Regularity for fully nonlinear integro-differential operators with regularly varying kernels
aa r X i v : . [ m a t h . A P ] A ug REGULARITY FOR FULLY NONLINEAR INTEGRO-DIFFERENTIALOPERATORS WITH REGULARLY VARYING KERNELS
SOOJUNG KIM, YONG-CHEOL KIM, AND KI-AHM LEEA bstract . In this paper, the regularity results for the integro-di ff erential operators of thefractional Laplacian type by Ca ff arelli and Silvestre [CS1] are extended to those for theintegro-di ff erential operators associated with symmetric, regularly varying kernels at zero.In particular, we obtain the uniform Harnack inequality and H¨older estimate of viscositysolutions to the nonlinear integro-di ff erential equations associated with the kernels K σ,β satisfying K σ,β ( y ) ≍ − σ | y | n + σ log 2 | y | ! β (2 − σ ) near zerowith respect to σ ∈ (0 ,
2) close to 2 (for a given β ∈ R ), where the regularity estimates donot blow up as the order σ ∈ (0 ,
2) tends to 2 . C ontents
1. Introduction 11.1. Introduction 11.2. Integro-di ff erential operators 31.3. Main results 62. Viscosity solutions 93. Regularity estimates for integro-di ff erential operators with regularly varyingkernels 123.1. Aleksandrov-Bakelman-Pucci type estimate 143.2. Barrier function 163.3. Power decay estimate of super-level sets 193.4. Harnack inequality 223.5. H¨older continuity 253.6. C ,α estimate 253.7. Truncated kernels at infinity 264. Uniform regularity estimates for certain integro-di ff erential operators as σ → − ntroduction Introduction.
In this paper, we are concerned with fully nonlinear elliptic integro-di ff erential operators associated with symmetric, regularly varying kernels at zero. Fromthe L´evy-Khinchine formula, the purely jump processes which allow particles to interact at large scales are generated by the integral operators in the form of(1) L u ( x ) = P . V . Z R n (cid:8) u ( x + y ) − u ( x ) − ( ∇ u ( x ) · y ) χ B (0) ( y ) (cid:9) d m ( y ) , where a so-called L´evy measure m satisfies Z R n min(1 , | y | ) d m ( y ) < + ∞ . Since the operators are given in too much generality, we therefore restrict ourselves toconsidering only the operators given by symmetric kernels. In this case, the operator (1)can be written as(2) L u ( x ) = Z R n { u ( x + y ) + u ( x − y ) − u ( x ) } K ( y ) dy , where a symmetric L´evy measure m in (1) is given by a symmetric kernel K ( y ) = K ( − y ) . We note that the value of L u ( x ) is well-defined when u is bounded in R n and C , at x (seeDefinition 2.1). Nonlinear integro-di ff erential operators associated with the linear integro-di ff erential operators above arise naturally in the study of the stochastic control theoryrelated to I u ( x ) = sup α L α u ( x ) , and game theory associated with I u ( x ) = inf β sup α L αβ u ( x ) . To study uniform regularity for such nonlinear integro-di ff erential operators, the concept ofellipticity for integro-di ff erential operators with respect to a class L of the linear, integro-di ff erential operators (2) was introduced by Ca ff arelli and Silvestre [CS1]; see [CC] forelliptic second-order di ff erential operators. In fact, the concept of ellipticity for integro-di ff erential operators I is characterized by the following property:inf L∈ L L v ( x ) ≤ I [ u + v ]( x ) − I u ( x ) ≤ sup L∈ L L v ( x ) . On the basis of this idea, the regularity theory for fully nonlinear elliptic integro-di ff erentialoperators has been developed by using analytic techniques along the lines of the Krylov andSafonov [KS, CC] which dealt with elliptic second-order di ff erential operators. We referto [CS1, CS2] and references therein for uniform regularity results for symmetric integro-di ff erential operators of the fractional Laplacian type, where the regularity estimates do notblow up as the order σ ∈ (0 ,
2) of the operators tends to 2. In the case when the kernels arenonsymmetric, the uniform regularity results can be found in [KL1, KL2, LD1]. We referto [KL3, LD2] for results on the regularity of the parabolic integro-di ff eretial operators.In this paper, we establish the uniform regularity of viscosity solutions to fully nonlinearelliptic integro-di ff erential equations associated with symmetric, regularly varying kernelsat zero. We are mainly interested in the kernels K for the integro-di ff erential operator (2)satisfying(3) Z R n min(1 , | y | ) K ( y ) dy < + ∞ and(4) (2 − σ ) λ l ( | y | ) | y | n ≤ K ( y ) ≤ (2 − σ ) Λ l ( | y | ) | y | n , < λ ≤ Λ < + ∞ , NTEGRO-DIFFERENTIAL OPERATORS WITH REGULARLY VARYING KERNELS 3 where l : (0 , + ∞ ) → (0 , + ∞ ) is a locally bounded, regularly varying function at zerowith index − σ ∈ ( − , ff erential operators of this type with probabilistic proofs.In particular, Kassmann and Mimica [KM] recently obtained H¨older type estimates forthe linear integro-di ff erential operators with regularly varying kernels at zero with index − σ ∈ [ − ,
0] based on intrinsic scaling properties; the H¨older type estimates blow upas the order of the operator approaches 2. As an extension of the regularity results byCa ff arelli and Silvestre [CS1], we obtain uniform regularity results of viscosity solutionsfor a certain class of fully nonlinear elliptic integro-di ff erential operators associated withsymmetric, regularly varying kernels at zero, which remain uniform as the order σ ∈ (0 , Integro-di ff erential operators. As mentioned above, we study uniform regularity ofviscosity solutions for a class of the fully nonlinear elliptic integro-di ff erential operatorsassociated with the kernels of the type: K ( y ) ≍ (2 − σ ) l ( | y | ) | y | n for a regularly varying function l at zero with index − σ ∈ ( − , l : (0 , + ∞ ) → (0 , + ∞ ) which stayslocally bounded away from 0 and + ∞ , will be commonly assumed to satisfy the followingproperties. Property 1.1.
Let a measurable function l : (0 , → (0 , + ∞ ) be locally bounded awayfrom and + ∞ . There exist positive constants σ ∈ (0 , , a ≥ , and ρ ∈ (0 , satisfyingthe following.(a) There exists δ ∈ h , min(2 − σ, σ ) (cid:17) ⊂ [0 , such thatl ( s ) l ( r ) ≤ a max ((cid:18) sr (cid:19) − σ + δ , (cid:18) sr (cid:19) − σ − δ ) for r , s ∈ (0 , . (b) Define L ( r ) : = σ Z r l ( s ) s ds . Then we have for any r ∈ (0 , ρ ) 12 ≤ L ( r ) l ( r ) ≤ . (c) We assume that l (1) = . Influenced by Kassmann and Mimica [KM], we introduce the monotone function L above defined by using the given function l in order to study scale invariant regularity es-timates for the integro-di ff erential operators associated with symmetric, regularly varyingkernels at zero.The function l at infinity will be commonly assumed to satisfy the following property. SOOJUNG KIM, YONG-CHEOL KIM, AND KI-AHM LEE
Property 1.2.
Let a measurable function l : (0 , + ∞ ) → (0 , + ∞ ) be locally bounded awayfrom and + ∞ , and satisfy Property 1.1. There exists a positive constant a ∞ ≥ such thatfor some δ ′ ∈ h , min(2 − σ, σ ) (cid:17) ⊂ [0 , l ( s ) l ( r ) ≤ a ∞ max ((cid:18) sr (cid:19) − σ + δ ′ , (cid:18) sr (cid:19) − σ − δ ′ ) for r , s ∈ [1 , + ∞ ) . Typical examples of the functions satisfying Property 1.1 are regularly varying functionsat zero with index − σ ∈ ( − , , and Property 1.2 is satisfied by assuming that the function l varies regularly at infinity with index − σ ∈ ( − , Example 1.3 (Regularly varying functions) . (a) Trivial examples of such regularly vary-ing functions at zero and infinity with index − σ ∈ ( − , are (2 − σ ) r − σ . The operator (2) with the choice above of the regularly varying function turns out tobe the well-known fractional Laplacian operator − ( − ∆ ) σ/ defined as − ( − ∆ ) σ/ u ( x ) : = (2 − σ ) Z R n u ( x + y ) + u ( x − y ) − u ( x ) | y | n + σ dy , which converges to the Laplacian operator as the order σ ∈ (0 , approaches . Wenote that the factor (2 − σ ) enables us to obtain second-order di ff erential operators asthe limits of integro-di ff erential operators (see [DPV, CS1] , for example) and henceuniform regularity results as the order σ ∈ (0 , goes to the classical one.(b) Among nontrivial examples of regularly varying functions l at zero with index − σ arefunctions which are equal to the following functions near zero (see [BGT] ):r − σ log 2 r ! β , r − σ log 2 r ! β , and r − σ log log 2 r ! β for β ∈ R . (c) The following functions are non-logarithmic regularly varying functions l at zero withindex − σ : r − σ exp log 2 r ! β for β ∈ (0 , ,and r − σ exp log 2 r , log log 2 r ! . For a certain class of regularly varying functions, the constants a ≥ , ρ ∈ (0 ,
1) and a ∞ in Properties 1.1 and 1.2 can be selected uniformly. Let σ ∈ (0 , , and let a locallybounded function l : (0 , + ∞ ) → (0 , + ∞ ) be a slowly varying function at zero and infinitywhich varies regularly at zero and infinity with index 0 from the definition; see AppendixA. For σ ∈ [ σ , , define a regularly varying function l σ at zero and infinity with index − σ ∈ ( − , − σ ] by(5) l σ ( r ) : = r − σ l ( r ) − σ , ∀ r > . Making use of a theory of regular variations, we shall prove in Proposition 4.1 that thefunction l σ satisfies Properties 1.1 and 1.2 with uniform constants a , a ∞ ≥ ρ ∈ (0 , σ ∈ [ σ , , where the uniform constants depend only on dimension n , σ ∈ (0 , NTEGRO-DIFFERENTIAL OPERATORS WITH REGULARLY VARYING KERNELS 5 and a given slowly varying function l at zero and infinity. Regarding the regularly varyingfunctions of the type (5), we remark that for σ ∈ (0 ,
2) and β ∈ ( − σ, − σ ) , the kernel K ( y ) ≍ (2 − σ ) | y | − n − σ log 1 | y | ! β/ near zeroassociated with the regularly varying function l ( r ) = r − σ (cid:16) log r (cid:17) β/ describes the asymp-totic behavior of the jumping kernel at zero of the subordinate process which has the char-acteristic exponent φ ( s ) : = s σ (1 + log s ) β/ ; refer to a potential theory of subordinateBrownian motions [KSV].To investigate a class of fully nonlinear elliptic integro-di ff erential operators associatedwith symmetric, regularly varying kernels at zero, let 0 < λ ≤ Λ < + ∞ , and let a function l : (0 , + ∞ ) → (0 , + ∞ ) satisfy Properties 1.1 and 1.2. Owing to Properties 1.1 and 1.2, thefollowing properties of the given function l concerning the symmetric integro-di ff erentialoperators are obtained; the proof can be found in Section 3. Lemma 1.4.
Let a measurable function l : (0 , + ∞ ) → (0 , + ∞ ) be locally bounded awayfrom and + ∞ , and satisfy Properties 1.1 and 1.2 with positive constants σ ∈ (0 , , a ≥ , a ∞ ≥ , and ρ ∈ (0 , . Then we have the following:(a) a r l ( r )2 − σ ≤ Z r sl ( s ) ds ≤ a r l ( r )2 − σ , ∀ r ∈ (0 , , (b) σ Z ∞ l ( s ) s ds ≤ a ∞ . Now, let L ( λ, Λ , l ) denote the class of the following linear integro-di ff erential operatorswith the kernels K : L u ( x ) = Z R n µ ( u , x , y ) K ( y ) dy , where µ ( u , x , y ) : = u ( x + y ) + u ( x − y ) − u ( x ) and(6) (2 − σ ) λ l ( | y | ) | y | n ≤ K ( y ) ≤ (2 − σ ) Λ l ( | y | ) | y | n . One can check that the kernels K satisfying (6) with Properties 1.1 and 1.2 satisfy (3) byusing Lemma 1.4. As mentioned in the introduction, we are concerned with the nonlinearintegro-di ff erential operator I in the form of(7) I u : = inf β sup α L αβ u for some L αβ ∈ L ( λ, Λ , l ) . As extremal cases of such nonlinear integro-di ff erential op-erators, the Pucci type extremal operators with respect to the class L ( λ, Λ , l ) are definedas M + L ( λ, Λ , l ) u : = sup L∈ L ( λ, Λ , l ) L u , M − L ( λ, Λ , l ) u : = inf L∈ L ( λ, Λ , l ) L u . (8)According to Lemma 2.6, the integro-di ff erential operator I of the inf-sup type in (7) iselliptic with respect to L ( λ, Λ , l ) in the nonlocal sense, which, in particular, implies M − L ( λ, Λ , l ) u ≤ I u ≤ M + L ( λ, Λ , l ) u , SOOJUNG KIM, YONG-CHEOL KIM, AND KI-AHM LEE where we refer to Definition 2.3 for the nonlocal notion of the ellipticity. Thus we shall dealwith a large class of the integro-di ff erential operators defined in terms of the Pucci typeextremal operators so as to establish uniform regularity estimates for the fully nonlinearelliptic integro-di ff erential operators associated with symmetric, regularly varying kernelsat zero.1.3. Main results.
Now we present our main results which extend the uniform regularityresults of Ca ff arelli and Silvestre [CS1]. Below and hereafter, we denote B R : = B R (0) for R > . Theorem 1.5 (Harnack inequality) . Let σ ∈ (0 , and let a measurable function l :(0 , + ∞ ) → (0 , + ∞ ) be locally bounded away from and + ∞ , and satisfy Properties 1.1and 1.2 with the positive constants σ ∈ [ σ , , a ≥ , a ∞ ≥ , and ρ ∈ (0 , . For < R < , and C > , let u ∈ C ( B R ) be a bounded, nonnegative function in R n such that M − L ( λ, Λ , l ) u ≤ C and M + L ( λ, Λ , l ) u ≥ − C in B R in the viscosity sense. Then there exist uniform constants C > and ρ ∈ (0 , such that sup B R u ≤ C inf B R u + C L ( ρ R ) ! , where L ( r ) : = σ Z r l ( s ) s ds ∀ < r < , and C > and ρ ∈ (0 , are uniform constants depending only on n , λ, Λ , σ , a , a ∞ , and ρ . Theorem 1.6 (H¨older estimate) . Under the same assumption as in Theorem 1.5, let u ∈ C ( B R ) be a bounded function in R n such that M − L ( λ, Λ , l ) u ≤ C and M + L ( λ, Λ , l ) u ≥ − C in B R in the viscosity sense. Then we haveR α [ u ] α, B R ≤ C k u k L ∞ ( R n ) + C L ( ρ R ) ! , where [ u ] α, B R stands for the α -H¨older seminorm on B R , and the uniform constants α ∈ (0 , , C > and ρ ∈ (0 , depend only n , λ, Λ , σ , a , a ∞ , and ρ . Remark 1.7. (i) According to Theorems 1.5 and 1.6, the Harnack inequality and H¨olderestimate hold for viscosity solutions to the fully nonlinear elliptic integro-di ff erential equa-tions with respect to L ( λ, Λ , l ) . In fact, if u is a viscosity solution to I u = f in B R for anelliptic integro-di ff erential operator with respect to L ( λ, Λ , l ) and f ∈ L ∞ ( B R ) , then u satisfies M − L ( λ, Λ , l ) u ≤ k f k L ∞ ( B R ) + |I | , and M + L ( λ, Λ , l ) u ≥ −k f k L ∞ ( B R ) − |I | in the viscosity sense. Thus, applying Theorems 1.5 and 1.6, the regularity results follow.(ii) For any regularly varying function l at zero and infinity with index − σ ∈ ( − , + ∞ , we obtain the Harnack inequality and H¨older estimatefor the elliptic integro-di ff erential operators with respect to L ( λ, Λ , l ) as a corollary. Here,the constants in the regularity estimates above depend only on n , λ, Λ , and the given regu-larly varying function l . NTEGRO-DIFFERENTIAL OPERATORS WITH REGULARLY VARYING KERNELS 7
Making use of Theorem 1.6, we establish the C ,α estimate for the fully nonlinear ellipticintegro-di ff erential operators associated with regularly varying kernels at zero and infinityprovided that the kernels satisfy a cancelation property at infinity; see Subsection 3.6.Furthermore, the H¨older estimate for the elliptic integro-di ff erential operators associatedwith truncated kernels at infinity is also obtained in Subsection 3.7, which is important forapplications. In fact, the assumption of the kernels at infinity for the H¨older estimate inTheorem 1.6 can be weakened replacing Property 1.2 by the boundedness of the integralat infinity(9) (2 − σ ) Z ∞ l ( s ) s ds ≤ a ∞ for some a ∞ >
0; in the following, we rephrase Theorem 3.14 by assuming (9).
Theorem 1.8.
Let σ ∈ (0 , and let l : (0 , + ∞ ) → [0 , + ∞ ) be a measurable functionwhich is locally bounded away from and + ∞ on (0 , , and satisfy Property 1.1 with thepositive constants σ ∈ [ σ , , a ≥ , and ρ ∈ (0 , . We assume that (2 − σ ) Z ∞ l ( s ) s ds ≤ a ∞ for some a ∞ > . For < R < , and C > , let u ∈ C ( B R ) be a bounded function in R n such that M − L ( λ, Λ , l ) u ≤ C and M + L ( λ, Λ , l ) u ≥ − C in B R in the viscosity sense. Then we haveR α [ u ] α, B R ≤ C k u k L ∞ ( R n ) + C L ( ρ R ) ! , where the uniform constants α ∈ (0 , , C > and ρ ∈ (0 , depend only n , λ, Λ , σ , a , ρ and a ∞ . With the help of Proposition 4.1, Theorems 1.5 , 1.6, and 1.8 yield the uniform Harnackinequality and H¨older estimate for the fully nonlinear elliptic integro-di ff erential operatorswith respect to the class L ( λ, Λ , l σ ) associated with symmetric, regularly varying kernelsof the type (5) for σ ∈ [ σ , ⊂ (0 , σ goes to 2 . Theorem 1.9 (Uniform estimates for the operators associated with the kernels of the type(5)) . Let σ ∈ (0 , , and let a measurable function l : (0 , + ∞ ) → (0 , + ∞ ) be locallybounded away from and + ∞ , and vary slowly at zero and infinity such that l (1) = . For σ ∈ [ σ , , define l σ ( r ) : = r − σ l ( r ) − σ , ∀ r ∈ (0 , + ∞ ) , L σ ( r ) : = σ Z r s − − σ l ( s ) − σ ds , ∀ r ∈ (0 , . (10) (a) For < R < , and C > , let u ∈ C ( B R ) be a bounded, nonnegative function in R n such that M − L ( λ, Λ , l σ ) u ≤ C and M + L ( λ, Λ , l σ ) u ≥ − C in B R in the viscosity sense. Then we have sup B R u ≤ C inf B R u + C L σ ( ρ R ) ! . SOOJUNG KIM, YONG-CHEOL KIM, AND KI-AHM LEE (b) Let u ∈ C ( B R ) be a bounded function in R n such that M − L ( λ, Λ , l σ ) u ≤ C and M + L ( λ, Λ , l σ ) u ≥ − C in B R in the viscosity sense. Then we haveR α [ u ] α, B R ≤ C k u k L ∞ ( R n ) + C L σ ( ρ R ) ! , where C > , ρ ∈ (0 , and α ∈ (0 , are uniform constants depending only onn , λ, Λ , σ and the slowly varying function l at zero and infinity. In Theorem 1.9, we establish the uniform Harnack inequality and H¨older estimate fora class of fully nonlinear elliptic integro-di ff erential operators associated with the kernels K σ in the form of (2 − σ ) λ l ( | y | ) − σ | y | n + σ ≤ K σ ( y ) ≤ (2 − σ ) Λ l ( | y | ) − σ | y | n + σ . In the case when l ≡ , we observe that for σ ∈ [ σ , , l σ ( r ) = r − σ L σ ( r ) = r − σ − ≥ (1 − − σ ) r − σ , ∀ r ∈ (0 , / . This implies that our results recover [CS1, Theorem 11.1, Theorem 12.1]. In particu-lar, considering the following example of slowly varying functions at zero: for l ( r ) : = (cid:16) log r (cid:17) , l β ( r ) = log 2 r ! β ∀ r ∈ (0 , , β ∈ R , Theorem 1.9 asserts the uniform Harnack inequality and H¨older estimate of the ellipticintegro-di ff erential operators associated with the regularly varying kernel K σ,β at zero withindex − σ ∈ ( − , − σ ] for β ∈ R :(2 − σ ) λ | y | n + σ log 2 | y | ! β (2 − σ ) ≤ K σ,β ( y ) ≤ (2 − σ ) Λ | y | n + σ log 2 | y | ! β (2 − σ ) near zero , where the uniform constants in the regularity estimates depend only on n , λ, Λ , σ , β andthe given slowly varying function l at zero.Lastly, we have the following theorem as a corollary of Theorem 1.8 by imposing (9)instead of Property 1.2. Theorem 1.10.
Let σ ∈ (0 , , and let a measurable function l : (0 , + ∞ ) → [0 , + ∞ ) belocally bounded away from and + ∞ on (0 , , and vary slowly at zero such that l (0) = .For σ ∈ [ σ , , define l σ : (0 , + ∞ ) → [0 , + ∞ ) and L σ : (0 , → (0 , + ∞ ) as (10) . Weassume that (2 − σ ) Z ∞ l σ ( s ) s ds ≤ a ∞ for some a ∞ > . For < R < , and C > , let u ∈ C ( B R ) be a bounded function in R n such that M − L ( λ, Λ , l ) u ≤ C and M + L ( λ, Λ , l ) u ≥ − C in B R in the viscosity sense. Then we haveR α [ u ] α, B R ≤ C k u k L ∞ ( R n ) + C L ( ρ R ) ! , where the uniform constants α ∈ (0 , , C > and ρ ∈ (0 , depend only n , λ, Λ , σ , a ∞ , and the slowly varying function l at zero. NTEGRO-DIFFERENTIAL OPERATORS WITH REGULARLY VARYING KERNELS 9
The rest of the paper is organized as follows. Section 2 contains an introduction toviscosity solutions, the ellipticity for integro-di ff erential operators and their properties.Section 3 is devoted to the proof of the uniform regularity estimates for a class of fullynonlinear elliptic integro-di ff erential operators associated with the kernels satisfying (6)with Properties 1.1 and 1.2. In Section 4, we prove Proposition 4.1 and obtain Theorems1.9 and 1.10 from Theorems 1.5, 1.6 and 1.8. In Appendix A, we give the definitions ofregularly and slowly varying functions and summarize their important properties which areused in the paper. 2. V iscosity solutions In this section, we give an introduction to the notions of viscosity solutions and theellipticity for integro-di ff erential operators as in [CS1]; see also [LD1, KL2]. Importantproperties of viscosity solutions such as the stabilities under uniform convergence and thecomparison principle are also provided; refer to [CC] for the local case. We begin with theconcept of C , at the point . Definition 2.1 ( C , at the point) . Let x ∈ R n . A function ϕ is said to be C , at the point x , denoted by ϕ ∈ C , ( x ) , if there exist a vector p ∈ R n and a number M > such that (11) | ϕ ( x + y ) − ϕ ( x ) − p · y | ≤ M | y | for small y ∈ R n .For a set Ω ⊂ R n , we say that ϕ is C , in Ω when (11) holds for any x ∈ Ω with a uniformconstant M > . Now, we recall the viscosity solutions for integro-di ff erential operators. Definition 2.2 (Viscosity solution) . Let Ω ⊂ R n be an open set and let f be a function in Ω . A bounded function u : R n → R which is upper (lower) semi-continuous in Ω is called aviscosity subsolution (supersolution) to the integro-di ff erential equation I u = f in Ω andwe write I u ≥ f in Ω ( I u ≤ f in Ω ) when the following holds: if a C -function ϕ touchesu from above (below) at x ∈ Ω in a small neighborhood N of x , i.e.,(i) ϕ ( x ) = u ( x ) , (ii) φ > u ( φ < u) in N \ { x } , then the function v defined as v : = ( ϕ in N,u in R n \ N , satisfies I v ( x ) ≥ f ( x ) ( I v ( x ) ≤ f ( x ) ). We say u is a viscosity solution if u is both a viscositysubsolution and a viscosity supersolution. Here, we consider bounded viscosity solutions for nonlocal operators for simplicity. Ourmethod to prove the Harnack inequality for viscosity solutions can be also employed underthe assumption that the viscosity solutions have a certain growth rate at infinity relatedto a class of integro-di ff erential operators to deal with; see [BI] for viscosity solutions tointegro-di ff erential equations in a general framework.The notion of ellipticity for integro-di ff erential operators is defined making use of anonlocal version of the Pucci extremal operators. For given 0 < λ ≤ Λ < + ∞ , and afunction l : (0 , + ∞ ) → [0 , + ∞ ) satisfying Property 1.1 and (9), which stays away from 0and + ∞ on (0 , L ( λ, Λ , l ) denote the class of the following linear integro-di ff erentialoperators with the kernels K satisfying (6): L u ( x ) = Z R n µ ( u , x , y ) K ( y ) dy , where µ ( u , x , y ) : = u ( x + y ) + u ( x − y ) − u ( x ) . We recall the Pucci type extremal operators: M + L ( λ, Λ , l ) u : = sup L∈ L ( λ, Λ , l ) L u , and M − L ( λ, Λ , l ) u : = inf L∈ L ( λ, Λ , l ) L u . One can check that M + L ( λ, Λ , l ) u ( x ) = (2 − σ ) Z R n (cid:8) Λ µ + ( u , x , y ) − λµ − ( u , x , y ) (cid:9) l ( | y | ) | y | n dy , M − L ( λ, Λ , l ) u ( x ) = (2 − σ ) Z R n (cid:8) λµ + ( u , x , y ) + − Λ µ − ( u , x , y ) (cid:9) l ( | y | ) | y | n dy , where µ ± ( u , x , y ) : = max {± µ ( u , x , y ) , } . In terms of the Pucci type operators with respect to the class L ( λ, Λ , l ) , the ellipticintegro-di ff erential operators with respect to L ( λ, Λ , l ) are defined as below, which we canapply our results to. Definition 2.3 (Ellipticity for nonlocal operators) . An operator I is said to be elliptic withrespect to the class L ( λ, Λ , l ) if it satisfies the following.(i) If a bounded function u in R n is of C , ( x ) , then I u ( x ) is defined classically.(ii) If a bounded function u in R n is of C , ( Ω ) for an open set Ω , then I u ( x ) is continuousin Ω . (iii) For bounded functions u ∈ C , ( x ) , and v ∈ C , ( x ) , we have M − L ( λ, Λ , l ) v ( x ) ≤ I [ u + v ]( x ) − I u ( x ) ≤ M + L ( λ, Λ , l ) v ( x ) . Remark 2.4.
If a bounded function u in R n is C , at the point x , then the Pucci operators M ± L ( λ, Λ , l ) u ( x ) are defined classically due to the properties (a) and (b) in Lemma 1.4.In Definition 2.2, a C -test function ϕ can be taken to be C , only at the contact point x for elliptic integro-di ff erential operators. We are led to consider a larger set of test functionsand a stronger concept of the viscosity solution, however, those approaches turn out to beequivalent thanks to the following lemma. The proof is similar to one of [CS1, Lemma4.3] with the help of Lemma 1.4; see also [LD1, Lemma 4.3]. Lemma 2.5.
Let Ω ⊂ R n be an open set and let I be an elliptic integro-di ff erential operatorwith respect to L = L ( λ, Λ , l ) . Let u : R n → R satisfy I u ≥ f in Ω in the viscosity sense.We assume that a function ϕ ∈ C , ( x ) (for a point x ∈ Ω ) touches u from above at x in asmall neighborhood N of x. Then the function v defined asv : = ( ϕ in N,u in R n \ N , satisfies I v ( x ) ≥ f ( x ) in the classical sense. Due to Property 1.1 and (9) (Lemma 1.4), the results of [CS1] on viscosity solutionsfor the elliptic integro-di ff erential operators hold true for our elliptic integro-di ff erentialoperators with respect to the class L ( λ, Λ , l ). First, the following lemma concerns thenonlinear integro-di ff erential operators of the inf-sup type (7); the proofs can be found in[CS1, Sections 3 and 4]. Lemma 2.6 (Properties of the inf-sup type operators) . Let I be the operator in the form of (7) . Then we have the following.(a) I is an elliptic integro-di ff erential operator with respect to L ( λ, Λ , l ) , that is, I satis-fies: NTEGRO-DIFFERENTIAL OPERATORS WITH REGULARLY VARYING KERNELS 11 (i) For bounded functions u and v which are C , at x, M − L ( λ, Λ , l ) v ( x ) ≤ I [ u + v ]( x ) − I u ( x ) ≤ M + L ( λ, Λ , l ) v ( x ) . (ii) If a bounded function u in R n is C , in an open set Ω , then I u ( x ) is continuousin Ω . (b) If u is a viscosity subsolution to I u = f in an open set Ω , and a function ϕ ∈ C , ( x ) (for x ∈ Ω ) touches u from above at x in a small neighborhood of x , then I u ( x ) isdefined classically and I u ( x ) ≥ f ( x ) . The viscosity solutions to the elliptic integro-di ff erential equations have nice stabilityproperties with respect to uniform convergence. Recalling the definition of Γ -convergence,a slightly stronger stability of viscosity solutions under Γ -convergence in Lemma 2.8 isquoted from [CS1, Lemma 4.5]. Definition 2.7 ( Γ -convergence) . We say a sequence of lower semi-continuous functions u k Γ -converges to u in a set Ω ⊂ R n if it satisfies the following conditions:(i) For every sequence x k → x in Ω , lim inf k →∞ u k ( x k ) ≥ u ( x ) . (ii) For every x ∈ Ω , there exists a sequence x k → x in Ω such that lim sup k →∞ u k ( x k ) = u ( x ) . Lemma 2.8 (Stability) . Let I be an elliptic operator with respect to the class L ( λ, Λ , l ) . For an open set Ω ⊂ R n , let u k be a sequence of functions that are uniformly bounded in R n such that(i) I u k ≤ f k in Ω in the viscosity sense,(ii) u k → u in the Γ -sense in Ω ,(iii) u k → u a.e. in R n ,(iv) f k → f locally uniformly in Ω for some continuous function f . Then I u ≤ f in Ω in the viscosity sense. As a corollary, we obtain the stability property under uniform convergence.
Corollary 2.9.
Let I be an elliptic operator with respect to the class L ( λ, Λ , l ) . For anopen set Ω ⊂ R n , let u k ∈ C ( Ω ) be a sequence of functions that are uniformly bounded in R n such that(i) I u k = f k in Ω in the viscosity sense,(ii) u k → u locally uniformly in Ω ,(iii) u k → u a.e. in R n ,(iv) f k → f locally uniformly in Ω for some continuous function f . Then I u = f in Ω in the viscosity sense. Lemma 2.10 quoted from [CS1, Theorem 5.9] states that the di ff erence of two viscositysolutions solves an equation in the same ellipticity class. In the proof, Jensen’s approach[J] using the inf- and sup-convolutions was employed to compare two viscosity solutionsto the fully nonlinear elliptic integro-di ff erential equations (see also [A]); we refer to [CC,Chapter 5] for the local case. Lemma 2.10.
Let I be an elliptic operator with respect to the class L ( λ, Λ , l ) . For an openset Ω ⊂ R n , let u and v be bounded in R n such that I u ≥ f , and I v ≤ g in Ω in the viscosity sense for two continuous functions f and g. Then M + L ( λ, Λ , l ) ( u − v ) ≥ f − g in Ω in the viscosity sense. The comparison principle for the elliptic integro-di ff erential operators as in [CS1, The-orem 5.2] follows from Lemma 2.10 with the help of a barrier function given in Lemma2.11; see also [CS1, Assumption 5.1 and Lemma 5.10]. Lemma 2.11.
For a given R ≥ , there exists δ R > such that the function ϕ R ( x ) : = min (cid:16) , | x | R (cid:17) satisfies M − L ( λ, Λ , l ) ϕ R ≥ δ R in B R .Proof. Let x ∈ B R . If x ± y ∈ B R , then we have µ ( ϕ R , x , y ) = | y | R . If x + y < B R , then µ ( ϕ R , x , y ) ≥ − | x | R ≥ . Thus it follows that for x ∈ B R , M − ϕ R ( x ) = (2 − σ ) λ Z R n µ ( ϕ R , x , y ) l ( | y | ) | y | n dy ≥ (2 − σ ) λ Z B R | y | R l ( | y | ) | y | n dy = : δ R > . (cid:3) Lastly, we state the comparison principle for the fully nonlinear elliptic integro-di ff erentialoperators with respect to L ( λ, Λ , l ); the proof is the same as one for Theorem 5.2 of [CS1]. Theorem 2.12 (Comparison principle) . Let I be an elliptic operator with respect to theclass L ( λ, Λ , l ) . For a bounded open set Ω ⊂ R n , let u and v be bounded in R n such that(i) I u ≥ f and I v ≤ f in Ω in the viscosity sense for some continuous functions f ,(ii) u ≤ v in R n \ Ω .Then u ≤ v in Ω .
3. R egularity estimates for integro - differential operators with regularly varyingkernels This section is mainly devoted to proving Theorems 1.5 and 1.6, which will providethe Harnack inequality and H¨older estimate for fully nonlinear elliptic integro-di ff erentialoperators associated with symmetric, regularly varying kernel at zero and infinity as men-tioned in Remark 1.7. Throughout this section, let 0 < λ ≤ Λ < + ∞ , and let a measurablefunction l : (0 , + ∞ ) → (0 , + ∞ ) be locally bounded away from 0 and + ∞ , and satisfy Prop-erties 1.1 and 1.2 with the positive constants σ ∈ [ σ , a ≥ , a ∞ ≥ , and ρ ∈ (0 ,
1) fora given σ ∈ (0 , L ( λ, Λ , l ) be the class of all linear integro-di ff erential operators L u ( x ) = Z R n µ ( u , x , y ) K ( y ) dy with the kernels K satisfying(2 − σ ) λ l ( | y | ) | y | n ≤ K ( y ) ≤ (2 − σ ) Λ l ( | y | ) | y | n , where µ ( u , x , y ) : = u ( x + y ) + u ( x − y ) − u ( x ) . In order to prove the uniform regularityestimates for a class of viscosity solutions to the elliptic integro-di ff erential equations withrespect to the class L ( λ, Λ , l ) , we will deal with the Pucci type extremal operators M ± L ( λ, Λ , l ) ,defined as (8) and simply denoted by M ± , since the elliptic operator I satisfies that M − L ( λ, Λ , l ) ≤ I − I [0] ≤ M + L ( λ, Λ , l ) . NTEGRO-DIFFERENTIAL OPERATORS WITH REGULARLY VARYING KERNELS 13
Before we proceed to regularity estimates for viscosity solutions to nonlocal equations,we study the important properties of the given function l satisfying Properties 1.1 and 1.2,which will be used later. Lemma 3.1.
Let a measurable function l : (0 , + ∞ ) → (0 , + ∞ ) be locally bounded awayfrom and + ∞ , and satisfy Properties 1.1 and 1.2 with positive constants σ ∈ (0 , , a ≥ , a ∞ ≥ , and ρ ∈ (0 , . Then we have the following.(a) For r ∈ (0 , , a r l ( r )2 − σ ≤ Z r sl ( s ) ds ≤ a r l ( r )2 − σ . (b) For r ∈ (0 , , Z r s l ( s ) ds ≤ a r l ( r ) . (c) For r ∈ (0 , , L ( r ) : = σ Z r l ( s ) s ds ≥ a (cid:16) r − σ/ − (cid:17) . In particular, for σ ∈ [ σ , , we have a (cid:16) r − σ / − (cid:17) ≤ L ( r ) ≤ a r − , ∀ r ∈ (0 , . (d) σ Z ∞ l ( s ) s ds ≤ a ∞ . Proof.
Using Property 1.1, we have that for r ∈ (0 , , Z r sl ( s ) ds = l ( r ) Z r s l ( s ) l ( r ) ds ≤ a r σ + δ l ( r ) Z r s − σ − δ ds ≤ a r σ + δ l ( r ) 12 − σ − δ r − σ − δ ≤ a − σ r l ( r )since δ ∈ h , min(2 − σ, σ ) (cid:17) . In a similar way, we deduce that for r ∈ (0 , a r l ( r )2 − σ ≤ Z r sl ( s ) ds ≤ a r l ( r )2 − σ and Z r s l ( s ) ds ≤ a r l ( r ) . Recalling from Property 1.1 that for r ∈ (0 , L ( r ) : = σ Z r l ( s ) s ds , it follows that L ( r ) = σ l ( r ) Z r s l ( s ) l ( r ) ds ≥ σ a r σ + δ l ( r ) Z r s − − σ − δ ds = σ a r σ + δ l ( r ) 1 σ + δ (cid:16) r − σ − δ − (cid:17) ≥ a r σ + δ l (1) a r − σ + δ (cid:16) r − σ − δ − (cid:17) = a (cid:16) r − σ + δ − r δ (cid:17) ≥ a (cid:16) r − σ/ − (cid:17) . Arguing in a similar way,, we have12 a (cid:16) r − σ/ − (cid:17) ≤ L ( r ) ≤ a r − , ∀ r ∈ (0 , . Since l (1) =
1, Property 1.2 yields that σ Z ∞ l ( s ) s ds ≤ σ a ∞ Z ∞ s − − σ + δ ′ ds ≤ a ∞ σσ − δ ′ ≤ a ∞ , completing the proof. (cid:3) Aleksandrov-Bakelman-Pucci type estimate.
First, we extend the nonlocal Aleksandrov-Bakelman-Pucci(ABP) estimate by Ca ff arelli and Silvestre [CS1, Lemma 8.1] for fullynonlinear elliptic integro-di ff erential operators with respect to L ( λ, Λ , l ) . Lemma 3.2.
Let R ∈ (0 , and ρ ∈ (0 , . Let r k : = ρ − − σ ) − k R , and R k ( x ) : = B r k ( x ) \ B r k + ( x ) for k ∈ N ∪ { } and x ∈ R n . Let u be a viscosity subsolution of M + L ( λ, Λ , l ) u = − f on B R such that u ≤ in R n \ B R , and let Γ be the concave envelope of u + : = max { u , } in B R . Then there exists a uniform constant ˜ C : = c n a λρ sup σ ∈ [ σ , − − − σ ) − σ ! > such that foreach x ∈ { u = Γ } and M > , we find k ∈ N ∪ { } satisfying (12) (cid:12)(cid:12)(cid:12)(cid:12)n y ∈ R k ( x ) : u ( y ) < u ( x ) + ( y − x ) · ∇ Γ ( x ) − Mr k o(cid:12)(cid:12)(cid:12)(cid:12) ≤ ˜ Cl ( R ) R f ( x ) M |R k ( x ) | , where ∇ Γ ( x ) stands for an element of the superdi ff erential of Γ at x , and c n > dependsonly on dimension n . Proof.
Let x be a contact point, that is, x ∈ { u = Γ } ⊂ B R . From Lemma 2.6, M + u ( x ) canbe defined classically and we have M + u ( x ) = (2 − σ ) Z R n (cid:8) Λ µ + ( u , x , z ) − λµ − ( u , x , z ) (cid:9) l ( | z | ) | z | n dz ≥ − f ( x ) , where µ ( u , x , z ) = u ( x + z ) + u ( x − z ) − u ( x ) , and µ ± ( u , x , z ) = max {± µ ( u , x , z ) , } . We note that u ( x ) = Γ ( x ) > . If x + z ∈ B R and x − z ∈ B R , then we have µ ( u , x , z ) ≤ Γ lies above u . If x + z < B R , then x + z and x − z do not belong to B R , which implies NTEGRO-DIFFERENTIAL OPERATORS WITH REGULARLY VARYING KERNELS 15 that µ ( u , x , z ) ≤ . Thus it follows that µ ( u , x , z ) ≤ z ∈ R n and hence f ( x ) ≥ (2 − σ ) λ Z R n µ − ( u , x , z ) l ( | z | ) | z | n dz ≥ (2 − σ ) λ Z B r (0) µ − ( u , x , z ) l ( | z | ) | z | n dz = (2 − σ ) λ + ∞ X k = Z R k (0) µ − ( u , x , z ) l ( | z | ) | z | n dz . The concavity of Γ implies that if u ( x + z ) < u ( x ) + z · ∇ Γ ( x ) − Mr k for some z ∈ B R , then µ − ( u , x , z ) ≥ Mr k . Indeed, we have that x ± z ∈ B R and µ ( u , x , z ) = u ( x + z ) + u ( x − z ) − u ( x ) ≤ u ( x + z ) + Γ ( x − z ) − u ( x ) < n u ( x ) + z · ∇ Γ ( x ) − Mr k o + { Γ ( x ) − z · ∇ Γ ( x ) } − u ( x ) = − Mr k . Suppose to the contrary that (12) is not true. Then we have f ( x )2 − σ ≥ λ + ∞ X k = Z R k (0) µ − ( u , x , z ) l ( | z | ) | z | n dz = λ l ( R ) + ∞ X k = Z R k (0) µ − ( u , x , z ) l ( | z | ) l ( R ) 1 | z | n dz ≥ λ l ( R ) + ∞ X k = Z R k (0) µ − ( u , x , z ) 1 a | z | R ! − σ + δ r nk dz ≥ λ l ( R ) 1 a + ∞ X k = (cid:18) r k R (cid:19) − σ + δ r nk Z R k (0) µ − ( u , x , z ) dz ≥ λ l ( R ) 1 a + ∞ X k = (cid:18) r k R (cid:19) − σ + δ r nk ˜ Cl ( R ) R f ( x ) M |R k ( x ) | Mr k = λ l ( R ) 1 a + ∞ X k = (cid:18) r k R (cid:19) − σ + δ r k r nk ˜ C f ( x ) l ( R ) R |R k (0) | = λ c n a ˜ C f ( x ) + ∞ X k = (cid:18) r k R (cid:19) − σ + δ ≥ λ c n a ˜ C f ( x ) + ∞ X k = (cid:18) r k R (cid:19) − σ + (2 − σ ) / ≥ λ c n a ˜ C f ( x ) + ∞ X k = (cid:18) r k R (cid:19) − σ ) = λ c n a ˜ C f ( x ) ρ − σ )0 (cid:0) − − − σ ) (cid:1) ≥ λ c n a ˜ C f ( x ) ρ (cid:0) − − − σ ) (cid:1) since 0 ≤ δ ≤ (2 − σ ) / . By choosing ˜ C ≥ a (cid:16) − − − σ ) (cid:17) λ c n ρ (2 − σ ) which is bounded above bya uniform constant for σ ∈ [ σ , , the result follows. (cid:3) In the proof of Lemma 3.2, we observe that f ( x ) is positive for x ∈ { u = Γ } . Lemma 3.3.
Under the same assumption as Lemma 3.2, there exists uniform constants ǫ n ∈ (0 , and ˜ M : = ˜ C /ǫ n > such that for each x ∈ { u = Γ } , we find some r = r k ≤ ρ − − σ ) R which satisfies the following:(a) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( y ∈ B r ( x ) \ B r / ( x ) : u ( y ) < u ( x ) + ( y − x ) · ∇ Γ ( x ) − ˜ M f ( x ) l ( R ) R r )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ n (cid:12)(cid:12)(cid:12) B r ( x ) \ B r / ( x ) (cid:12)(cid:12)(cid:12) , (b) Γ ( y ) ≥ u ( x ) + ( y − x ) · ∇ Γ ( x ) − ˜ M f ( x ) l ( R ) R r ∀ y ∈ B r / ( x ) , (c) (cid:12)(cid:12)(cid:12) ∇ Γ ( B r / ( x )) (cid:12)(cid:12)(cid:12) ≤ c n ˜ M f ( x ) l ( R ) R ! n | B r / ( x ) | , where ˜ C > is the uniform constant as in Lemma 3.2, and ǫ n ∈ (0 , and c n > areuniform constant depending only on n . Proof.
For a small ǫ n > , let ˜ M : = ˜ C /ǫ n . We apply Lemma 3.2 with M = ˜ M f ( x ) l ( R ) R to have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( y ∈ B r ( x ) \ B r / ( x ) : u ( y ) < u ( x ) + ( y − x ) · ∇ Γ ( x ) − ˜ M f ( x ) l ( R ) R r )(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ n (cid:12)(cid:12)(cid:12) B r ( x ) \ B r / ( x ) (cid:12)(cid:12)(cid:12) for some r = r k , which proves (a). With the help of (a), we employ the same arguments asthe proofs of Lemma 8.4 and Corollary 8.5 in [CS1] to show (b) and (c). (cid:3) Now we obtain a nonlocal version of the ABP estimate in the following theorem makinguse of Lemma 3.3 together with a dyadic cube decomposition; we refer to [CS1, Theorem8.7] for the proof.
Theorem 3.4 (ABP type estimate) . Let R ∈ (0 , , and let ρ ∈ (cid:16) , / (32 √ n ) i be a con-stant. Let u be a viscosity subsolution of M + L ( λ, Λ , l ) u = − f on B R such that u ≤ in R n \ B R , and let Γ be the concave envelope of u + in B R . Then there existsa finite, disjoint family of open cubes Q j with diameters d j ≤ ρ − − σ ) R such that n Q j o covers the contact set { u = Γ } , and satisfies the following:(a) { u = Γ } ∩ Q j , ∅ for any Q j , (b) |∇ Γ ( Q j ) | ≤ c n ˜ Cl ( R ) R ! n max Q j ∩{ u =Γ } f n | Q j | , (c) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y ∈ √ nQ j : u ( y ) ≥ Γ ( y ) − c n ˜ Cl ( R ) R max Q j ∩{ u =Γ } f d j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ µ | Q j | for µ : = − ǫ n ∈ (0 , , where ˜ C > is the uniform constant as in Lemma 3.2, and ǫ n ∈ (0 , (appearing in Lemma 3.3) and c n > are uniform constant depending only on n . Barrier function.
As in [CS1] and [KL2], we construct the barrier function at eachscale, where the monotone function L associated with l given in Property 1.1 plays a roleto obtain scale invariant estimates. Lemma 3.5.
Let R ∈ (0 , / . For κ ∈ (0 , , there exist uniform constants p = p ( n , λ, Λ ) > n , and ǫ ∈ (0 , / such that the function ϕ ( x ) : = min {| κ R | − p , | x | − p } for κ : = ǫ κ > satisfies M − L ( λ, Λ , l ) ϕ ( x ) ≥ , ∀ x ∈ B R \ B κ R . NTEGRO-DIFFERENTIAL OPERATORS WITH REGULARLY VARYING KERNELS 17
Proof.
Assume without loss of generality that x = R e for κ R ≤ R < R . We need tocompute M − ϕ ( x ) = (2 − σ ) Z R n (cid:8) λµ + ( ϕ, x , y ) − Λ µ − ( ϕ, x , y ) (cid:9) l ( | y | ) | y | n dy = (2 − σ ) Z R n λµ + l ( | y | ) | y | n dy + (2 − σ ) Z B ρ R (cid:18) λ µ + − Λ µ − (cid:19) l ( | y | ) | y | n dy + (2 − σ ) Z B c ρ R (cid:18) λ µ + − Λ µ − (cid:19) l ( | y | ) | y | n dy ≥ (2 − σ ) Z R n λµ + l ( | y | ) | y | n dy + (2 − σ ) Z B ρ R (cid:18) λ µ + − Λ µ − (cid:19) l ( | y | ) | y | n dy − − σ ) Λ R − p Z B c ρ R l ( | y | ) | y | n dy = : I + I + I , where ρ ≤ min { ρ, / } will be chosen su ffi ciently small later .For | y | < R , we have | x + y | − p + | x − y | − p − | x | − p = R − p ((cid:12)(cid:12)(cid:12)(cid:12)(cid:12) xR + yR (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − p + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) xR − yR (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − p − ) ≥ R − p p ( −| y | + ( p + y −
12 ( p + p + y | y | ) (13)for y : = y / R ; see [CS1, Lemma 9.1]. We choose N ∋ p > n large enough so that(14) ( p + λ Z ∂ B y d σ ( y ) − Λ | ∂ B | = : δ > . We use (13), (14), Lemma 3.1 and Property 1.1 to obtain I = (2 − σ ) Z B ρ R (cid:18) λ µ + − Λ µ − (cid:19) l ( | y | ) | y | n dy ≥ (2 − σ ) pR − p Z B ρ R λ p + y R − Λ | y | R + ( p + p + y | y | R l ( | y | ) | y | n dy ≥ (2 − σ ) pR − p δ R Z ρ R sl ( s ) ds − Λ ( p + p + ω n R Z ρ R s l ( s ) ds ≥ (2 − σ ) pR − p δ a (2 − σ ) ρ l ( ρ R ) − Λ ( p + p + ω n a ρ l ( ρ R ) = pR − p δ a ρ − (2 − σ ) Λ ( p + p + ω n a ρ l ( ρ R ) ≥ pR − p δ a ρ − (2 − σ ) Λ ( p + p + ω n a L ( ρ R ) . We select a uniform constant ρ = ρ ( a , a ∞ , σ ) ≤ min( ρ, /
2) small so that(15) 2 a ∞ ≤ a (cid:16) ρ − σ / − (cid:17) ≤ L ( ρ ) , and then we use Lemma 3.1 and Properties 1.1 and 1.2 again to have − I = − σ ) Λ R − p Z B c ρ R l ( | y | ) | y | n dy ≤ − σ ) Λ R − p ω n ( a ∞ σ + σ L ( ρ R ) ) ≤ − σσ Λ ω n R − p a (cid:16) ρ − σ / − (cid:17) + L ( ρ R ) ≤ − σσ Λ ω n R − p L ( ρ R ) , where we note that L is monotone. Thus we deduce that I + I ≥ pR − p δ a ρ − (2 − σ ) Λ ( p + p + ω n a L ( ρ R ) − − σσ Λ ω n R − p L ( ρ R ) = R − p L ( ρ R ) p δ a ρ − (2 − σ ) Λ p ( p + p + ω n a − (2 − σ ) 8 Λ ω n σ ≥ σ ∈ [ σ , , where σ ∈ [ σ ,
2) depends only on n , λ, Λ , a , a ∞ , ρ, and σ . Thus thelemma holds true for σ ∈ [ σ , . For σ ∈ [ σ , σ ), we will make I su ffi ciently large by selecting κ > x = R e with κ R ≤ R < R , we have that for κ : = ǫ κ ∈ (0 , κ / I ≥ (2 − σ ) Z R n λµ + l ( | y | ) | y | n dy ≥ (2 − σ ) λ Z B R / ( x ) n | x − y | − p − R − p o l ( | y | ) | y | n dy ≥ (2 − σ ) λ Z B R / ( x ) \ B κ R ( x ) | x − y | − p l ( | y | ) | y | n dy = (2 − σ ) λ Z B R / (0) \ B κ R (0) | z | − p l ( | x + z | ) | x + z | n dz ≥ (2 − σ ) λ n + R n min s ∈ [ R / , R / l ( s ) ! ω n Z R / κ R s − p + n − ds ≥ (2 − σ ) λ n + R n p − n (cid:8) ( κ R ) − p + n − ( R / − p + n (cid:9) min s ∈ [ R / , R / l ( s ) ≥ (2 − σ ) λ n + R n R − p + n p − n ( κ κ ! p − n − p − n ) min s ∈ [ R / , R / l ( s ) ≥ (2 − σ ) λ n + R n R − p + n p − n κ κ ! p − n min s ∈ [ R / , R / l ( s ) ≥ (2 − σ ) λ n + R n R − p + n p − n ǫ − p + n a ρ ! σ + δ l ( ρ R ) ≥ (2 − σ ) λ n + R − p p − n ǫ a ρ ! L ( ρ R ) . From the argument above, we notice that for σ ∈ [ σ , , I + I ≥ − CR − p L ( ρ R ) , where a uniform constant C > n , λ, Λ , a , a ∞ , ρ, and σ . Therefore, wechoose a uniform constant ǫ = κ κ ∈ (0 , /
8) su ffi ciently small to conclude that M − ϕ ( x ) ≥ I + I + I ≥ x ∈ B R \ B κ R in the case when σ ∈ [ σ , σ ) . This finishes the proof. (cid:3)
NTEGRO-DIFFERENTIAL OPERATORS WITH REGULARLY VARYING KERNELS 19
Lemma 3.6.
Let R ∈ (0 , / and < δ < δ < . Assume δ ≤ ρ , where ρ = ρ ( a , a ∞ , ρ, σ ) ≤ min( ρ, / is the constant satisfying (15) . There exists a continuousfunction Φ in R n such that(a) Φ is nonnegative and uniformly bounded in R n ,(b) Φ = outside B R , (c) Φ ≥ in B δ R , (d) L ( δ R ) − M − L ( λ, Λ , l ) Φ ≥ − ψ in R n for some nonnegative, uniformly bounded function ψ such that supp( ψ ) ⊂ B δ R . Proof.
Let κ : = δ / . According to Lemma 3.5, the function ϕ ( x ) : = min {| κ R | − p , | x | − p } satisfies M − ϕ ( x ) ≥ ∀ x ∈ B R \ B κ R for some 0 < κ < κ / p > n . Now we define Φ : R n → [0 , + ∞ ) by Φ ( x ) : = c P ( x ) ∀ x ∈ B κ R ( κ R ) p (cid:8) min (cid:0) | κ R | − p , | x | − p (cid:1) − R − p (cid:9) ∀ x ∈ B R \ B κ R B R , where P ( x ) : = − a | x | + b with a = p ( κ R ) − and b : = − κ p + p . Thus Φ is a C , -functionon B R . By setting c : = κ p ( δ − p − , the property (b) follows. Note that Lemma 3.5 impliesthat M − Φ ( x ) ≥ ∀ x ∈ B R \ B δ R / . It remains to show that L ( δ R ) − M − Φ ≥ − C in B δ R . We use Properties 1.1 and 1.2 and Lemma 3.1 to deduce that for x ∈ B δ R , M − Φ ( x ) ≥ − (2 − σ ) Λ Z R n µ − ( Φ , x , y ) l ( | y | ) | y | n dy ≥ − (2 − σ ) Λ Z B δ R µ − ( Φ , x , y ) l ( | y | ) | y | n dy − (2 − σ ) Λ c b Z R n \ B δ R l ( | y | ) | y | n dy ≥ − (2 − σ ) Λ Z B δ R µ − ( Φ , x , y ) l ( | y | ) | y | n dy − Λ c b ω n σ { L ( δ R ) + a ∞ }≥ − (2 − σ ) Λ c R − ω n a δ R − σ l ( δ R ) − bc Λ ω n σ L ( δ R ) ≥ − Λ ω n ( a c δ + bc σ ) L ( δ R )since D Φ ≥ − c R − I a.e. in B δ R for some c = c ( δ , δ ) > , where we recall that0 < δ ≤ ρ , and ρ = ρ ( a , a ∞ , ρ, σ ) ≤ min( ρ, /
2) satisfies (15). (cid:3)
Power decay estimate of super-level sets.
We use the ABP type estimate in Theorem3.4 and the barrier functions constructed in Lemma 3.6 to obtain the measure estimates ofsuper-level sets of the viscosity supersolutions to fully nonlinear elliptic integro-di ff erentialoperators with respect to L ( λ, Λ , l ). Lemma 3.7.
Let < R < / and let Q r = Q r (0) denote a dyadic cube of side r centeredat for r > . There exist uniform constants ε , ρ , µ ∈ (0 , and M > , dependingonly on n , λ, Λ , a , a ∞ , ρ, and σ , such that if(a) u ≥ in R n , (b) inf Q R √ n u ≤ , (c) M − L ( λ, Λ , l ) u ≤ ε L ( ρ R ) on Q R in the viscosity sense,then (cid:12)(cid:12)(cid:12)(cid:12) { u ≤ M } ∩ Q R √ n (cid:12)(cid:12)(cid:12)(cid:12) > µ (cid:12)(cid:12)(cid:12)(cid:12) Q R √ n (cid:12)(cid:12)(cid:12)(cid:12) . Proof.
Let ρ ∈ (0 ,
1) be a constant to be chosen later depending only on n and ρ > , where the constant ρ satisfies (15). Let Φ be the barrier function in Lemma 3.6 with δ : = ρ (cid:16) ≤ min n / (32 √ n ) , ρ o(cid:17) , and δ : = . Then v : = Φ − u satisfies that v ≤ B R , max B R v ≥ M + L ( λ, Λ , l ) v ≥ M − L ( λ, Λ , l ) Φ − M − L ( λ, Λ , l ) u ≥ − ( ψ + ε ) L ( ρ R ) in B R in the viscosity sense. For the concave envelope Γ of v + in B R , Theorem 3.4 with the helpof Property 1.1 yields that1 R ≤ R max B R v ≤ c n |∇ Γ ( B R ) | / n ≤ c n X j |∇ Γ ( Q j ) | / n ≤ c n ˜ Cl ( R ) R L ( ρ R ) X j max Q j ( ψ + ε ) n | Q j | / n ≤ c n ˜ C L ( ρ R ) l ( ρ R ) R a ρ − σ − δ X j max Q j (cid:16) ψ n + ε n (cid:17) | Q j | / n ≤ c n ˜ C a ρ R X j max Q j (cid:16) ψ n + ε n (cid:17) | Q j | / n , so we have 1 R ≤ CR X j max Q j (cid:16) ψ n + ε n (cid:17) | Q j | / n for a uniform constant C > n , λ, a , ρ, and σ , Recalling that thenonnegative function ψ is uniformly bounded with supp ψ ⊂ B ρ R in Lemma 3.6, and P j | Q j | ≤ c n | B R | , it follows that1 R ≤ C ε R + CR X Q j ∩ B ρ R , ∅ | Q j | / n . By selecting ε > CR X Q j ∩ B ρ R , ∅ | Q j | / n ≥ . NTEGRO-DIFFERENTIAL OPERATORS WITH REGULARLY VARYING KERNELS 21
Now we select ρ > ffi ciently small such that 32 √ n ρ ≤ √ n in order to show that32 √ nQ j ⊂ B R √ n for any Q j satisfying Q j ∩ B ρ R , ∅ . Thus [ Q j ∩ B ρ R , ∅ Q j is covered by n √ nQ j : Q j ∩ B ρ R , ∅ o contained in B R √ n . On the other hand, according to Theorem 3.4 together with the previous argument, wehave(17) (cid:12)(cid:12)(cid:12)(cid:12)n y ∈ √ nQ j : u ( y ) ≤ M o(cid:12)(cid:12)(cid:12)(cid:12) ≥ µ | Q j | for some M > . Indeed, from the previous argument, we see that c n ˜ Cl ( R ) R L ( ρ R ) max Q j ( ψ + ε ) d j ≤ M for a uniform constant M > σ ∈ [ σ ,
2) since d j ≤ ρ R . Then it followsfrom Theorem 3.4 that µ | Q j | ≤ (cid:12)(cid:12)(cid:12)(cid:12)n y ∈ √ nQ j : v ( y ) ≥ Γ ( y ) − M o(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)n y ∈ √ nQ j : u ( y ) ≤ k Φ k L ∞ ( R n ) + M = : M o(cid:12)(cid:12)(cid:12)(cid:12) since Φ is uniformly bounded in R n , and Γ is positive in B R . Taking a subcover of n √ nQ j : Q j ∩ B ρ R , ∅ o with finite overlapping, we deduce from (16) and (17) thatfor uniform constants M > µ ∈ (0 , , (cid:12)(cid:12)(cid:12)(cid:12) { u ≤ M } ∩ Q R √ n (cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:12)(cid:12)(cid:12)(cid:12) { u ≤ M } ∩ B R √ n (cid:12)(cid:12)(cid:12)(cid:12) > µ (cid:12)(cid:12)(cid:12)(cid:12) Q R √ n (cid:12)(cid:12)(cid:12)(cid:12) , which finishes the proof. (cid:3) The Calder´on-Zygmund technique combined with Lemma 3.7 implies the followingdecay measure estimate of super-level sets making use of the monotonicity of the function L . Corollary 3.8.
Under the same assumption as Lemma 3.7, we have (cid:12)(cid:12)(cid:12)(cid:12)n u > M k o ∩ Q R √ n (cid:12)(cid:12)(cid:12)(cid:12) ≤ (1 − µ ) k (cid:12)(cid:12)(cid:12)(cid:12) Q R √ n (cid:12)(cid:12)(cid:12)(cid:12) , ∀ k = , , · · · , and hence (cid:12)(cid:12)(cid:12)(cid:12) { u > t } ∩ Q R √ n (cid:12)(cid:12)(cid:12)(cid:12) ≤ CR n t − ǫ ∀ t > , where C > and ǫ > are uniform constants depending only on n , λ, Λ , a , a ∞ , ρ, and σ . Using a standard covering argument, we deduce the weak Harnack inequality as follows.
Theorem 3.9 (Weak Harnack inequality) . For < R < , and C > , let u be a nonnega-tive function in R n such that M − L ( λ, Λ , l ) u ≤ C in B R in the viscosity sense. Then we have |{ u > t } ∩ B R | ≤ CR n u (0) + C L ( ρ R ) ! ǫ t − ǫ ∀ t > , and hence ? B R | u | p ! / p ≤ C ( u (0) + C L ( ρ R ) ) , where C > , ǫ > , ρ ∈ (0 , , and p > are uniform constants depending only on n , λ, Λ , a , a ∞ , ρ and σ . Harnack inequality.
Making use of the weak Harnack inequality in Theorem 3.9, weprove the scale invariant Harnack inequality for fully nonlinear elliptic integro-di ff erentialoperators with respect to L ( λ, Λ , l ), where the constant in the Harnack estimate dependsonly on on n , λ, Λ , a , a ∞ , ρ (in Properties 1.1 and 1.2) and σ . The proof of [CS1, The-orem 11.1] has been adapted to our elliptic integro-di ff erential operators associated withregularly varying kernels at zero and infinity. Theorem 3.10.
For < R < , and C > , let u ∈ C ( B R ) be a nonnegative function in R n such that M − L ( λ, Λ , l ) u ≤ C L ( ρ R ) , and M + L ( λ, Λ , l ) u ≥ − C L ( ρ R ) in B R in the viscosity sense, where ρ ∈ (0 , is the constant as in Theorem 3.9. Then we have sup B R u ≤ C ( u (0) + C ) , where a uniform constant C > depends only on n , λ, Λ , a , a ∞ , ρ and σ .Proof. We may assume that u > , u (0) ≤ , and C = . Let ǫ > γ : = ( n + /ǫ. Consider the minimal value of α > u ( x ) ≤ h α ( x ) : = α − | x | R ! − γ ∀ x ∈ B R . We claim that α > x be a point such that u ( x ) = h α ( x ) . Wemay assume that x ∈ B R , otherwise α is small. Let d : = R − | x | and r : = d / . Let A : = { u > u ( x ) / } . According to the weak Harnack inequality in Theorem 3.9, wehave | A ∩ B R | ≤ CR n u ( x ) ! ǫ ≤ CR n α − ǫ dR ! γǫ = C α − ǫ dR ! d n ≤ C α − ǫ d n . This implies that(18) |{ u > u ( x ) / } ∩ B r ( x ) | ≤ C α − ǫ | B r ( x ) | since B r ( x ) ⋐ B R and r = d / . Now we will show that there is a uniform number θ ∈ (0 ,
1) such that |{ u < u ( x ) / } ∩ B θ r ( x ) | ≤ | B θ r ( x ) | for a large constant α > , from which (18) yields that α > x ∈ B θ r ( x ) , u ( x ) ≤ h α ( x ) ≤ α d − θ rR ! − γ = α dR ! − γ (cid:18) − θ (cid:19) − γ = (cid:18) − θ (cid:19) − γ u ( x ) . For θ ∈ (0 , , consider v ( x ) : = (cid:18) − θ (cid:19) − γ u ( x ) − u ( x ) . Note that v is nonnegative in B θ r ( x ) . To apply the weak Harnack inequality to w : = v + , NTEGRO-DIFFERENTIAL OPERATORS WITH REGULARLY VARYING KERNELS 23 we will compute M − w in B θ r ( x ) . First, we see that for x ∈ B θ r ( x ) , M − w ( x ) = (2 − σ ) Z R n (cid:8) λµ + ( v + , x , y ) − Λ µ − ( v + , x , y ) (cid:9) l ( | y | ) | y | n dy ≤ M − v ( x ) + (2 − σ ) Z R n (cid:8) Λ v − ( x + y ) + Λ v − ( x − y ) (cid:9) l ( | y | ) | y | n dy ≤ L ( ρ R ) + (2 − σ ) Z R n (cid:8) Λ v − ( x + y ) + Λ v − ( x − y ) (cid:9) l ( | y | ) | y | n dy = L ( ρ R ) + − σ ) Z { v ( x + y ) < } − Λ v − ( x + y ) l ( | y | ) | y | n dy ≤ L ( ρ R ) + − σ ) Λ Z R n \ B θ r ( x − x ) ( u ( x + y ) − (cid:18) − θ (cid:19) − γ u ( x ) ) + l ( | y | ) | y | n dy in the viscosity sense, where v satisfies M − v = M − [ − u ] ≤ L ( ρ R ) on B R in the viscositysense.Consider the largest number β > u ( x ) ≥ g β ( x ) : = β − | x | R ! + , and let x ∈ B R be a point such that u ( x ) = g β ( x ) . We observe that β ≤ u (0) ≤ . Using Lemma 3.1, we have(2 − σ ) Z R n µ − ( u , x , y ) l ( | y | ) | y | n dy ≤ (2 − σ ) Z R n µ − ( g β , x , y ) l ( | y | ) | y | n dy = (2 − σ ) Z B ρ R µ − ( g β , x , y ) l ( | y | ) | y | n dy + Z R n \ B ρ R µ − ( g β , x , y ) l ( | y | ) | y | n dy ≤ C (2 − σ ) β Z B ρ R | y | R l ( | y | ) | y | n dy + − σ ) β Z R n \ B ρ R l ( | y | ) | y | n dy ≤ C β R a ρ R l ( ρ R ) + β − σσ { L ( ρ R ) + a ∞ }≤ C β a ρ L ( ρ R ) + βσ L ( ρ R ) ≤ C β L ( ρ R ) ≤ CL ( ρ R ) , where we recall that 0 < ρ ≤ ρ ; see (15). Since M − u ≤ L ( ρ R ) on B R in the viscositysense, it follows that (2 − σ ) Z R n µ + ( u , x , y ) l ( | y | ) | y | n dy ≤ CL ( ρ R ) , which asserts that(19) (2 − σ ) Z R n { u ( x + y ) − } + l ( | y | ) | y | n dy ≤ CL ( ρ R ) , where we note that u ( x ) ≤ β ≤ u ( x − y ) > y ∈ R n . We may assume that u ( x ) ≥ , otherwise α is uniformly bounded. In order to estimate M − w in B θ r ( x ) , we consider that for x ∈ B θ r ( x )(2 − σ ) Z R n \ B θ r ( x − x ) ( u ( x + y ) − (cid:18) − θ (cid:19) − γ u ( x ) ) + l ( | y | ) | y | n dy = (2 − σ ) Z R n \ B θ r ( x − x ) ( u ( x + x + y − x ) − (cid:18) − θ (cid:19) − γ u ( x ) ) + l ( | x + y − x | ) | x + y − x | n · | x + y − x | n | y | n l ( | y | ) l ( | x + y − x | ) ! dy . Since we see that for x ∈ B θ r ( x ) and y ∈ R n \ B θ r ( x − x ) , | x + y − x | n | y | n l ( | y | ) l ( | x + y − x | ) dy ≤ a a ∞ R θ r ! n + σ + max( δ,δ ′ ) , it follows from (19) that for x ∈ B θ r ( x ) M − w ( x ) ≤ L ( ρ R ) + Λ a a ∞ n + R θ r ! n + σ + max( δ,δ ′ ) CL ( ρ R ) ≤ C R θ r ! n + σ + δ L ( ρ R ) ≤ C R θ r ! n + L (cid:18) ρ θ r (cid:19) owing to monotonicity of the function L . Now we apply the weak Harnack inequality to w in B θ r ( x ) to obtain that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( u < u ( x )2 ) ∩ B θ r / ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( w > (cid:18) − θ (cid:19) − γ − ! u ( x ) ) ∩ B θ r / ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( θ r ) n w ( x ) + C R θ r ! n + σ + δ ǫ (cid:18) − θ (cid:19) − γ − ! − ǫ u ( x ) − ǫ = C ( θ r ) n (cid:18) − θ (cid:19) − γ − ! u ( x ) + C R θ r ! n + ǫ (cid:18) − θ (cid:19) − γ − ! − ǫ u ( x ) − ǫ ≤ C ( θ r ) n (cid:18) − θ (cid:19) − γ − ! ǫ + C R θ r ! ( n + ǫ u ( x ) ǫ ≤ C ( θ r ) n (cid:18) − θ (cid:19) − γ − ! ǫ + C R θ r ! ( n + − γ ) ǫ (cid:18) θ (cid:19) − γǫ α − ǫ ≤ C ( θ r ) n (cid:18) − θ (cid:19) − γ − ! ǫ + θ − γǫ α − ǫ ! since u ( x ) = α ( R / r ) γ and γ = ( n + /ǫ. We choose a uniform constant θ > ffi cientlysmall so that C ( θ r ) n (cid:18) − θ (cid:19) − γ − ! ǫ ≤ | B θ r / ( x ) | . If α > ffi ciently large, then we have C ( θ r ) n θ − γǫ α − ǫ ≤ | B θ r / ( x ) | , which implies that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( u < u ( x )2 ) ∩ B θ r / ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | B θ r / ( x ) | . NTEGRO-DIFFERENTIAL OPERATORS WITH REGULARLY VARYING KERNELS 25
On the other hand, according to (18), we have that for large α > (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)( u > u ( x )2 ) ∩ B θ r / ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C α − ǫ | B θ r / ( x ) | < | B θ r / ( x ) | , which is a contradiction. Therefore, we conclude that α > B R u ≤ α γ , which completes the proof. (cid:3) H¨older continuity.
From the Harnack inequality, we obtain the following H¨olderregularity of the viscosity solutions to fully nonlinear elliptic integro-di ff erential equationswith respect to L ( λ, Λ , l ). Theorem 3.11 (H¨older continuity) . For < R < , and C > , let u ∈ C ( B R ) be anonnegative function in R n such that M − L ( λ, Λ , l ) u ≤ C L ( ρ R ) , and M + L ( λ, Λ , l ) u ≥ − C L ( ρ R ) in B R in the viscosity sense. Then we haveR α [ u ] α, B R ≤ C (cid:0) k u k L ∞ ( R n ) + C (cid:1) where [ u ] α, B R stands for the α -H¨older seminorm on B R , and uniform constants ρ , α ∈ (0 , and C > depend only on n , λ, Λ , a , a ∞ , ρ and σ . C ,α estimate. In this subsection, we present an interior C ,α estimate for viscosity so-lutions to the elliptic integro-di ff eretial operators as a important consequence of the H¨olderestimate. To apply the incremental quotients technique iteratively in the nonlocal setting,the cancellation condition (20) below for the kernels at infinity is assumed; refer to [CS1,Section 13]. For contants θ > D > , we define L ( λ, Λ , l ; θ , D ) by the class ofthe following linear integro-di ff erential operators with the kernels K : L u ( x ) = Z R n µ ( u , x , y ) K ( y ) dy , such that (2 − σ ) λ l ( | y | ) | y | n ≤ K ( y ) ≤ (2 − σ ) Λ l ( | y | ) | y | n , and(20) Z R n \ B θ | K ( y ) − K ( y − h ) || y | dy ≤ D , ∀| h | < θ . Theorem 3.12.
There is a uniform constant θ > (depending only on n , λ, Λ , a , a ∞ , ρ,σ ) such that if u ∈ C ( B ) is a bounded, nonnegative function in R n such that I u = in B in the viscosity sense for an elliptic operator I with respect to L ( λ, Λ , l ; θ , D ) , then wehave k u k C ,α ( B / ) ≤ C (cid:0) k u k L ∞ ( R n ) + |I | (cid:1) , where α ∈ (0 , and C > depend only on n , λ, Λ , a , a ∞ , ρ, σ , and D . Making use of the incremental quotients, we obtain C ,α -estimate for nonlocal opera-tors. The uniform H¨older estimate in Theorem 3.11 is applicable to w h ( x ) : = u ( x + h ) − u ( x ) | h | α for any small vector h ∈ R n when (20) holds. Indeed, we introduce the cut-o ff function η supported in a smaller ball, and divide the incremental quotient w h into two functions w h , : = η w h and w h , : = (1 − η ) w h . With the help of (20), we deal with the incremental quotient of the kernel K replacing the incremental quotient of u in order to show that |L w h , | is bounded by C k u k L ∞ ( R n ) . So we apply Theorem 3.11 to w h , to deduce C α for the H¨olderexponent α > /α ] times, it follows that u is Lipschitz continuous. By applying the previous argument to the Lipschitz quotient of u , we deduce the uniform C ,α -estimate. Note that the C ,α -estimate is not scale-invariantsince it relies on the values θ and D . Truncated kernels at infinity.
In this subsection, we are concerned with the ellipticintegro-di ff erential operators associated with the symmetric kernels satisfying Property 1.1near zero which may not satisfy Property 1.2 at infinity. This subsection corresponds to[CS1, Section 14] which involves in the results of the fractional Laplacian type integro-di ff erential operators. Consider the linear integro-di ff erential operator LL u ( x ) = Z R n µ ( u , x , y ) K ( y ) dy , with the nonnegative kernel K which is split by K ( y ) = K ( y ) + K ( y ) ≥ R n , where the linear integro-di ff erential operator L with the kernel K belongs to L ( λ, Λ , l ) , and k K k L ( R n ) ≤ κ for κ ≥ . For κ ≥ , we denote by ˜ L ( λ, Λ , l , κ ) the class of all the linearintegro-di ff erential operators above. Using Lemma 3.1, we see that the truncated kernel K at infinity satisfying (2 − σ ) λ l ( | y | ) | y | n χ B (0) ≤ K ( y ) ≤ (2 − σ ) Λ l ( | y | ) | y | n χ B (0) is one of the typical kernels for the linear integro-di ff erential operators belonging to theclass ˜ L ( λ, Λ , l , − σ ) a ∞ /σ ). It is obvious that the larger class ˜ L ( λ, Λ , l , κ ) coincides with L ( λ, Λ , l ) for κ = . The Pucci type extremal operators with respect to the class ˜ L ( λ, Λ , l , κ )are defined as M + ˜ L ( λ, Λ , l ,κ ) u : = sup L∈ ˜ L ( λ, Λ , l ,κ ) L u , M − ˜ L ( λ, Λ , l ,κ ) u : = inf L∈ ˜ L ( λ, Λ , l ,κ ) L u . The same argument as in [CS1, Lemma 14.1] provides the following lemma regardingthe relation between the Pucci type operators with respect to the classes ˜ L ( λ, Λ , l , κ ) and L ( λ, Λ , l ) . Lemma 3.13.
Let u be a bounded function in R n and C , at x . Then we have M − ˜ L ( λ, Λ , l ,κ ) u ( x ) ≥ M − L ( λ, Λ , l ) u ( x ) − κ k u k L ∞ ( R n ) , and M + ˜ L ( λ, Λ , l ,κ ) u ( x ) ≤ M + L ( λ, Λ , l ) u ( x ) + κ k u k L ∞ ( R n ) . Applying Theorem 3.11 combined with Lemma 3.13, we deduce the H¨older estimatefor the elliptic integro-di ff erential operators associated with truncated kernels at infinity. Theorem 3.14.
For < R < , and C > , let u ∈ C ( B R ) be a bounded, nonnegativefunction in R n such that M − ˜ L ( λ, Λ , l ,κ ) u ≤ C L ( ρ R ) , and M + ˜ L ( λ, Λ , l ,κ ) u ≥ − C L ( ρ R ) in B R in the viscosity sense. Then we haveR α [ u ] α, B R ≤ C (cid:8) (1 + κ ) k u k L ∞ ( R n ) + C (cid:9) , NTEGRO-DIFFERENTIAL OPERATORS WITH REGULARLY VARYING KERNELS 27 where uniform constants ρ , α ∈ (0 , and C > depend only on n , λ, Λ , a , a ∞ , ρ and σ .
4. U niform regularity estimates for certain integro - differential operators as σ → − In this section, we study the uniform Harnack inequality and H¨older estimate for theelliptic integro-di ff erential operators associated with the certain regularly varying kernels atzero, where the regularity estimates remain uniform as the order σ ∈ (0 ,
2) of the operatortends to 2 . Consider a measurable function l : (0 , + ∞ ) → (0 , + ∞ ) which stays locallybounded away from 0 and + ∞ , and is slowly varying at zero and infinity. We may assumethat l (1) = . For σ ∈ (0 , , we define(21) l σ ( r ) : = r − σ l ( r ) − σ , ∀ r > . It is easy to check that the function l σ varies regularly at zero and infinity with index − σ ∈ ( − , . As seen in Subsection 1.2, let L ( λ, Λ , l σ ) denote the class of all linearintegro-di ff erential operators L u ( x ) = Z R n µ ( u , x , y ) K ( y ) dy with the kernels K satisfying(2 − σ ) λ l σ ( | y | ) | y | n ≤ K ( y ) ≤ (2 − σ ) Λ l σ ( | y | ) | y | n , where µ ( u , x , y ) : = u ( x + y ) + u ( x − y ) − u ( x ) . For a given constant σ ∈ (0 ,
2) and afunction l , the uniform Harnack inequality and H¨older estimate for fully nonlinear ellipticintegro-di ff erential operators with respect to the class L ( λ, Λ , l σ ) for σ ∈ [ σ ,
2) are estab-lished. In fact, once it is proved that the function l σ satisfies Properties 1.1 and 1.2 withuniform constants a , a ∞ ≥ ρ ∈ (0 ,
1) with respect to σ ∈ [ σ , , the uniform regu-larity estimates follow from the results of Section 3. Here, the regularity estimates dependonly on n , λ, Λ , σ , and the given function l . Therefore, it su ffi ces to prove the followingproposition in order to obtain Theorem 1.9. Proposition 4.1.
For a given σ ∈ (0 , , let δ : = min σ − σ ) , ! . For σ ∈ [ σ , , let l σ be defined as (21) . Then l σ satisfies Properties 1.1 and 1.2 withuniform constants a , a ∞ ≥ and ρ ∈ (0 , with respect to σ ∈ [ σ , , where the constantsa , a ∞ and ρ depend only on σ , and the slowly varying function l .Proof. Since l varies slowly at zero and infinity, there exist c ≥ c ∞ ≥ l ( s ) l ( r ) ≤ c max ((cid:18) sr (cid:19) δ , (cid:18) sr (cid:19) − δ ) ∀ r , s ∈ (0 , , l ( s ) l ( r ) ≤ c ∞ max ((cid:18) sr (cid:19) δ , (cid:18) sr (cid:19) − δ ) ∀ r , s ∈ [1 , + ∞ )from Potter’s theorem; see Appendix A. This implies that l ( s ) l ( r ) ≤ c − σ max ((cid:18) sr (cid:19) − σ + δ (2 − σ ) , (cid:18) sr (cid:19) − σ − δ (2 − σ ) ) ∀ r , s ∈ (0 , , and l ( s ) l ( r ) ≤ c − σ ∞ max ((cid:18) sr (cid:19) − σ + δ (2 − σ ) , (cid:18) sr (cid:19) − σ − δ (2 − σ ) ) ∀ r , s ∈ [1 , + ∞ ) . By choosing a : = c , a ∞ : = c ∞ , and δ = δ ′ = δ (2 − σ ) , the property (a) in Property 1.1and Property 1.2 hold since δ = δ ′ ≤ σ − σ ) (2 − σ ) ≤ σ ≤ σ ∀ σ ∈ [ σ , . Lastly, the property (b) in Property 1.1 will be proved in the following Lemma 4.2. (cid:3)
Lemma 4.2.
Under the same assumption as in Proposition 4.1, we have that σ R r s − l σ ( s ) dsl σ ( r ) → as r → + uniformly with respect to σ ∈ [ σ , . In particular, there exists a uniform constant ρ ∈ (0 , such that for any σ ∈ [ σ , , ≤ σ R r s − l σ ( s ) dsl σ ( r ) = L σ ( r ) l σ ( r ) ≤ , ∀ r ∈ (0 , ρ ) , where L σ ( r ) : = σ Z r s − l σ ( s ) ds . Proof.
For 0 < r < , we rewrite R r s − − σ l ( s ) − σ dsr − σ l ( r ) − σ = Z / r t − − σ l ( tr ) l ( r ) ! − σ dt . Note that for t ∈ (1 , / r ) l ( tr ) l ( r ) ! − σ ≤ c − σ t δ (2 − σ ) , in the proof of Proposition 4.1. So the integrand is bounded by c t − − σ + δ (2 − σ ) which is integrable since σ − δ (2 − σ ) ≥ σ/ ≥ σ / . Thus it follows from the DominatedConvergence Theorem that R / r t − − σ (cid:16) l ( tr ) l ( r ) (cid:17) − σ dt converges to R + ∞ t − − σ dt = σ as r → + since l varies slowly at zero. Now, it remains to show the uniform convergence withrespect to σ ∈ [ σ , . Let ε ∈ (0 ,
1) be given. For a small uniform constant r ∈ (0 ,
1) tobe chosen later, we have σ Z / r t − − σ l ( tr ) l ( r ) ! − σ dt − = σ Z / r t − − σ l ( tr ) l ( r ) ! − σ − dt + σ Z / r / r t − − σ l ( tr ) l ( r ) ! − σ dt − σ Z ∞ / r t − − σ dt = : I + I + I . We select r > ffi ciently small so that for 0 < r < r , I ≤ σ c Z / r / r t − − σ + δ (2 − σ ) dt = σ c σ − δ (2 − σ ) n r σ − δ (2 − σ )0 − r σ − δ (2 − σ ) o ≤ σ c σ r σ/ ≤ c r σ / < ε , and hence | I | = σ Z ∞ / r t − − σ dt ≤ r σ ≤ r σ / < ε . NTEGRO-DIFFERENTIAL OPERATORS WITH REGULARLY VARYING KERNELS 29
Now we claim that for a fixed r > , I : = σ Z / r t − − σ l ( tr ) l ( r ) ! − σ − dt → r → + uniformly with respect to σ ∈ [ σ , . According to the Uniform Convergence Theorem in[BGT, Theorem 1.5.2], we have that l ( tr ) l ( r ) → r → + uniformly for t ∈ [1 , / r ] . Then it follows that l ( tr ) l ( r ) ! − σ uniformly converges to 1 as r → + for t ∈ [1 , / r ] and σ ∈ (0 , . Namely, there exists a uniform constant ρ ∈ (0 ,
1) with respect to σ ∈ [ σ , , depending only on l and σ , such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) l ( tr ) l ( r ) ! − σ − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε , ∀ r ∈ (0 , ρ ) , t ∈ [1 , / r ] . Thus, we have that for any 0 < r < ρ,σ Z / r t − − σ l ( tr ) l ( r ) ! − σ − dt < σε Z / r t − − σ dt ≤ ε (cid:16) − r σ (cid:17) < ε . Therefore, for any 0 < r < ρ, we conclude that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ Z / r t − − σ l ( tr ) l ( r ) ! − σ dt − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ε, which finishes the proof. (cid:3) A ppendix A. R egular variations
We recall regularly varying functions at zero and their properties. The results of regu-larly varying functions at infinity are established in [BGT], which are simple inversions ofthose for regularly varying functions at zero.
Definition A.1 (Regular and slow variations) . Let l : (0 , → (0 , + ∞ ) be a measurablefunction.(i) A function l : (0 , → (0 , + ∞ ) is said to vary regularly at zero with index α ∈ R if forevery κ > r → + l ( κ r ) l ( r ) = κ α . (ii) A regularly varying function is called to be slowly varying if its index α is zero. We state the important properties of regularly and slowly varying functions used in thispaper as a lemma. The proofs and more details for regular and slow variations can be foundin [BGT]; see also [KM, Appendix A].
Lemma A.2.
Let l : (0 , → (0 , + ∞ ) be a measurable function.(i) Any function l that varies regularly with index α ∈ R is of the forml ( r ) = r α l ( r ) for some slowly varying function l . (ii) Let l be a regularly varying function with index − α ≤ which is locally boundedaway from and + ∞ . Then Potter’s theorem [BGT, Theorem 1.5.6] asserts that forany δ > , there exists A δ ≥ such that for < r , s < l ( s ) l ( r ) ≤ A δ max ((cid:18) sr (cid:19) − α + δ , (cid:18) sr (cid:19) − α − δ ) . (iii) Let l be slowly varying and β > − . Then Karamata’s theorem [BGT, Proposition1.5.8] asserts that lim r → + R r s β l ( s ) dsr β + l ( r ) = β + . (iv) Let l be a regularly varying function with index − α < . Then [BGT, Theorem 1.5.11] states that lim r → + R r s − l ( s ) dsl ( r ) = α . This implies that if l varies regularly with index α < , so does the function r R r s − l ( s ) ds . (v) Let l be a regularly varying function with index α ∈ R . Then the Uniform ConvergenceTheorem [BGT, Theorem 1.5.2] asserts thatl ( κ r ) l ( r ) → κ α as r → + uniformly in κ ∈ [ a , b ] for each [ a , b ] ⊂ (0 , + ∞ ) . Acknowledgement
The authors would like to thank Prof. Moritz Kassmann for sug-gesting the problem considered in this paper. Ki-Ahm Lee was supported by the Na-tional Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP)(No.2014R1A2A2A01004618). Ki-Ahm Lee also hold a joint appointment with the Re-search Institute of Mathematics of Seoul National University.R eferences [A] S. Awatif, ´Equations d’Hamilton-Jacobi du premier ordre avec termes int´egro-di ff ´erentiels. I. Unicit´e dessolutions de viscosit´e, Comm. Partial Di ff erential Equations (1991), 1057–1074.[BI] G. Barles and C. Imbert, Second-order elliptic integro-di ff erential equations: Viscosity solutions’ theoryrevisited , Ann. Inst. H. Poincar´e Anal. Non Lin´eaire (2008), 567–585.[BGT] N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation,
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