of De Giorgi to this nonlocal and nonlinear setting. The same authors ursued their investigation in the field by roving Harnack s ineuality for solutions of the homogeneous euation in [17]. There are also many other recent regularity results. In [9] the second author studied the local Hölder regularity using viscosity methods. In [13] the results of [16 17] are generalized to functions belonging to a fractional analogue of the so-called De Giorgi classes we refer to [19 Chater 7] for the classical definition. This can be seen as the nonlocal counterart of the celebrated results by Giauinta and Giusti contained in [18] for minima of local functionals with growth. It is worth ointing out that a ioneering use of fractional De Giorgi classes has been done in [31] in a local context. Finally in [1] Hölder regularity u to the boundary was obtained using barrier arguments. Such a result has been recently enhanced by the same authors in [0] for. Here we seize the oortunity to mention the aer [6] in which regularity for euations of the tye 1.1 with a measure datum f is studied. In articular in [6 Corollary 1.] the authors obtained the remarkable shar result u C 0 whenever f L N s 1 1 loc + 1 N.

3 HIGHER HÖLDER REGULARITY 3 where the latter is a Lorentz sace. There has also been some recent rogress in terms of higher differentiability of the solutions: in [7] and [14] the case = is treated also for a more general kernel Kx y not necessarily given by x y N s while for a general results in this sense have been obtained by the third author in [35 Theorem 1.3] and by the first and second author in [7 Theorem 1.5]. See also the more recent contributions [3 4]. However in none of the above mentioned aers an exlicit Hölder exonent has been obtained. This is tyical when for instance using techniues in the vein of De Giorgi and Moser: one can just rove the existence of some Hölder exonent. Remark 1.. It is worth ointing out that in the linear case i.e. when = the regularity of solutions is well-understood. For instance it follows from the integral reresentation in terms of Poisson kernels that solutions of s u = 0 are C see [8 age 15 formula ]. In addition solutions of s u = f for f L are C s whenever s 1 see for examle [ Theorem 6.4]. Remark 1.3. We also mention that there are other ways of defining fractional versions of the Lalace oerator. This has been done in [5 6 11] [36] and [37]. In [5] and [6] an interesting connection to a nonlocal tug-of-war is found while the oerator introduced in [11] is connected to Lévy rocesses. We oint out that the oerators considered in these aers differ substantially from the one considered in the resent aer Main results. Here we describe our main result. This is valid for local weak solutions of 1.1. However a roer definition of weak solutions is uite technical and we therefore ostone this definition and other technicalities to Section.4. Below is our main theorem: Theorem 1.4. Let Ω R N be a bounded and oen set. Assume < 0 < s < 1 and > N if s N s We define the exonent 1. ΘN s := min 1 if s > N { 1 s N } 1. 1 Let u W s loc Ω L loc Ω L 1 s R N be a local weak solution of s u = f in Ω where f L loc Ω. Then u Cδ loc Ω for every 0 < δ < ΘN s. More recisely for every 0 < δ < ΘN s and every ball B 4R x 0 Ω there exists a constant C = CN s δ > 0 such that [u] C δ B R/8 x 0 C 1 [ u R δ L B R x 0 + R s uy 1 1 dy x 0 y N+s + R s N 1 1 f L B R x 0 R N \B R x 0 We first observe that we ask the solution to be locally bounded. This may seem to be a restriction. However this kind of mild regularity for a weak solution comes for free under the standing assumtions on f see Theorem 3. and Remark 3.3 below. Also the reader may observe that in the theorem above there is the assumtion u Ls 1 R N. This guarantees that the term R N \B R x 0 ] uy 1 dy x 0 y N+s.

4 4 BRASCO LINDGREN AND SCHIKORRA is finite see Section.1 for the recise definition of this sace. It is noteworthy to oint out that u L 1 s R N whenever u is globally bounded or grows slower that x s / 1 at infinity. Before going further and giving the ideas and the details of the roof of Theorem 1.4 we give an examle showing that for every finite satisfying the assumtions of Theorem 1.4 and every 0 < ε 1 there exists u ε W s loc L loc L 1 s such that s u ε L loc u ε C Θ+ε but u ε C. >Θ+ε This shows that in general the exonent Θ found in Theorem 1.4 can not be imroved. Examle 1.5 Sharness of the Hölder exonent <. For N we take < N + 1 and 0 < s 1. Observe that with these choices we automatically get s N. We then fix an exonent N/s < < and observe that { 1 ΘN s = min s N } 1 = 1 s N. 1 1 again thanks to our choices. Finally we consider the function u ε x = x Θ+ε where By construction we have that 0 < ε < 1 N 1. Θ + ε > s and thus u ε C Θ+ε loc R N W s loc RN and Θ + ε 1 < s and thus u ε Ls 1 R N. By using the homogeneity and radial symmetry of u ε and the roerties of the oerator s it is not difficult to see that u ε W s loc RN L loc RN Ls 1 R N is a local weak solution of We observe that f L loc RN N s u = f with fx = C sα x Θ+ε s 1 s. > s Θ + ε 1 Θ + ε > 1 1 and the latter is true thanks to the definition of Θ. We also observe that since in this case u ε C loc RN for any > Θ + ε s N u ε x u ε y u ε x u ε 0 [u ε ] C 0 B R = su xy B R x y su x B R x = su x Θ+ε = +. x B R As for the case = we observe that for s = 1/ the exonent Θ reduces to 1. We now give an examle showing that in general this exonent can not be reached already in the linear case =. Namely for N one can find u such that 1 u L loc u C 01 loc. This the very same reason 1 for the failure of the Calderón-Zygmund estimates for the Lalacian in the borderline case of L u L loc u C 11 loc. 1 Essentially both statements are a conseuence of the fact that the Riesz transforms do not ma L into L.

5 HIGHER HÖLDER REGULARITY 5 Examle 1.6 Sharness of the Hölder exonent s = 1/ = and =. For s = 1/ = and = we have ΘN s = 1 as already noticed. We denote by B R N the unit ball and set for N 3 the following Riesz otential ux := x y 1 N dy for x R N B i.e. the convolution between the singular kernel I 1 x = x 1 N and the characteristic function 1 B. We first observe that by [39 Theorem 1 Chater V] and using the fact that 1 B L 1 R N L R N we have u L m R N for any N N 1 < m. In articular u L R N for N 3. By [34 Theorem 3.] 1/ u coincides u to a multilicative constant with 1 B. However in view of [3 Lemma.15] we find u L Ω = for any bounded domain Ω comactly containing B. In articular u is not a Lischitz function A glimse of the roof. The starting oint is to rove Theorem 1.4 for the homogeneous euation 1.3 s u = 0 and then use a harmonic relacement argument to transfer the regularity to solutions of 1.1 under the standing assumtions on the right-hand side f. The main idea to rove Theorem 1.4 for f 0 is uite simle: we differentiate euation 1.3 in a discrete sense and then test the differentiated euation against monotone ower functions of fractional derivatives of the solution i.e. uantities like δ h ux h ϑ 1 δ h ux h ϑ where δ hux := ux + h ux. By choosing ϑ > 0 and 1 in a suitable way this establishes a recursive gain in integrability see Proosition 4.1 of the tye δ h ux h s δ h ux L +1 B 1/ h s. This can be iterated finitely many times in order to obtain L B 1 δ h ux h s L loc for every < uniformly in h 1 and thus u Cloc δ for any 0 < δ < s. This art of the roof can be considered as a nonlocal counterart of the method based on the Moser s iteration that can be used to rove Lischitz regularity for harmonic functions see for examle [15 Proosition 3.3]: differentiate the euation in discrete sense of order one; test with owers of first order differential uotients; use Sobolev ineuality to get reverse Hölder ineualities and iterate infinitely many times to get δ h ux L h loc uniformly in h 1 i.e. local Lischitz regularity of the solutions Once almost s Hölder regularity is established we are able to refine the estimates used to rove Proosition 4.1 and obtain a more owerful yet more comlicated iterative self-imroving scheme. This is Proosition 5.1 giving an estimate of the tye δh 1.4 su u + 1 δh u 1 + ϑ whenever < 1. 1+s +ϑ h 1+ L + 1 B 1/ h 1+ϑ L B 1

6 6 BRASCO LINDGREN AND SCHIKORRA This art of the roof is uite eculiar of the nonlocal setting and it seems not to have a local counterart. This new scheme allows for an iteration both on the differentiability ϑ and the integrability giving in the end { } s u Cloc α for every α < min 1 1. We oint out that the method can never give better than Lischitz regularity. This is due to the simle fact that we can only test with first order differential uotients. This can also be seen in the reuirement that the order of differentiability 1 + ϑ / on the right-hand side in 1.4 must be less than 1. As announced at the beginning once we roved this regularity for the homogeneous euation we can transfer the regularity to the inhomogeneous euation by uite a standard erturbation argument. However in order not to lose the Hölder exonent in this transfer we need to emloy a blow-u argument similar to that used by Caffarelli and Silvestre in [1] see Proosition 6. below. Noteworthy is that the methods in this aer are uite similar to those in [7] by the first two authors where we only test with δ h ux h ϑ and iterate to obtain higher differentiability results. Remark 1.7 Deendence of the constant. If one would carefully kee track of all the constants one could erhas iterate the revious scheme infinitely many times and arrive at exactly C Θ regularity. Also if great care was taken in estimating the constants the results should be stable as s 1. However we believe that the roofs are comlicated and long enough without exlicitly estimating the s deendence at each ste. About the case s 1 we only oint out that the integrability hyothesis on f of Theorem 1.4 becomes in the limit > N if N f L loc with 1 if > N which is exactly the shar assumtion on f on the scale of Lebesgue saces in order to get Hölder regularity of solutions in the local case i.e. local weak solutions of u = f. We refer to [5 Corollary 1 age 6] for this last result. This coincidence and a careful insection of the results in [5] seem to suggest that one could still rove Theorem 1.4 by slightly weakening the assumtions on f and taking it to belong to some suitable Lorentz or Marcinkiewicz sace. However we refer not to insist on this oint Plan of the aer. In Section we introduce the exedient definitions and notations needed in the aer. We also state and rove some reliminary results such as existence of solutions and functional sace embeddings. In Section 3 we recall some known results and rove a first Hölder regularity result for the non-homogeneous euation. The hard work is carried out in Section 4 where the almost C s regularity is roved for the homogeneous euation. It is based on Proosition 4.1 which exresses a gain of integrability of second order differential uotients. This self-imroving estimate is then iterated to obtain Theorem 4.. In Section 5 we use Theorem 4. to obtain an enhanced version of Proosition 4.1 namely Proosition 5.1 which exresses an interlinked gain between integrability and differentiability of second order differential uotients. The iteration of this then results in Theorem 5. which is nothing but Theorem 1.4 for the homogeneous euation. Finally in Section 6 we use erturbation arguments in order transfer the regularity to the inhomogeneous euation and rove the main theorem in full generality. The Aendix A contains the roof of several crucial ointwise ineualities.

7 HIGHER HÖLDER REGULARITY 7 Acknowledgements. We thank Shigehiro Sakata University of Miyazaki for having drawn our attention on [3 Lemma.15]. Part of this work has been carried out during a visit of L. B. & E. L. to Freiburg in January 017 a visit of L. B. to Stockholm in February 017 and of E. L. to Ferrara in May 017. Hosting institutions are kindly acknowledged. E. L. is suorted by the Swedish Research Council grant no and A. S. is suorted by the German Research Foundation grant no. SCHI and Daimler-Benz Foundation grant no. 3-11/16. A. S. was Heisenberg fellow.. Preliminaries.1. Notation. Let 1 < <. We define the monotone function J : R R by J t = t t. We denote by ω N the measure of the N dimensional oen ball of radius 1. For a measurable function ψ : R N R and a vector h R N we define ψ h x = ψx + h δ h ψx = ψ h x ψx δ hψx = δ h δ h ψx = ψ h x + ψx ψ h x. We recall that we have the discrete Leibniz rule.1 δ h ϕ ψ = ψ h δ h ϕ + ϕ δ h ψ. Let 1 < and let ψ L R N for 0 < 1 we set and for 0 < < We then introduce the two Besov-tye saces N R N = [ψ] N R N := su h >0 [ψ] B R N := su h >0 δ h ψ h δh ψ h L R N L R N { ψ L R N : [ψ] N R N < + } 0 < 1 and { } B R N = ψ L R N : [ψ] B R N < + 0 < <. We also need the Sobolev-Slobodeckiĭ sace W R N = { ψ L R N : [ψ] W R N < + } 0 < < 1 where the seminorm [ ] W R N is defined by RN ψx ψy 1 [ψ] W R N = x y N+ dx dy. RN We endow these saces with the norms and ψ N R N = ψ L R N + [ψ] N R N ψ B R N = ψ L R N + [ψ] B R N ψ W R N = ψ L R N + [ψ] W R N. A few times we will also work with the sace W Ω for a subset Ω R N W Ω = { ψ L Ω : [ψ] W Ω < + } 0 < < 1.

8 8 BRASCO LINDGREN AND SCHIKORRA where we define [ψ] W Ω = Ω Ω ψx ψy 1 x y N+ dx dy. We will also denote the average of a function ψ over the ball B r x 0 by 1 ψ x0r = ψ dx = B rx 0 B r x 0 B rx 0 ψ dx... Tail saces. We introduce the tail sace { RN L αr N = u L u } loc RN : dx < + > 0 and α > x N+α and define for every x 0 R N R > 0 and u L αr N Tail α u; x 0 R = [R α R N \B R x 0 ] 1 u dx. x x 0 N+α It is not difficult to see that the uantity above is always finite for a function u L αr N. Lemma.1. Let α > 0 and 0 < < m <. Then L m α R N L αr N. Proof. This is an easy conseuence of Jensen s ineuality and the fact that the measure 1 + x N+α 1 dx is finite on R N. We leave the details to the reader. The following easy technical results contain comutations that will be used many times. We state them as searate results for ease of readability. Lemma.. Let α > 0 and 0 < <. For every 0 < r < R and x 0 R N we have R α uy N+α R dy Tail x y N+α αu; x 0 R. R r su x B rx 0 R N \B R x 0 Proof. It is sufficient to observe that for x B r x 0 and y R N \ B R x 0 we have x y y x 0 x x 0 y x 0 r y x 0 r R y x 0 = R r R y x 0. This gives the desired conclusion. Lemma.3. Let α > 0 and 0 < <. Suose that B r x 0 B R x 1. Then for every u L αr N we have r α N+α Tail α u; x 0 r R Tail αu; x 1 R + r N u R R x 1 x 0 L B R x. 1 If in addition u L m loc RN for some < m then r α N+α Tail α u; x 0 r R Tail αu; x 1 R R R x 1 x 0. m m m + N ω N r N m u α m + N L m B R x. 1

9 Proof. We first observe that and x x 0 x x 1 x 1 x 0 R x 1 x 0 R Then we decomose the tail as follows Tail α u; x 0 r = r α HIGHER HÖLDER REGULARITY 9 x x 0 r for every x B r x 0. R N \B R x 1 r α R x x 1 for every x B R x 1 u u dx + rα x x 0 N+α B R x 1\B rx 0 x x 0 N+α Tail αu; x 1 R + r N R R x 1 x 0 N+α dx u dx. B R x 1 This gives the first estimate. In order to get. we roceed in a slightly different way to estimate the second integral: by using Hölder ineuality B R x 1\B rx 0 Thus we obtain. as well. u x x 0 N+α dx u L m B R x 1 R N \B rx 0 = N ω N m m m α m + N 1 x x 0 N+α m m dx r α m+n m u L m B R x 1. m m.3. Embedding ineualities. The following result can be found for examle in [10 Lemma.3]. Lemma.4. Let 0 < < 1 and 1 <. Then we have the continuous embedding More recisely for every ψ B R N we have for some constant C = CN > 0. The next one is contained in [10 Proosition.4]. B R N N R N. [ψ] N R N C 1 [ψ] B R N Proosition.5. Let 1 < and 0 < α < < 1. Then we have the continuous embedding More recisely for every ψ N R N we have [ψ] W α R N C for some constant C = CN > 0. N R N W α R N. α α [ψ] α α ψ N R N L R N The following result is a sort of localized version of the revious estimate. This embedding should be seen as a Sobolev-tye embedding. For the embedding roerties of Triebel-Lizorkin and Besov saces the interested reader is referred to e.g. [33 Section.]

10 10 BRASCO LINDGREN AND SCHIKORRA Lemma.6. Suose 1 < and 0 < α < < 1. Let u L loc RN be such that for some h 0 > 0 and some ball B r R N with r > h 0 we have su h < +. L B r Then for every ϱ > 0 such that ϱ + h 0 r we have and su [u] W α B ϱ C 1 δ h u h L B ϱ su C h su L B r h + u L B r L B r + u L B r where B ϱ is concentric with B r. Here C 1 = C 1 N α h 0 > 0 and C = C N h 0 are constants that blow u as 1 α or h 0 0. Proof. We take χ C 0 B ϱ+h0/3 such that 0 χ 1 χ = 1 in B ϱ χ C and D χ C h 0 h. 0 Note that we have δ h χ C h and δ h hχ C 0 h h. 0 We start by observing that.3 [u] W α B ϱ [u χ] W α R N. By using [7 Proosition.7] we obtain [u χ] W α R N C h α 0 α su δ h u χ h where C = CN > 0. We now use [7 Lemma.3] to find.4 [u χ] W α R N C su δh u χ h where 3 C = Ch 0 N α. We now observe that therefore for 0 < h < h 0 δh u χ h.5 L R N L R N L R N δ hu χ = χ h δ hu + δ h u δ h χ h + u δ hχ χ h δh C u h δh C u h δh C u h L B r L B r L R N + δ h u δ h χ h L R N + h α 0 α u χ L R N + u χ L R N + u δh χ h L R N + h 1 0 δ h u L B ϱ+ h0 /3 + h 0 u L B r + u L B r 3 The constant blows-u as 1 α and h0 0.

11 HIGHER HÖLDER REGULARITY 11 where we used the estimates on χ and D χ and the triangle ineuality to estimate the L norm of δ h u. The constant C deends on α N and h 0 as before. By combining.3.4 and.5 we obtain [u] W α B C ϱ su h + u L B r thus concluding the roof of the first ineuality. L B r In order to rove the second ineuality we still use the cut-off function χ and the Leibniz rule.1 for δ h u χ. We may write for 0 < h < h 0 δ h u h δ h u χ L B ϱ h L R N δ h u χ C h + u h δ h χ L R N h L R N δ h u χ C h + u h L B R+ h0 /3 + u L B R+h0 L R N δ h u χ C h + u L B R+h0 L R N where we have used the bound on χ and D χ and [7 Lemma.3]. Finally we aly.5 to treat the term containing δh u χ. In what follows we set N if s < N N s s = + if s > N and s = N if s < N N N + s 1 if s > N.. Proosition.7 Poincaré & Sobolev. Suose 1 < < and 0 < s < 1. Let Ω R N bounded set. For every u W s R N such that u = 0 almost everywhere in R N \ Ω we have u C L s Ω 1 [u] if s < N W s R N be an oen and u L Ω C Ω s N 1 [u] if s > N W s R N u L Ω C 3 Ω + s N 1 [u] W s R N for every 1 < if s = N for constants C 1 = C 1 N s > 0 C = C N s > and C 3 = C 3 N s > 0. In articular we also have.6 u L Ω C Ω s N for some C = CN s > 0. [u] W s R N In what follows we will use the Camanato sace L λ we refer to [19 Chater ] for the relevant definition. Theorem.8 Morrey-tye embedding. Let 1 < and 0 < < 1 be such that > N. If ψ N R N then ψ C 0α loc RN for every 0 < α < N/. More recisely we have ψx ψy α +N su x y x y α C [ψ] N R N ψ L R α N N with C = CN α > 0 which blows-u as α N/.

12 1 BRASCO LINDGREN AND SCHIKORRA Proof. Let us fix a ball B and for every x 0 B we consider the ball B δ x 0 with δ > 0. We are going to show that ψ belongs to the Camanato sace L γ B for every N/ < γ <. By a slight abuse of notation we denote by Then for N/ < γ < we have B B δ x 0 ψ ψ x0δ dx = C C We now recall that from Proosition.5 Finally we observe that Thus we obtain [ψ] W γ R N C ω N δ N ψ x0δ = B B δ x 0 B Bδ x 0 1 B B δ x 0 ψ dx. B B δ x 0 δ N+γ B B δ x 0 δ N+γ B B δ x 0 γ γ [ ψx ψy ] dy B B δ x 0 B B δ x 0 B B δ x 0 B B δ x 0 dx RN RN ψx ψy x y N+γ dx dy. γ γ [ψ] ψ. N R N L R N B B δ x 0 for every x 0 B. su δ γ ψ ψ x0δ dx C x 0 B δ>0 B B δ x 0 γ γ ψx ψy dx dy ψx ψy x y N+γ dx dy γ γ [ψ] ψ. N R N L R N This imlies that ψ L γ B for every ball B R N and every N/ < γ <. By using that L γ B is isomorhic to C 0α B see [19 Theorem.9] with α = γ N/ and [ψ] C 0α B C su δ γ ψ ψ x0δ dx x 0 B δ>0 B B δ x 0 we get the desired conclusion on the ball B. The constant C > 0 above does not deend on the size of the ball B. Thus by the arbitrariness of B we get the conclusion..4. Existence. We start with the following 1 Definition.9 Local weak solution. Suose 1 < < and 0 < s < 1. Let Ω R N bounded set and f L loc Ω with We say that u W s loc.7 Ω L 1 s for every ϕ W s Ω comactly suorted in Ω. s if s N or > 1 if s = N. R N is a local weak solution of s u = f in Ω if RN J ux uy ϕx ϕy x y N+s dx dy = f ϕ dx RN Ω be an oen and

13 HIGHER HÖLDER REGULARITY 13 We now want to detail the notion of weak solutions to a Dirichlet boundary value roblem for s. With this aim we introduce the following sace: given Ω Ω R N oen and bounded sets and g Ls 1 R N we define Xg s Ω Ω := {v W s Ω Ls 1 R N : v = g a. e. in R N \ Ω}. Definition.10 Dirichlet roblem. Suose 1 < < and 0 < s < 1. Let Ω Ω R N be two oen and bounded sets f L Ω with s if s N or > 1 if s = N and g L 1 s R N. We say that u Xg s Ω Ω is a weak solution of the boundary value roblem {.8 s u = f in Ω u = g in R N \ Ω if.7 holds for every ϕ X s 0 Ω Ω. About the sace X s 0 Ω Ω the following simle technical result will be useful. Lemma.11. Let 1 < < and 0 < s < 1. Let Ω Ω R N be two oen and bounded sets. Then for every u X s 0 Ω Ω we have [u] W s R N [u] W s Ω + C s distω RN \ Ω s u L Ω for some C = CN > 0. In articular we have the continuous embedding X s 0 Ω Ω W s R N. Proof. We take u X s 0 Ω Ω and then write [u] W s R N = ux uy dx dy R N R x y N+s N Ω Ω ux uy = dx dy + x y N+s where we used that u 0 outside Ω. It is only left to observe that ux dx dy = u Ω R N \Ω x y N+s Ω u Ω Ω R N \Ω R N \Ω R N \B dx x ux dx dy x y N+s dy x y N+s dx dy x y N+s where dx = distx R N \ Ω. By comuting the last integral and using that d distω R N \ Ω. we get the desired conclusion. By enforcing the assumtions on g a solution to.8 does exist. This is the content of the next result. Proosition.1 Existence. Suose 1 < < and 0 < s < 1. Let Ω Ω R N be two oen and bounded sets f L Ω with s if s N or > 1 if s = N and g W s Ω L 1 s R N. Then roblem.8 admits a uniue weak solution u Xg s Ω Ω. Proof. We can adat the roof of [4 Theorem 1 & Remark 3] concerning the case f = 0. In what follows whenever X is a normed vector sace we denote by X its toological dual. We endow the vector sace X s 0 Ω Ω = {v W s Ω Ls 1 R N : v = 0 a. e. in R N \ Ω} dx

14 14 BRASCO LINDGREN AND SCHIKORRA with the norm of W s Ω. This is a searable reflexive Banach sace. We now introduce the oerator A : Xg s Ω; Ω W s Ω defined by Ω Ω J vx vy ϕx ϕy Av ϕ = x y N+s dx dy J vx gy ϕx + x y N+s dx dy v Xg s Ω; Ω ϕ W s Ω Ω R N \Ω where denotes the relevant duality roduct. We know that Av W s Ω for every v Xg s Ω; Ω see [4 Remark 1]. Moreover we have that A has the following roerties see [4 Lemma 3]: 1 for every v u Xg s Ω Ω we have if {u n } n N X s g Au Av u v 0; Ω Ω converges in W s Ω to u Xg s Ω Ω then lim Au n Au v = 0 for all v W s Ω ; n 3 we have Au Ag u g lim = +. u W s Ω + u g W s Ω Finally we introduce the modified functional A 0 v := Av + g for every v X s 0 Ω Ω. We observe that A 0 : X s 0 Ω; Ω W s Ω and that X s 0 Ω; Ω W s Ω with continuous injection. This imlies that there holds W s Ω X s 0 Ω; Ω as well still with continuous injection. Thus A 0 can be considered as an oerator from X s 0 Ω; Ω to its toological dual. Moreover roerties 1 and 3 above imly that A 0 is monotone coercive and hemicontinuous see [38 Chater II Section ] for the relevant definitions. It is only left to observe that under the standing assumtions the linear functional T f : v Ω f v dx v X s 0 Ω Ω belongs to the toological dual of X s 0 Ω Ω. Indeed for every v X s 0 Ω; Ω we have 4 T f v = f v dx f L Ω v L Ω Ω 1 1 s f L Ω v L s Ω Ω and the last term can be controlled by means of the Sobolev ineuality in Proosition.7 thanks to the fact that X s 0 Ω Ω W s R N by Lemma.11. Then by [38 Corollary.] we get existence of v X s 0 Ω Ω such that A 0 v ϕ = T f ϕ for every ϕ X s 0 Ω Ω. By definition this is euivalent to i.e. Av + g ϕ = T f ϕ for every ϕ X s 0 Ω Ω Ω Ω J vx + gx vy gy ϕx ϕy x y N+s dx dy + Ω R N \Ω J vx + gx gy ϕx x y N+s dx dy = Ω f ϕ dx. 4 We assume for simlicity that s N. The borderline case s = N can be treated in the same manner we leave the details to the reader.

15 HIGHER HÖLDER REGULARITY 15 By observing that v = 0 in R N \ Ω and that J vx + gx gy ϕx Ω R N \Ω x y N+s dx dy J vx + gx vy gy ϕx = Ω R N \Ω x y N+s dx dy J vx + gx vy gy ϕy x y N+s dx dy R N \Ω Ω this is the same as.7. Then v +g is the desired solution. Uniueness now follows from the strict monotonicity of the oerator A 0. Remark.13 Variational solutions. We observe that under the slightly stronger assumtion g W s Ω L s R N existence of the solution to.8 can be obtained by solving the following strictly convex variational roblem min { Fv : v X s g Ω L s R N } where the functional F is defined by Fv = 1 Ω Ω vx vy x y N+s dx dy + Ω R N \Ω vx gy dx dy f v dx. x y N+s Ω Existence of a minimizer can be easily inferred by using the Direct Methods in the Calculus of Variations see for examle [19]. 3. Basic regularity estimates 3.1. Known results. In this section we list some known regularity results for weak solutions. We start with the following imortant result contained in [16 Theorem 1.]. The roof in [16] is erformed under the stronger assumtion that the boundary datum g is in W s R N. However a closer insection of the roof reveals that this is not needed see also [6 Remarks 1.1 &.1]. We use the standard notation osc E ψ = su ψ inf ψ. E E Theorem 3.1 Hölder continuity for s harmonic functions. Let 1 < < 0 < s < 1 and E E R N be oen and bounded sets. Let v Xg s E E be the solution to { s v = 0 in E v = g in R N \ E for some g W s E L 1 s R N. Then there exists α = αn s > 0 such that for every B R x 0 E we have r 3.1 osc Brx 0v C R α v dx B R x 0 for every 0 < r R. In articular [ 3. v v x0r dx C r N r α R B rx 0 Here C = CN s > Tail 1s v; x 0 R v dx + Tail 1s v; x 0 R ]. B R x 0

16 16 BRASCO LINDGREN AND SCHIKORRA Proof. The estimate 3.1 is roved in [16 Theorem 1.]. Here we just show how to get the excess decay estimate 3. from 3.1. We have v v x0r = vx vy dy dx B rx 0 B rx 0 Brx 0 vx vy dy dx B rx 0 B rx 0 B rx 0 oscbrx 0v dx = ωn r N osc Brx 0v. By using 3.1 we get the desired conclusion. A local L -bound for solutions with a right-hand side will be needed. The result below is [9 Theorem 3.8]. Just like for the revious result the stronger assumtion g W s R N stated in [9] is not needed. Theorem 3. Local boundedness. Let 1 < < and 0 < s < 1. Let E E R N be oen and bounded sets. For f L E with > N if s N s 1 if s > N take u Xg s E E to be the weak solution to { s u = f in E u = g in R N \ E for some g W s E L 1 s R N. For every R > 0 such that B R x 0 E and every 0 < σ < 1 the following scaling invariant estimate holds u L B σr x 0 C where C = CN s σ > 0. u dx B R x Tail 1s u; x 0 σr + R s N 1 f L B R x 0 Remark 3.3. From the revious result we get in articular that if f L loc Ω with > N if s N s 1 1 if s > N and u W s loc Ω L 1 s R N is a local weak solution of s u = f in Ω then u L loc Ω. Indeed for every B R x 0 Ω we have that u is the weak solution to { s u = f in B R x 0 u = u in R N \ B R x 0 and the boundary datum which is u itself lies in W s B R x 0 L 1 s R N. Then Theorem 3. imlies u L B σr x 0 for every 0 < σ < 1.

17 HIGHER HÖLDER REGULARITY Hölder regularity for non-homogeneous euations. Here we rove that local weak solutions of s u = f are in C 0 for some 0 < < s rovided that f is integrable enough. We establish the result by transferring the excess decay estimate of Theorem 3.1 from an s harmonic function to the solution u. Lemma 3.4. Let 0 < s < 1 and Ω R N be an oen and bounded set. For f L loc Ω with s if s N or > 1 if s = N we consider a local weak solution u W s loc Ω L 1 s R N of the euation s u = f in Ω. We take a air of concentric balls B B Ω and define v Xu s B B the uniue weak solution of { s v = 0 in B v = u in R N \ B. Then for s N we have 3.3 [u v] W s R N C B and also 3.4 B u v dx C B 1 1 N s N f dx B N s N + s N 1 B f dx for a constant C = CN s > 0. In the case s = N we have the same estimates with N /N s relaced by an arbitrary exonent m < and the constant C deending on m as well. Proof. We first observe that v exists thanks to Proosition.1 since u W s B Ls 1 R N. Subtracting the weak formulations of the euations solved by u and v we get ϕx J ux uy J vx vy ϕy x y N+s dx dy = f ϕ dx R N R N for every ϕ X s 0 B B. With ϕ = u v which is admissible by definition of Xu s B B we use ineuality A.4 to get 5 1 B 1 [u v] W s R N C f u v dx C f dx u v dx B B 1 C B 1 1 s f dx u v L s B B C B 1 1 s f dx B 1 [u v]w s R N where we used the Sobolev ineuality for the sace X s 0 B B see Proosition.7 and Lemma.11. By simlifying a factor [u v] W s R N we obtain 1 [u v] 1 W s R N C B 1 1 s f dx B which in turn gives 3.3. Estimate 3.4 now follows by alying Poincaré s ineuality in 3.3 see again Proosition.7. 5 As usual for simlicity we only treat the case s < N. The case s N can be handled similarly.

18 18 BRASCO LINDGREN AND SCHIKORRA Lemma 3.5 Decay transfer. Let 0 < s < 1 and Ω R N be an oen and bounded set. For f L loc Ω with we consider a local weak solution u W s loc s if s N or > 1 if s = N Ω L 1 s R N of the euation s u = f in Ω. If B 4R x 0 Ω then we have the excess decay estimate N R u u x0r dx C R γ f B rx 0 r L B 4Rx0 ] r α + C [R γ f R L B 4R x + 0 u dx + Tail 1s u; x 0 4R B 4R x 0 for every 0 < r R. Here s + N 3.5 γ := N 1 1 m if s N for an arbitrary < m < if s = N. and C = CN s m > 0. The exonent α is the same as in Theorem 3.1. Proof. As always for simlicity we work with the case s < N. Take a ball B 4R x 0 Ω and define v Xu s B 3R x 0 B 4R x 0 the uniue solution to the Dirichlet boundary value roblem { s v = 0 in B 3R x 0 v = u in R N \ B 3R x 0. We start by observing that u x0r v x0r = thus Brx 0 [u v] dx u v dx B rx 0 u u x0r dx C B rx 0 + C u v dx + C B rx 0 v v x0r dx B rx 0 u x0r v x0r dx B rx 0 C u v dx + C B rx 0 v v x0r dx B rx 0 where C = C > 0. By using Lemma 3.4 and Theorem 3.1 for v with E = B 3R x 0 and E = B 4R x 0 we obtain for 0 < r R u u x0r dx C B rx 0 + C N R R s +N 1 s f dx r B 3R x 0 [ r α R v dx + Tail 1s v; x 0 R B R x 0 ]

19 HIGHER HÖLDER REGULARITY 19 with C = CN s. For the second term we use Lemma.3 with x 1 = x 0 and deduce Tail 1s v; x 0 R C Tail 1s u; x 0 4R + v dx. B 4R x 0 Finally we use Lemma 3.4 again and obtain v dx C B 4R x 0 v u dx + C B 4R x 0 C R s +N 1 s Here C = CN s. This concludes the roof. B 4R x 0 u dx B 4R x 0 f dx We can now obtain Hölder regularity of solutions to the non-homogeneous euation. Theorem 3.6. Let and 0 < s < 1. For f L loc Ω with { } N > max s 1 if s N 1 if s > N we consider a local weak solution u W s loc Ω L 1 s R N of the euation s u = f in Ω. + C u dx. B 4R x 0 We set γ = N + α + γ α where γ is as in 3.5 and α as in Theorem 3.1. Then u C 0 loc Ω. More recisely for every ball B R0 z Ω we have the estimate [ ] [u] C 0 B R0 z C f L B R1 z + u L B R1 z + Tail 1s u; z R where R 1 = R 0 + distb R 0 z Ω. The constant C deends only on N s R 0 and distb R0 z Ω. Proof. Take a ball B R0 z Ω and set d = distb R0 z Ω > 0 and R 1 = d + R 0. Choose a oint x 0 B R0 z and consider the ball B 4R x 0 with R < min{1 d/8}. We observe that > s. If 6 s N we may aly Lemma 3.5 and obtain N R u u x0r dx C R γ f B rx 0 r L B 4Rx0 ] r α + C [R γ f R L B 4R x + 0 u dx + Tail 1s u; x 0 4R B 4R x 0 N R C R γ f r L B R1 z r α + C [d γ f R L B R1 z + u L B R1 z + Tail 1s u; x 0 4R ] 6 For the conformal case s = N it is sufficient to reroduce the roof by using Lemma 3.5 with an exonent m >.

20 0 BRASCO LINDGREN AND SCHIKORRA for every 0 < r R < min{1 d/8}. Here we used that u L loc Ω thanks to the hyothesis on f and Theorem 3.. For the tail term we can use. of Lemma.3 with m = with B 4R x 0 B R1 z then for R < min{1 δ/8} we have s N+s Tail 1s u; x 0 4R 1 4 R 4 R Tail 1s u; z R C u 1 R 1 x 0 z L B R1 z R 1 In the second estimate we used that In conclusion 4 R R 1 < Tail 1s u; z R C u 1 L B R1 z. d R 0 + d u u x0r dx C B rx 0 + C < 1 and 4 R R 1 x 0 z 4 R < 1. R 1 R 0 N R R γ f r L B R1 z r α [d γ f R L B R1 z + u L B R1 z + Tail 1s u; z R 1 ] ossibly with a different constant C = CN s > 0. Observe that thanks to the hyothesis on we have γ := s + N 1 > 0. s Then we make the choice r = R θ with θ = 1 + γ N + α. For every 0 < r min{1 d/8 θ } and every x 0 B R0 z we obtain r u u x0r dx C [d γ + 1 f B rx 0 B R0 z L B R1 z + u L B R1 z + Tail 1s u; z R 1 ] where = γ α N + α + γ > 0. This shows that u belongs to the Camanato sace L N+ B R0 z which is isomorhic to C 0 B R0 z. This imlies the desired conclusion. 4. Almost C s -regularity: homogeneous case In what follows we use the notation B R for the N dimensional oen ball of radius R centered at the origin. The cornerstone of our main result is the following integrability gain for the s derivative of an s harmonic function. δ h u h s Proosition 4.1. Assume and 0 < s < 1. Let u W s loc B L 1 s R N be a local weak solution of s u = 0 in B. Suose that RN 4.1 u L B 1 1 and Tail 1s u; uy 1 = dy 1 \B1 y N+s

21 and that for some and 0 < h 0 < 1/10 we have su h s HIGHER HÖLDER REGULARITY 1 Then for every radius 4 h 0 < R 1 5 h 0 we have su +1 h s C L +1 B R 4 h0 Here C = CN s h 0 > 0 and C + as h 0 0. L B 1 su < +. h s L B R+4 h0 Proof. We first observe that u L loc B thanks to Theorem 3. and Remark 3.3. Thus the first hyothesis does make sense. We divide the roof into five stes. Ste 1: Discrete differentiation of the euation. For notational simlicity set dx dy r = R 4 h 0 and dµx y = x y N+s. Take a test function ϕ W s B R vanishing outside B R+r/. By testing.7 with ϕ h for h R N \ {0} with h < h 0 and then changing variables we get 4. J u h x u h y ϕx ϕy dµx y = 0. R N R N We subtract.7 from 4. and divide by h > 0 to get J u h x u h y J ux uy 4.3 h R N R N + 1 ϕx ϕy dµ = 0 for every ϕ W s B R vanishing outside B R+r/. Let 1 and ϑ > 0 and use 4.3 with the test function uh u ϕ = J +1 h ϑ η 0 < h < h 0 where η is a non-negative standard Lischitz cut-off function such that η 1 on B r η 0 on R N \ B R+r/. η C R r = C. 4 h 0 Note that these assumtions imlies δ h η h C. h 0 We get J u h x u h y J ux uy R N R N h 1+ϑ J +1 u h x ux ηx J +1 u h y uy ηy dµ = 0. The double integral is now divided into three ieces: J u h x u h y J ux uy I 1 := I := B R B R B R+r R N \B R h 1+ϑ J +1 u h x ux ηx J +1 u h y uy ηy dµ J u h x u h y J ux uy J +1 u h x ux ηx dµ h 1+ϑ.

22 BRASCO LINDGREN AND SCHIKORRA and J u h x u h y J ux uy I 3 := R N \B R B R+r h 1+ϑ J +1 u h y uy ηy dµ where we used that η vanishes identically outside B R+r/. We will estimate I 1 in what follows. We start by observing that J +1 u h x ux ηx J +1 u h y uy ηy J +1 u h x ux J +1 u h y uy = ηx + ηy J +1 u h x ux + J +1 u h y uy + ηx ηy. Thus J u h x u h y J ux uy J u h x ux ηx J u h y uy ηy J u h x u h y J ux uy ηx + ηy J +1 u h x ux J +1 u h y uy J u h x u h y J ux uy u h x ux + u h y uy ηx ηy.

23 HIGHER HÖLDER REGULARITY 3 The first term in the last exression has a ositive sign. For the negative term we roceed like this: we use A. Young s ineuality and then A.7 J u h x u h y J ux uy u h x ux + u h y uy ηx ηy 1 u h x u h y + ux uy u h x u h y u h x u h y ux uy ux uy u h x ux + u h y uy ηx + ηy ηx ηy C u h x u h y + ux uy ε u h x ux +1 + u h y uy +1 ηx ηy + C ε u h x u h y u h x u h y ux uy ux uy u h x ux 1 + u h y uy 1 ηx + ηy C u h x u h y + ux uy ε u h x ux +1 + u h y uy +1 ηx ηy + C ε J u h x u h y J ux uy J +1 u h x ux J +1 u h y uy ηx + ηy where C = C > 0 and ε > 0 is arbitrary. By utting all the estimates together and choosing ε sufficiently small we then get for a different C = C > 0 I 1 1 J u h x u h y J ux uy C B R B R h 1+ϑ J +1 u h x ux J +1 u h y uy ηx + ηy dµ C u h x u h y + ux uy ηx ηy B R B R u hx ux +1 + u h y uy +1 h 1+ϑ We can further estimate the ositive term by using A.5. This leads us to δ h ux 1 δ h ux I 1 c δ huy 1 δ h uy ηx + ηy dµ B R B R h 1+ϑ h 1+ϑ C u h x u h y + ux uy ηx ηy δ h ux +1 + δ h uy +1 dµ B R B R where c = c > 0 and C = C > 0. We now observe that if we set for simlicity A = δ hux 1 h 1+ϑ δ h ux and dµ. B = δ huy 1 h 1+ϑ δ h uy h 1+ϑ

24 4 BRASCO LINDGREN AND SCHIKORRA and then use the convexity of τ τ we have A ηx B ηy = ηx + ηy A B + A + B We thus get the following lower bound for I 1 ] [ δ h u 1 δ h u I 1 c η 4.4 C h 1+ϑ B R B R 1 A B ηx + ηy + 1 A + B ηx ηy A B ηx + ηy + A + B ηx ηy. W s B R ηx ηy u h x u h y + ux uy ηx ηy u hx ux +1 + u h y uy +1 h 1+ϑ dµ δh ux 1+ C h 1+ϑ + δ huy 1+ h 1+ϑ ηx ηy dµ B R B R where c = c > 0 and C = C > 0. By recalling that I 1 + I + I 3 = 0 and using the estimate for I 1 we arrive at ] [ δ h u 1 where we set 4.5 I 11 := and B R B R h 1+ϑ δ h u η W s B R C I 11 + I 1 + I + I 3 for C = C > 0 u h x u h y + ux uy ηx ηy δ h ux +1 + δ h uy +1 dµ I 1 := B R B R δh ux 1+ h 1+ϑ + δ huy 1+ h 1+ϑ ηx ηy dµ. Ste : Estimates of the local terms I 11 and I 1. We first treat I 11 by estimating ux uy ηx ηy δ h ux +1 B R B R h 1+ϑ dµ. h 1+ϑ The other terms of I 11 can be dealt with similarly. By using that η is Lischitz and that we get ux uy ηx ηy δ h ux +1 B R B R h 1+ϑ dµ C h 0 ux uy BR BR δ h ux +1 x y N+s h 1+ϑ dx dy.

25 For = the last term reduces to 1 δ h ux +1 B R B R x y N+ s h 1+ϑ dx dy = for some C = CN s > 0. For > we take instead 4.6 ε = min HIGHER HÖLDER REGULARITY 5 B R B R δ h ux C BR +1 dx dy δh ux +1 x y N+ s h 1+ϑ dx h 1+ϑ δ h ux C u L B R+h0 BR BR dx C δ h ux dx { 1 } s 1 > 0 h 1+ϑ then by Young s ineuality with exonents / and / + we have 7 ux uy BR BR δ h ux +1 x y N+s h 1+ϑ dx dy C [u] W C [u] W C [u] W C s ε + BR h 0 s ε + BR+h0 s ε BR+h0 + C h 0 + C u x y s ε s + B R B R + + s ε s R s ε s + + L B R+h0 B R δ h u h x h + 1+ϑ + h 1+ϑ N δh ux +1 + dy 1+ϑ h + dx B R +1 δ h u h x + h 1+ϑ + where C = CN h 0 s > 0. We also used that. Observe that the choice 4.6 of ε assures that In order to estimate the term s ε s + [u] W we use the Sobolev embedding namely Lemma.6 with α = s > 0. s ε BR+h0 ε = s r = R + 4 h 0 ϱ = R + h 0. By further using that u is locally bounded more recisely by 4.1 and since R + 4h 0 1 we get [u] s ε C su W BR+h0 h s + 1. Therefore for we get I 11 C where C = CN h 0 s > 0. B R δ h ux h + 1+ϑ + dx + su L B R+4 h0 h s L B R+4h0 + 1 dx dx 7 We realize that the second term of I11 can be estimated by exactly the same uantity excet that B R is relaced by B R+h0 in the first term. This is the reason why we changed B R to B R+h0 above so that both terms in I 11 enjoy the same estimate.

26 6 BRASCO LINDGREN AND SCHIKORRA 4.7 As for I 1 we have BR BR δ h ux 1+ h 1+ϑ ηx ηy dx dy C x y N+s C C δ h ux BR 1+ dx B R h 1+ϑ δ h ux 1+ϑ h δ h u B R h 1+ϑ dx + δ h ux 1 B R dx + 1 where we used the Lischitz character of η treated the integral in y as we did above used the fact that 1 + Young s ineuality with exonents 8 / and / + and that u is bounded. Here C = CN h 0 s > 0. Thus we obtain I 1 C δ h u B R h 1+ϑ + + dx + 1 with C = CN h 0 s > 0. If we now use these estimates in 4.4 we get ] [ δ h u 1 δ h u η C δ h u + dx + su h 1+ϑ W s B B R R h 1+ϑ h s C I + I 3 with C = Ch 0 N s > 0. L B R+4h0 + 1 Ste 3: Estimates of the nonlocal terms I and I 3. Both nonlocal terms I and I 3 can be treated in the same way. We only estimate I for simlicity. Since u 1 in B 1 we have J u h x u h y J ux uy J +1 δ h ux C 1 + u h y 1 + uy 1 δ h ux where C = C > 0. For x B R+r/ we have B R r/ x B R and thus 1 dy R N \B R x y N+s R N \B R r x 1 x y N+s dy CN h 0 s by recalling that R r = 4 h 0. By using Lemma. we get for x B R+r/ and then Lemma.3 RN uy 1 N+s R RN uy 1 dy \BR x y N+s R r \BR y N+s dy N+s R RN uy 1 N+s R dy + R N u 1 dy R r \B1 y N+s R r B 1 CN h 0 s. In the last estimate we have used the bounds assumed on u and 4 h 0 < R 1. The term involving u h can be estimated similarly. Recall also that η = 0 outside B R+r/. Hence we have δ h u δ h u I + I 3 C dx C 1 + h 1+ϑ dx B R+r B R h 1+ϑ dx 8 Only in the case > for = it is not needed we simly estimate δ h ux +1 u L B R+h0 δ h ux.

27 HIGHER HÖLDER REGULARITY 7 by Young s ineuality. Here C = CN h 0 s > 0 as always. By inserting this estimate in 4.8 we obtain ] [ δ h u 1 δ h u η C δ h u + dx + su 4.10 h s + 1 h 1+ϑ W s B R B R h 1+ϑ L B R+4h0 Ste 4: Going back to the euation. For ξ h R N \ {0} such that h ξ < h 0 we use ineuality A.3 with the choices A = ux + h + ξ ux + ξ B = ux + h ux γ = 1 + to arrive at 4.11 ξ δ ξ δ h u s 1+ϑ 1+ h L 1+ B r C δ ξ δ h u 1 δ h u ξ s h 1+ϑ L B r C η δ ξ δ h u 1 δ h u ξ s h 1+ϑ where C = C > 0. Here we used that η 1 on B r. Now by the discrete Leibniz rule.1 we write η δ ξ δ h u 1 δ h u = δ ξ η δ h u 1 δ h u δ h u 1 δ h u δ ξ η and thus find 4.1 δ ξ δ h u s ξ 1+ϑ 1+ h L 1+ B r δ ξ C ξ s δ h u 1 h 1+ϑ δ h u η L R N δ ξ η + C ξ s ξ δ h u 1 δ h u h 1+ϑ where C = C > 0. For the first term in 4.1 we aly [7 Proosition.6] with the choice and get δ ξ δ h u 4.13 su ξ s ξ >0 1 h 1+ϑ δ h u η ψ = δ hu 1 L R N h 1+ϑ δ h u η C 1 s [ δ h u 1 h 1+ϑ where C = CN h 0 > 0. Here we also used that R+r + h 0 = R. As for the second term in 4.1 we observe that for every 0 < ξ < h δ ξ η ξ s δ h u 1 δ h u h 1+ϑ ξ L R N C C δ h u 1 δ h u h 1+ϑ B R+r +h 0 δ h u C BR dx h 1+ϑ C δ h u B R h 1+ϑ δ h u η ξ δ h u 1+ h 1+ϑ dx + ] W s B R L B R+r +h 0 dx + 1 ξ L R N L R N where C = CN h 0 s > 0. Here we have used the estimates of η the fact that u is bounded and Young s ineuality with exonents / and / +.

28 8 BRASCO LINDGREN AND SCHIKORRA From and 4.14 we get for any 0 < ξ < h 0 1+ δ ξ δ h u [ δ h u 1 δ h u 4.15 s 1+ϑ C η ξ 1+ h 1+ h 1+ϑ L 1+ B r ] W s BR + C B R δ h u h 1+ϑ + dx + 1 with C = CN h 0 s > 0. We then choose ξ = h and take the suremum over h for 0 < h < h 0. Then 4.15 together with 4.10 imly su B r δ h u 1+s +ϑ h 1+ dx C su B R δ h u h 1+ϑ dx + su h s L B R+4h0 + 1 where C = CN h 0 s > 0. By observing that 1 + ϑ / = s < 1 we can use the second estimate of Lemma.6 and relace the first order difference uotients in the right-hand side of 4.16 with second order ones. This gives su B r δ h u 1+s +ϑ h 1+ dx C for some constant C = CN h 0 s > 0. su δ h u h 1+ϑ L + B R+4h0 Ste 5: Conclusion. We now secify our choices for and ϑ. We set 4.18 = + and ϑ = In articular we obtain and Then 4.17 becomes su h s 1 + s + ϑ = s +1 L +1 B r 1 + ϑ C = s s = s. + su + s = su h s L B R+4h0 where C = CN h 0 s > 0. In articular u to modifying the constant C we obtain su +1 h s C su h s + 1 L +1 B R 4 h0 as desired where we recall that r = R 4 h 0. L B R+4h0 + 1 h s L B R+4 h0 + 1 By iterating the revious result we can obtain the following regularity result for s harmonic functions. This is the main outcome of this section. Theorem 4. Almost C s regularity for s harmonic functions.. Let Ω R N be a bounded and oen set and 0 < s < 1. Suose u W s loc Ω L 1 s R N is a local weak solution of Then u Cloc δ Ω for every 0 < δ < s. s u = 0 in Ω.

29 HIGHER HÖLDER REGULARITY 9 More recisely for every 0 < δ < s and every ball B R x 0 Ω there exists a constant C = CN s δ > 0 such that we have the scaling invariant estimate 4.19 [u] Cδ B R/ x 0 C u R δ L B R x 0 + R s N [u]w s B R x 0 + Tail 1s u; x 0 R. Proof. We first observe that u L loc Ω by Theorem 3. and Remark 3.3. We assume for simlicity that x 0 = 0 then we set M R = u L B R + R s N [u]w s B R + Tail 1s u; 0 R > 0. We oint out that it is sufficient to rove that the rescaled function satifies the estimate u R x := 1 M R ur x for x B [u R ] Cδ B 1/ C. By scaling back we would get the desired estimate. Observe that by definition the function u R is a local weak solution of s u = 0 in B and satisfies 4.0 u R L B 1 1 RN \B1 u R y 1 y N+s dy 1 [u R ] W s B 1 1. In what follows we forget the subscrit R and simly write u in lace of u R in order not to overburden the resentation. We fix 0 < δ < s and choose i N \ {0} such that Then we define the seuence of exonents We define also We note that By alying Proosition 4.1 with 9 s δ > N + i. i = + i i = 0... i. h 0 = 1 64 i R i = i + 1 h 0 = 7 8 i i for i = 0... i. R h 0 = 7 8 and R i 1 4 h 0 = 3 4. R = R i and = i = + i for i = 0... i 1 and observing that R i 4 h 0 = R i h 0 we obtain the iterative scheme of ineualities su δh h s C su u L 1 BR1 +4h 0 h s + 1 L B 7/8 su h h 0 h s L i +1 B Ri+1 +4h 0 C su h s + 1 for i = 1... i L i BRi +4h 0 9 We observe that by construction we have Thus these choices are admissible in Proosition h 0 < R i 1 5 h 0 for i = 0... i 1.

30 30 BRASCO LINDGREN AND SCHIKORRA and finally su h s = su L i B3/4 h s C L +i B Ri 1 4h 0 su h s + 1. L +i 1 B Ri 1 +4h 0 Here C = CN δ s > 0 as always. We note that by [7 Proosition.6] together with the relation δ h u = 1 δh u δ hu we have 4.1 su h s C su δ h u L B 7/8 0< h < h 0 h s L B 7/8 C [u] W s B 7/8+ h0 + u L B 7/8+ h0 C [u] W s B 1 + u L B 1 CN δ s where we also have used the assumtions 4.0 on u. Hence the iterative scheme of ineualities leads us to 4. su h s CN δ s. L i B3/4 Take now χ C 0 B 5/8 such that In articular we have for all h > 0 We also recall that 0 χ 1 χ = 1 in B 1/ χ C D χ C. δ h χ h s C δ h χ h s C. δ hu χ = χ h δ hu + δ h u δ h χ h + u δ hχ. Hence for 0 < h < h 0 [u χ] s B i R N = δh u χ h s C L i RN 4.3 C C χ h h s + δ h u δ h χ L i RN h s + u δh χ L i RN h s L i RN h s + δ h u L i B5/8+h0 + u L i B5/8+h0 L i B5/8+ h0 h s + u L i B3/4 CN δ s L i B3/4 by 4.. By Lemma [u χ] N s i Finally by noting that thanks to the choice of i we have R N CN δ s [u χ] B s i CN δ s. R N s i > N and δ < s N i we may aly Theorem.8 with = s α = δ and = i to obtain δ i +N s i s δ i N [u] Cδ B 1/ = [u χ] Cδ B 1/ C [u χ] s N i R N u χ L R N s i This concludes the roof. CN δ s.

31 HIGHER HÖLDER REGULARITY 31 Remark 4.3. Under the assumtions of the revious theorem a covering argument combined with 4.19 imlies the more flexible estimate: for every 0 < σ < 1 [u] C δ B σr x 0 C u R δ L B R x 0 + R s N [u]w s B R x 0 + Tail 1s u; x 0 R with C now deending on σ as well and blowing-u as σ 1. Indeed if σ 1/ then this is immediate. If 1/ < σ < 1 then we can cover B σ R x 0 with a finite number of balls B r/ x 1... B r/ x k where x i B R/ x 0 and r = R. By using 4.19 on every ball B r x i B R x 0 Ω we get [u] Cδ B r/ x i C u r δ L B rx i + r s N [u]w s B rx i + Tail 1s u; x i r. By observing that B r x i B R x 0 summing over i and using Lemma.3 for the tail term we get the desired conclusion. 5. Imroved Hölder regularity: the homogeneous case Now that we know that a solution to the homogeneous euation is locally Hölder δ continuous for any 0 < δ < s we can obtain the following imrovement of Proosition 4.1 which rovided an increased integrability estimate. In contrast the following roosition also increases the differentiability. Proosition 5.1. Assume and 0 < s < 1. Let u W s loc B Ls 1 R N be a local weak solution of s u = 0 in B. Suose that u L B 1 1 and RN uy 1 dy 1. \B1 y N+s Assume further that for some 0 < h 0 < 1 0 < ϑ < 1 and > 1 such that 1 + ϑ / < 1 we have δh su u < +. 0< h h 0 Then for every radius 4 h 0 < R 1 5 h 0 we have δh su u 1+ 1+s +ϑ h 1+ Here C deends on the N h 0 s and. L 1+ B R 4 h0 h 1+ϑ C L B 1 su δ h u h 1+ϑ L B R+4 h Proof. We will go back to the estimates in the roof of Proosition 4.1. The only difference is that we estimate the term I 11 defined in 4.5 differently. From Theorem 4. and Remark 4.3 we can choose { 0 < ε < min s 1 s } such that [u] C s ε B R CN h 0 s. Using this together with the assumed regularity of η we have ux uy ηx ηy x y N+s C x y N+ 1 s ε. Thanks to the choice of ε the last exonent is strictly less than N and we may conclude ux uy BR ηx ηy x y N+s dy CN h 0 s