Regularity estimates for fractional orthotropic p-Laplacians of mixed order

1 Introduction

In this paper, we investigate regularity estimates for weak solutions to nonlocal equations

where L is a nonlinear integro-differential operator of the form

for p > 1 and f ∈ L q ( Q ) for some sufficiently large q . The operator L is clearly determined by the family of measures ( μ ( x , d y ) ) x ∈ R n . In the special case, when L is the generator of a Lévy process, μ ( x , A ) measures the number of expected jumps from x into the set A within the unit time interval. However, the class of operators that we consider in this paper is more involved, and for that reason, we first take a look at an important example. Let n ∈ N . For s 1 , … , s n ∈ ( 0 , 1 ) , we define

This family plays a central role in our paper since admissible operators resp. families of measures will be defined on the basis of μ axes . Given x ∈ R n , the measure μ axes ( x , ⋅ ) only charges differences that occur along the axes

Hence, we can think of the operator Lu for μ ( x , ⋅ ) = μ axes ( x , ⋅ ) as the sum of one-dimensional fractional p-Laplacian in R n with orders of differentiability s 1 , … , s n ∈ ( 0 , 1 ) depending on the respective direction. In particular, μ axes ( x , ⋅ ) does not possess a density with respect to the Lebesgue measure. An interesting phenomenon for the case p = 2 and s = s 1 = ⋯ = s n is that on the one hand the corresponding energies for the fractional Laplacian and L are comparable. On the other hand (for sufficiently good functions), both operators converge to the Laplace operator as s ↗ 1 . It is known that the fractional p-Laplacian converges to the p-Laplacian (see [11, Theorem 2.8] or [20, Lemma 5.1] for details), which is defined by

However, the operator L for μ ( x , ⋅ ) = μ axes ( x , ⋅ ) converges for any p > 1 and s = s 1 = ⋯ = s n to the following local operator (up to a constant depending on p only)

as s ↗ 1 , where a : R n → R n with a ( z ) = ( ∣ z i ∣ p − 2 z i ) i ∈ { 1 , … , n } . This convergence is a direct consequence of the convergence for the one-dimensional fractional p-Laplacian and the summation structure of the operator for μ axes . For details, we refer the reader to Proposition D.1. The operator A loc p is known as orthotropic p-Laplacian and is a well-known operator in the analysis (see, for instance, [41, Chapter 1, Section 8]). This operator is sometimes also called pseudo p-Laplacian. Minimizers for the corresponding energies have been studied in [6], where the authors prove Hölder continuity of minimizers. In [9], local Lipschitz regularity for weak solutions to orthotropic p-Laplace equations for p ≥ 2 and every dimension is proved. The case, when p is allowed to be different in each direction, is also studied in several papers. For instance, in [3], the authors introduce anisotropic De Giorgi classes and study related problems. Another interesting paper studying such operators with nonstandard growth condition is [8], where the authors show that bounded local minimizers are locally Lipschitz continuous. For further results, we refer the reader to the references given in the previously mentioned papers.

The two local operators Δ p and A loc p are substantially different, as Δ p is invariant under orthogonal transformation, while A loc p is not. One strength of our results is that they are robust, and we can recover results for the orthotropic p-Laplacian by taking the limit.

One way to deal with the anisotropy of μ axes is to consider for given s 1 , … , s n ∈ ( 0 , 1 ) a class of suitable rectangles instead of cubes or balls. Hence, we define s max = max { s 1 , … , s n } .

The advantage of taking these cubes is that they take the anisotropy of the measures resp. operators into account and the underlying metric measure space is a doubling space. The choice of s max in the definition of M r ( x ) is not important. It can be replaced by any positive number ς ≥ s max . We only need to ensure that M r ( x ) are balls in a metric measure space with radius r > 0 and center x ∈ R n . This allows us to use known results on doubling spaces like the John-Nirenberg inequality or results on the Hardy-Littlewood maximal function.

In the spirit of [15], for each k ∈ { 1 , … , n } , we define E r k ( x ) = { y ∈ R n : ∣ x k − y k ∣ < r s max / s k } . Note, that

We consider families of measures μ ( x , d y ) , which are given through certain properties regarding the reference family μ axes ( x , d y ) . Let us introduce and briefly discuss our assumptions on the families ( μ ( x , ⋅ ) ) x ∈ R n .

Assumption 1 provides integrability and symmetry of the family of measures. Furthermore, we assume the following tail behavior of ( μ ( x , ⋅ ) ) x ∈ R n .

Note that Assumption 2 is a stronger assumption than an assumption on the volume on the complement of every M r ( x 0 ) . It gives an appropriate tail behavior for the family of measures in each direction separately and allows us to control the appearing constants in our tail estimate in all directions. This is necessary to prove robust estimates for the corresponding operators.

Note that by Assumption 2 and (1.4), we have

Hence, (1.5) shows that Assumption 2 implies μ ( x 0 , R n ⧹ M ρ ( x 0 ) ) ≤ c μ axes ( x 0 , R n ⧹ M ρ ( x 0 ) ) for all x 0 ∈ R n .

Finally, we assume the local comparability of corresponding functionals. Hence, we define for any open and bounded Ω ⊂ R n

and ℰ μ ( u , v ) = ℰ R n μ ( u , v ) whenever these quantities are finite.

Local comparability of the functionals is an essential assumption on the family of measures. It tells us that our family of measures can vary from our reference family in the given sense of local functionals without losing crucial information on ( μ ( x , ⋅ ) ) x ∈ R n like functional inequalities, which we deduce for the explicitly known family ( μ axes ( x , ⋅ ) ) x ∈ R n . This assumption allows us, for instance, to study operators of the form (1.2) for μ axes in the general framework of bounded and measurable coefficients. We emphasize that further examples of families of measures satisfying (1.6) can be constructed similarly to the case p = 2 (see [15, Section 9]). In this paper, we study nonlocal operators of the form (1.2) for families of measures that satisfy the previously given assumptions.

It is not hard to see that the family ( μ axes ( x , ⋅ ) ) x ∈ R n is admissible in the aforementioned sense. Note that Assumptions 1 and 3 are clearly satisfied. Furthermore, for every x 0 ∈ R n , k ∈ { 1 , … , n } and all r > 0 ,

which shows Assumption 2 for Λ = 2 p .

The purpose of this paper is to study weak solutions to nonlocal equations governed by the class of operators L as in (1.2). In order to study weak solutions, we need appropriate Sobolev-type function spaces that guarantee regularity and integrability with respect to μ .

The space V p , μ ( Ω ∣ R n ) can be seen as a nonlocal analog of the space H 1 , p ( Ω ) . It provides fractional regularity (measured in terms of μ ) inside of Ω and integrability on R n ⧹ Ω . The space V p , μ ( Ω ∣ R n ) will serve as solution space. On the other hand, the space H Ω p , μ ( R n ) can be seen as a nonlocal analog of H 0 1 , p ( Ω ) . See [28] and [24] for further studies of these spaces in the case p = 2 .

We are interested in finding robust regularity estimates for weak solutions to a class of nonlocal equations. This means that the constants in the regularity estimates do not depend on the orders of differentiability of the integro-differential operator itself but only on a lower bound of the orders. Let us formulate the main results of this paper. For this purpose, we define s ¯ to be the harmonic mean of the orders s 1 , … , s n , that is,

It is well known that the Harnack inequality fails for weak solutions to singular equations of the type (1.1). Even in the case p = 2 and s 1 = ⋯ = s n , the Harnack inequality does not hold (see, for instance, [4,7]). Our first main result is a weak Harnack inequality for weak supersolutions to equations of the type (1.1). Throughout the paper, we denote by p ⋆ = n p / ( n − p s ¯ ) the Sobolev exponent, which will appear in Theorem 2.7.

Although the weak Harnack inequality provides an estimate on the infimum only, it is sufficient to prove a decay of oscillation for bounded weak solutions and therefore a local Hölder estimate.

Note that the result needs global boundedness of weak solutions. The same assumption is also needed in the previous works [14,15,25]. Global and local boundedness of weak solutions to anisotropic nonlocal equations are nontrivial open questions.

Furthermore, note that the general case, replacing M 1 4 , M 1 2 , M 15 16 , M 1 by M r 4 ( x 0 ) , M r 2 ( x 0 ) , M 15 r 16 ( x 0 ) , M r ( x 0 ) , for x 0 ∈ R n and r ∈ ( 0 , 1 ] follows by a translation and anisotropic scaling argument introduced in Section 4. See also [14].

Let us comment on related results in the literature. The underlying ideas in developing regularity results for uniformly elliptic operators in divergence form with bounded and measurable coefficients go back to the influential contributions by De Giorgi, Nash, and Moser (see [19,45,46]). These works led to many further results in various directions. Similar results for nonlocal operators in divergence form have been obtained by several authors including these works: [1, 2,5,12,16,17,18,22,24,26,27, 33,34,38,42,43,44,54,55]. See also the references therein. For further regularity results concerning nonlocal equations governed by fractional p-Laplacians, we refer the reader to [10,38,39,48,49,50,51].

In [22], the authors extend the De Giorgi-Nash-Moser theory to a class of fractional p-Laplace equations. They provide the existence of a unique minimizer to homogenous equations and prove local regularity estimates for weak solutions. Moreover, in [21], the same authors prove a general Harnack inequality for weak solutions.

Nonlocal operators with anisotropic and singular kernels of the type μ axes are studied in various mathematical areas such as stochastic differential equations and potential theory. In [4], the authors study regularity estimates for harmonic functions for systems of stochastic differential equations d X t = A ( X t − ) d Z t driven by Lévy processes Z t with Lévy measure μ axes ( 0 , d y ) , where 2 s 1 = ⋯ = 2 s n = α and p = 2 . See also [13,56,52]. Sharp two-sided bounds for the heat kernels are established in [35,37]. In [36], the authors prove the existence of transition density of the process X t and establish semigroup properties of solutions. The existence of densities for solutions to stochastic differential equations with Hölder continuous coefficients driven by Lévy processes with anisotropic jumps has been proved in [30]. Such types of anisotropies also appear in the study of the anisotropic stable JCIR process, see [29].

Our approach follows mainly the ideas of [14,25] and [15]. In [25], the authors develop a local regularity theory for a class of linear nonlocal operators, which covers the case s = s 1 = ⋯ = s n ∈ ( 0 , 1 ) and p = 2 . Based on the ideas of [25], the authors in [14] establish regularity estimates in the case p = 2 for weak solutions in a more general framework, which allows the orders of differentiability s 1 , … , s n to be different. In [15], parabolic equations in the case p = 2 and possible different orders of differentiability are studied. That paper provides robust regularity estimates, which means the constants in the weak Harnack inequality and Hölder regularity estimate do not depend on the orders of differentiability but on their lower one, only. This allows us to recover regularity results for local operators from the theory of nonlocal operators by considering the limit.

The purpose of this paper is to provide local regularity estimates as in [14] for operators which are allowed to be nonlinear. This nonlinearity leads to several difficulties like the need for a different proof for the discrete gradient estimate (see Lemma 3.4). Since we cannot use the helpful properties of Hilbert spaces (like Plancherel’s theorem), we also need an approach different from the one in [14] to prove the Sobolev-type inequality. One strength of this paper is the robustness of all results. This allows us to recover regularity estimates for the limit operators such as for the orthotropic p-Laplacian.

Finally, we would like to point out that it is also interesting to study such operators in nondivergence form. We refer the reader to [53] for regularity results concerning the fractional Laplacian and to [40] for the fractional p-Laplacian. See also [23] for the anisotropic case.

Even in the most simple case, that is, p = 2 and s = s 1 = s 2 = ⋯ = s n , regularity estimates for operators in nondivergence form of the type (1.2) with μ = μ axes lead to various open problems such as an Alexandrov-Bakelmann-Pucci estimate.

The authors wish to express their thanks to Lorenzo Brasco for helpful comments.

Outline: This paper is organized as follows. In Section 2, we introduce appropriate cut-off functions and prove auxiliary results concerning functionals for admissible families of measures. One main result of that section is the Sobolev-type inequality (see Theorem 2.7). In Section 3, we prove the weak Harnack inequality, and Section 4 presents the proof of the local Hölder estimate. In Appendix A, we prove some auxiliary algebraic inequalities, and in Appendix B, we briefly sketch the construction of appropriate anisotropic “dyadic” rectangles. In Appendix C, we use the anisotropic dyadic rectangles to sketch the proof of a suitable sharp maximal function theorem.

2 Auxiliary results

This section is devoted to providing some general properties for the class of nonlocal operators that we study in the scope of this paper. The main auxiliary result is a robust Sobolev-type inequality.

Let us first introduce a class of suitable cut-off functions that will be useful for appropriate localization.

For brevity, we simply write τ for any such function from ( τ x 0 , r , λ ) x 0 , r , λ , if the respective choice of x 0 , r and λ is clear. The existence of such functions is standard.

Recall the definition of admissible families of measures K ( p , s 0 , Λ ) from Definition 1.2.

For future purposes, we deduce the following observation. It is an immediate consequence of the foregoing lemma.

Note that the constants in Lemma 2.2 and Corollary 2.3 do not depend on s 0 . Therefore, the lower bound s 0 ≤ s k for all k ∈ { 1 , … , n } can be dropped here.