On the H\"older regularity for solutions of integro-differential equations like the anisotropic fractional Laplacian

TL;DR

This paper investigates the regularity properties of solutions to anisotropic fractional Laplacian equations, adapting classical techniques to establish H"older and $C^{1, eta}$ regularity for solutions.

Contribution

It extends the De Giorgi method and geometric techniques to anisotropic fractional Laplacian equations, providing new regularity results for viscosity solutions.

Findings

Established $C^{eta}$-regularity for solutions of class $C^{2}$.

Derived an ABP estimate and Harnack inequality for viscosity solutions.

Proved interior $C^{1, eta}$ regularity for solutions.

Abstract

In this paper we study integro-differential equations like the anisotropic fractional Laplacian. As in [Silvestre, Indiana University Mathematics Journal 55, 2006], we adapt the De Giorgi technique to achieve the $C^{\gamma}$-regularity for solutions of class $C^{2}$ and use the geometry found in [Caffarelli, Leit\~ao, and Urbano, Math. Ann. 360, 2014] to get an ABP estimate, a Harnack inequality and the interior $C^{1, \gamma}$ regularity for viscosity solutions.

Equations

$$(-\Delta {)}^{\beta ,s}f(x)=C_{\beta ,s}{\int }_{{\mathbb{R}}^n}\frac{f(x)-f(\varsigma )}{{\left({\sum }_{i=1}^n|{\varsigma }_i-x_i{|}^{b_i}\right)}^{\frac{c+s}{2}}}d\varsigma ,$$

$$Lu\left(x\right)=0,$$

$$Lu\left(x\right):={\int }_{{\mathbb{R}}^n}\left(u\left(x+y\right)-u\left(x\right)-{\chi }_{B_1}\left(y\right)\nabla u\left(x\right)\cdot y\right)\mathcal{K}\left(y\right)dy,$$

$$\frac{\lambda q_{\max ,s}}{\Vert y{\Vert }^{c+s}}\le \mathcal{K}\left(y\right)\le \frac{\Lambda q_{\max ,s}}{\Vert y{\Vert }^{c+s}},\forall y\in {\mathbb{R}}^n\setminus \left\{0\right\},$$

$$\Vert y{\Vert }^2=\sum _{i=1}^n|y_i{|}^{b_i}\text{and}{q}_{\max ,s}=\frac{4}{b_{\max }}-s.$$

$$\mathcal{K}(y)=\frac{1}{\Vert y{\Vert }^{c+s}}.$$

$$\kappa L_r\eta (x)+2{\int }_{{\mathbb{R}}^n\setminus B_{\frac{1}{4}}}(|8y{|}^{\tau }-1)\mathcal{K}(ry)r^{n}dy<\frac{1}{2}\underset{\mathcal{B}\subset B_2,|\mathcal{B}|<\delta }{\inf }{\int }_{\mathcal{B}}\mathcal{K}(ry)r^{n}dy,$$

$$L_{r}v(x):={\int }_{{\mathbb{R}}^n}\left(v\left(x+y\right)-v\left(x\right)-{\chi }_{B_1}\left(ry\right)\nabla v\left(x\right)\cdot y\right)\mathcal{K}\left(ry\right)r^{n}dy.$$

$$-C\le L_r\eta (x)\le C,$$

$$T_{\beta ,r}{e}_i=r^{\frac{2}{b_i}}{e}_i,$$

$$\mathcal{I}u\left(x\right):=\underset{\alpha }{\inf }\underset{\beta }{\sup }{L}_{\alpha \beta }u\left(x\right)=0,$$

$$b_i=n+{\sigma }_i\text{and}s=2-c$$

$$c_{\sigma }=q_{\max ,s}\text{and}\Vert y{\Vert }^{c+s}=\sum _{i=1}^n|y_i{|}^{n+{\sigma }_i}$$

$${\Theta }_r\left(x\right):=\left\{\left(y_1,\dots ,y_n\right)\in {\mathbb{R}}^n:\Vert y-x\Vert <r\right\}.$$

$$E_{r,l}\left(x\right):=\left\{\left(y_1,\dots ,y_n\right)\in {\mathbb{R}}^n:\sum _{i=1}^n\frac{{\left(y_i-x_i\right)}^2}{r^{\frac{4}{b_i}}}<l^2\right\}.$$

$$R_{r,l}\left(x\right):=\left\{\left(y_1,\dots ,y_n\right)\in {\mathbb{R}}^n:|y_i-x_i|<l^{\frac{2}{b_{\min }}}{r}^{\frac{2}{b_i}}\right\}$$

$$E_{r,l}^{\max }\left(x\right):=\left\{\left(y_1,\dots ,y_n\right)\in {\mathbb{R}}^n:\sum _{i=1}^n\frac{{\left(y_i-x_i\right)}^2}{r^{2\frac{b_{\max }}{b_i}}}<l^2\right\}.$$

$$R(x):=\left\{\left(y_1,\dots ,y_n\right)\in {\mathbb{R}}^n:|y_i-x_i|<l_i\right\}$$

$$R\left(x\right)\subset \left\{\left(y_1,\dots ,y_n\right)\in {\mathbb{R}}^n:|y_i-x_i|<2^{-\mathfrak{C}(k+1)}{r}^{\frac{2}{b_i}}\right\},$$

$$\tilde{R}\left(x\right):=\left\{\left(y_1,\dots ,y_n\right)\in {\mathbb{R}}^n:|y_i-x_i|<{\left[2^{-\mathfrak{C}\left(\frac{b_{\min }}{2}\right)k}r\right]}^{\frac{2}{b_i}}\right\}.$$

$$B_r=B_r(0),{\Theta }_r={\Theta }_r(0)\text{and}{E}_{r,l}^{\max }=E_{r,l}^{\max }(0).$$

$$T_{\beta ,r}{e}_i=r^{\frac{2}{b_i}}{e}_i\text{or}{T}_{\max ,r}{e}_i=r^{\frac{b_{\max }}{b_i}}{e}_i,$$

$$\displaystyle\eta(y)=\left\{\begin{array}[]{ll}(1-|y|^{2})^{2},\ \text{ if }\ y\in B_{1},\\ \\ 0,\ \text{ if }\ y\in(\mathbb{R}^{n}\setminus B_{1}).\end{array}\right.$$

$$|(-\Delta {)}^{\beta ,s}\eta (x)|\le C\text{for all}x\in B_{3/4}.$$

$$|\eta (x)-\eta (x+y)+⟨\nabla \eta (x),y⟩|\le C|y{|}^2\text{for all}(x,y)\in B_{3/4}\times E_{r_0,1},$$

$${\int }_{E_{r_0,1}}\frac{|\eta (x)-\eta (x+y)+⟨\nabla \eta (x),y⟩|}{\Vert y{\Vert }^{c+s}}dy$$

$$\sum _{k=0}^{\infty }{r}_k^{-s}{\int }_{B_1\setminus E_{1/2,1}}\frac{|T_{r_k}y{|}^2}{\Vert y{\Vert }^{c+s}}dy$$

$${\int }_{{\mathbb{R}}^n\setminus E_{r_0,1}}\frac{|\eta (x)-\eta (x+y)|}{\Vert y{\Vert }^{c+s}}dy$$

$$|(-\Delta {)}^{\beta ,s}\eta (x)|$$

$$\kappa (-\Delta {)}^{\beta ,s}\eta (x)+2{\int }_{{\mathbb{R}}^n\setminus B_{\frac{1}{4}}}(|8y{|}^{\tau }-1){\mathcal{K}}_0(y)dy<\frac{1}{2}\underset{\mathcal{B}\subset B_2,|\mathcal{B}|<\delta }{\inf }{\int }_{\mathcal{B}}{\mathcal{K}}_0(y)dy,$$

Full text

Key words: Fractional Laplacian, integro-differential equations, regularity theory, anisotropy.

AMS Subject Classification MSC 2010: 26A33; 35J70; 47G20, 35J60, 35D35, 35D40, 35B65

1. 1 Introduction

2. 2 Preliminaries

3. 2.1 Smooth solutions

4. 2.2 Viscosity solutions and extremal operators

5. 3 Hölder Regularity: smooth solutions

6. 4 Hölder Regularity: viscosity solutions

7. 4.1 Nonlocal anisotropic ABP estimate

8. 4.2 A barrier function

9. 4.3 Harnack inequality and regularity

1 Introduction

In [19], the second author presents the anisotropic fractional Laplacian

[TABLE]

where $\beta=\left(b_{1},\dots,b_{n}\right)\in\mathbb{R}^{n}$ represents the different homogeneities in different directions, $b_{i}>0$, $0<s<2$, $c=\displaystyle\sum_{i=1}^{n}\frac{2}{b_{i}}$ and $C_{\beta,s}>0$ is a normalization constant. In this work we develop a regularity theory for integro-differential equations like the anisotropic fractional Laplacian

[TABLE]

where

[TABLE]

$0<s<\frac{4}{b_{\max}}$, and the kernel $\mathcal{K}$ is symmetric, $\mathcal{K}(y)=\mathcal{K}(-y)$, and satisfy the anisotropic bounds

[TABLE]

where $0<\lambda\leq\Lambda$ and we denote $b_{\max}=\max\left\{b_{1},\dots,b_{n}\right\}$,

[TABLE]

Integro-differential equations appear in the context of discontinuous stochastic processes. For example, competitive stochastic games with two or more players, which are allowed to choose from different strategies at every step in order to maximize the expected value of some function at the first exit point of a domain. Integral operators like (1.1) correspond to purely jump processes when diffusion and drift are neglected. The anisotropic setting we consider also appears in the context of magnetic resonance imaging (MRI) of the human brain (cf. [20, 15]), anomalous diffusion (cf. [23]), biological tissues (cf. [23, 14]), financial mathematics (see [24, 8]).

The main difference between the fractional Laplacian $(-\Delta)^{s}$ and the anisotropic fractional Laplacian $(-\Delta)^{\beta,s}$ is the geometry determined by the kernel

[TABLE]

In the seminal work [8], this anisotropic geometry required a refinement of the techniques presented in [9]: for example, a new covering lemma and a suitable scaling. Recently, in [19], the second author studied an extension problem related to anisotropic fractional Laplacian and a riemannian metric $g$ was crucial to get an anisotropic version of the Almgren’s frequency formula obtained in [10].

The paper is divided into two parts. In the sequel, we comment on the strategies to achieve our results:

1. (Smooth solution). In the first part of the paper, we will show that the De Giorgi’s approach, see [12, 17], allows us to reach the $C^{\gamma}$-regularity for smooth solutions $u$ of (1.2), where the estimates do not depend on the norm of any derivative or modulus of continuity of $u$. As in [25], we will control the behavior of a solution $u$ of (1.2) away from the origin to obtain a Growth Lemma and use an iterate argument to get the desired regularity. In this analysis, two tools are crucial: barrier function and suitable scaling. In fact, in order to find an appropriate way to control the behavior of $u$ away from the origin in the isotropic case [25], Silvestre established an interesting inequality involving radial barriers $\eta$ and the kernel $\mathcal{K}$:

Silvestre inequality. Given a $\delta>0$, there exist $\kappa>0$ and $\tau>0$ only depending on $\beta$, dimension $n$, $s$ and $\delta$ such that for all $r>0$ and $x\in\mathbb{R}^{n}$:

[TABLE]

where

[TABLE]

The Silvestre inequality reveals the appropriate scaling for our analysis: the scaling determined by the kernel $\mathcal{K}$. Furthermore, the barrier functions $\eta$ should satisfy the bounds:

[TABLE]

for some positive constant $C$ depending on $\beta$, dimension $n$, and $s$. In our case, we will use radial functions as barrier functions and the anisotropic scaling $T_{\beta,r}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ defined by

[TABLE]

where $e_{i}$ is the $i$-th canonical vector, to get the anisotropic Silvestre inequality and access to the $C^{\gamma}$-regularity.

2. (Viscosity solution). In the second part of the paper, we get the regularity theory established in [9, 8] for viscosity solutions of non-local Isaac’s equation like the anisotropic fractional Laplacian

[TABLE]

where $L_{\alpha,\beta}$ is as in (1.2). An important example of the equation (1.8) was studied in [8]. In fact, if

[TABLE]

where $\sigma_{i}\in(0,2)$ we have

[TABLE]

for $\sigma=\left(\sigma_{1},\dots,\sigma_{n}\right)$. In [9, 8], the key that gives access to the regularity theory to viscosity solutions $u$ of the equation (1.8) is a non-local ABP estimate. In [8], the correct geometry to reach a non-local ABP estimate for integro-differential equation governed by anisotropic kernels $\mathcal{K}_{\alpha,\beta}$ was discovered. More precisely, the geometry determined by the level sets of the kernels $\mathcal{K}_{\alpha\beta}$:

[TABLE]

With this geometry at hand, three steps are fundamental to obtain a non-local ABP estimate, a Harnack Inequality and the desired regularity:

$u$ stays quadratically close to the tangent plane to concave envelope $\Gamma$ of $u$ in a (large) portion of the neighbourhoods of the contact points and such that, in smaller neighbourhoods (with the same geometry), the concave envelope $\Gamma$ has quadratic growth: here, our neighbourhoods are ellipses $E_{r,1}$ with the same geometry of $\Theta_{r}$. 2. 2.

Covering Lemma. Since our neighbourhoods will be ellipses $E_{r,1}$, our covering is naturally made of $n$-dimensional rectangles $\mathcal{R}_{r}$ and we invoke a covering lemma from [5]. 3. 3.

A barrier function. We use the natural anisotropic scaling $T_{\beta,r}$ and a radial function to build an adequate barrier function and, together with the nonlocal anisotropic version of the ABP estimate, we get a lemma that links a pointwise estimate with an estimate in measure, Lemma 4.13. This is the crucial step towards a regularity theory. The iteration of Lemma 4.13 implies the decay of the distribution function $\lambda_{u}:=\left|\left\{u>t\right\}\right|$ and the tool that makes this iteration possible is the so called Calderón -Zygmund decomposition. Since our scaling is anisotropic we need a Calderón -Zygmund decomposition for $n$-dimensional rectangles generated by our scaling. A fundamental device we use for that decomposition is the Lebesgue differentiation theorem for $n$-dimensional rectangles that satisfy the condition of Caffarelli-Calderón in [5]. Hence we obtain the Harnack inequality and, as a consequence, we achieve the interior $C^{\gamma}$ regularity for a solution $u$ of equation (1.8) and, under additional assumptions on the kernels $\mathcal{K}_{\alpha\beta}$, interior $C^{1,\gamma}$ estimates.

Finally, we emphasize that the restriction $0<s<4/b_{\max}$ in our results comes from the class of solutions $u$ we are studying: solutions of class $C^{2}$ or viscosity solutions ($u$ is touched by a $C^{2}$ function). However, we believe that the results obtained here can naturally be extended for $0<s<2$ if we consider an appropriate class of solutions $u$ and change the metric of $\mathbb{R}^{n}$, a namely, $(\mathbb{R}^{n},g$), where $g$ is the metric determined by kernel $\|\cdot\|$, see [19]. We plan to address this issue in a forthcoming paper. Furthermore, the Lemma [6] allows the homogeneity degrees $b_{i}$ depend on $x$, see [11]. We would also like to mention that in [7] an important regularity theory for integro-differential equations was developed, where the kernels are singular, and only charge the coordinate axes for the jumps, and each axis may charge jumps with a different exponent.

The paper is organized as follows. In section 2 we gather all the necessary tools for our analysis: fundamental geometry, Silvestre inequality, the notion of viscosity solution for the problem (1.8), the extremal operators of Pucci type associated with the family of kernels $K_{\alpha\beta}$ and some notation. In Section 3 we present the proof of $C^{\gamma}$-regularity of smooth solutions and as a corollary we get a result type Liouville. The Section 4 is divided in three subsections: 4.1, where the nonlocal ABP estimate for a solution $u$ of equation (1.8) is obtained, is the most important of the paper. Sections 4.2 and 4.3 are devoted to the proof of the Harnack inequality and its consequences.

2 Preliminaries

In this section we gather anisotropic versions of some results obtained in [25, 8]. We begin with geometric informations that we will systematically use along the work.

Given $r,l>0$ and $x\in\mathbb{R}^{n}$, we will denote

[TABLE]

If $b_{\min}=\min\left\{b_{1},\dots,b_{n}\right\}$ and $b_{\max}=\max\left\{b_{1},\dots,b_{n}\right\}$ we define

[TABLE]

and

[TABLE]

Furthermore, if $\mathfrak{C}=\mathfrak{C}>0$ is a natural number and the $n$-dimensional rectangle

[TABLE]

satisfies

[TABLE]

for some number natural $k$, we define the corresponding $n$-dimensional rectangle $\tilde{R}(x)$ by

[TABLE]

We will also consider the notation

[TABLE]

The geometric properties of the sets defined above will be crucial in our analysis. We collect them in the following Lemma.

Lemma 2.1** (Fundamental Geometry).**

Let $r>0$ and $l>0$. Then, given $x\in\mathbb{R}^{n}$, we have the following relations:

$E_{r,1}(x)\subset\Theta_{r\sqrt{n}}\subset E_{r\mathfrak{C},1}(x)$* and $E_{2^{-\mathfrak{C}}r}(x)\subset E_{r,\frac{1}{4}}(x)$, for some natural number $\mathfrak{C}=\mathfrak{C}\left(n,b_{\max}\right)>0$.* 2. 2.

If $R$ is a $n$-dimensional rectangle, then $R(x)\subset\tilde{R}(x)$. Moreover, $R_{r,l}(x)\subset E_{(rl)c_{\max},1}(x)$, where $c_{\max}=n^{\frac{b_{\max}}{4}}$, if $r,l\in(0,1)$. 3. 3.

$E^{\max}_{\frac{r}{2},1}(x)\subset E^{\max}_{r,1/2}(x)$* and $E^{\max}_{r,l}(x)\subset E^{\max}_{rl,1}(x)$, if $l\geq 1$.* 4. 4.

If $\tau_{1}$ is the topology generated by Euclidean balls $B_{r}(z)$ and $\tau_{2}$ is the topology generated by anisotropic balls $\Theta_{r}(z)$, then $\tau_{1}=\tau_{2}$. 5. 5.

If $T_{\beta,r}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is defined by

[TABLE]

where $e_{i}$ is the i-th canonical vector, then $T_{\beta,r}(B_{l})=E_{r,l}$ or $T_{\max,r}(B_{l})=E^{\max}_{r,l}$.

Next we will divide this section into two subsections: Smooth solutions and Viscosity solutions and extremal operators.

2.1 Smooth solutions

Without loss of generality, we consider $L=(-\Delta)^{\beta,s}$. In this subsection, we establish the tools to get the regularity $C^{\gamma}$ for $\Delta^{\beta,s}$-harmonic smooth functions. Precisely, we show that the operator $\Delta^{\beta,s}$ applied to radial functions $\eta$ is bounded for $s\in\left(0,4/b_{\max}\right)$ and we get the Silvestre inequality for $\Delta^{\beta,s}\eta$.

Lemma 2.2** (Barrier function).**

Let $\eta:\mathbb{R}^{n}\rightarrow\mathbb{R}$ defined by

[TABLE]

There exist $C>0$ only depending on $\beta$ , dimension $n$ and $s$ such that

[TABLE]

Proof.

Choose $r_{0}=r_{0}(n,\beta)\in(0,1)$ such that

[TABLE]

where $C$ is a positive constant only depending on $\beta$, and dimension $n$. Denote $T_{r_{k}}:=T_{\beta,r_{k}}$, where $r_{k}=r_{0}2^{-k}$. Then, we get

[TABLE]

and we can estimate

[TABLE]

where $C(n,\beta,s)=\displaystyle\int_{B_{1}\setminus E_{1/2,1}}\dfrac{|y|^{2}}{\|y\|^{c+s}}dy$. On the other hand, if $r_{1}=r_{1}(r_{0})>0$ is such that $\Theta_{r_{1}}\subset E_{r_{0},1}$, we obtain

[TABLE]

Then, we find

[TABLE]

∎

Taking into account (2.9) we get the Silvestre inequality for $\Delta^{\beta,s}$:

Lemma 2.3** (Silvestre inequality).**

Given a $\delta>0$, there exist $0<\kappa<\frac{1}{4}$ and $\tau>0$ only depending on $\beta$, dimension $n$, $s$ and $\delta$ such that

[TABLE]

for all $x\in B_{3/4}$, where $\mathcal{K}_{0}(y):=\frac{1}{\|y\|^{c+s}}$ for all $y\in\mathbb{R}^{n}\setminus\left\{0\right\}$.

2.2 Viscosity solutions and extremal operators

In this subsection we collect the technical properties of the operator $\mathcal{I}$ that we will use throughout the paper. Since $\mathcal{K}_{\alpha\beta}$ is symmetric and positive, we obtain

[TABLE]

and

[TABLE]

For convenience of notation, we denote

[TABLE]

and we can write

[TABLE]

for some kernel $\mathcal{K}_{\alpha\beta}$.

We now define the adequate class of test functions for our operators.

Definition 2.4**.**

A function $\varphi$ is said to be $C^{1,1}$ at the point $x$, and we write $\varphi\in C^{1,1}\left(x\right)$, if there is a vector $v\in\mathbb{R}^{n}$ and numbers $M,\eta_{0}>0$ such that

[TABLE]

for $|x|<\eta_{0}$. We say that a function $\varphi$ is $C^{1,1}$ in a set $\Omega$, and we denote $\varphi\in C^{1,1}\left(\Omega\right)$, if the previous holds at every point, with a uniform constant $M$.

Remark 2.5*.*

Let $u\in C^{1,1}\left(x\right)\cap L^{\infty}\left(\mathbb{R}^{n}\right)$ and $M>0$ and $\eta_{0}>0$ be as in definition 2.4. Then, by Lemma 2.2, we find

[TABLE]

We now introduce the notion of viscosity subsolution (and supersolution) $u$ in a domain $\Omega$, with $C^{2}$ test functions that touch $u$ from above or from below. We stress that $u$ is allowed to have arbitrary discontinuities outside of $\Omega$.

Definition 2.6**.**

Let $f$ be a bounded and continuous function in $\mathbb{R}^{n}$. A function $u:\mathbb{R}^{n}\rightarrow\mathbb{R}$, upper (lower) semicontinuous in $\overline{\Omega}$, is said to be a subsolution (supersolution) to equation $\mathcal{I}u=f$, and we write $\mathcal{I}u\geq f$ ($\mathcal{I}u\leq f$), if whenever the following happen:

$x_{0}\in\Omega$ is any point in $\Omega$; 2. 2.

$B_{r}\left(x_{0}\right)\subset\Omega$, for some $r>0$; 3. 3.

$\varphi\in C^{2}\left(\overline{B_{r}\left(x_{0}\right)}\right)$; 4. 4.

$\varphi\left(x_{0}\right)=u\left(x_{0}\right)$; 5. 5.

$\varphi\left(y\right)>u\left(y\right)$ ($\varphi\left(y\right)<u\left(y\right)$) for every $y\in B_{r}\left(x_{0}\right)\setminus\left\{x_{0}\right\}$;

then, if we let

[TABLE]

we have $\mathcal{I}v\left(x_{0}\right)\geq f\left(x_{0}\right)$ ($\mathcal{I}v\left(x_{0}\right)\leq f\left(x_{0}\right)$).

Remark 2.7*.*

Functions which are $C^{1,1}$ at a contact point $x$ can be used as test functions in the definition of viscosity solution (see Lemma 4.3 in [9]).

Next, we define the class of linear integro-differential operators that will be a fundamental tool for the regularity analysis.

Definition 2.8**.**

Let $\mathfrak{L}_{0}$ be the collection of linear operators $L_{\alpha\beta}$. We define the maximal and minimal operator with respect to $\mathfrak{L}_{0}$ as

[TABLE]

and

[TABLE]

By definition, if $\mathcal{M}^{+}u\left(x\right)<\infty$ and $\mathcal{M}^{-}u\left(x\right)<\infty$, we get

[TABLE]

and

[TABLE]

The proofs of the results that we now present can be found in the sections $3$, $4$ and $5$ of [9]. The first result ensures that if $u$ can be touched from above, at a point $x$, with a paraboloid then $Iu\left(x\right)$ can be evaluated classically.

Lemma 2.9**.**

If we have a subsolution, $\mathcal{I}u\geq f$ in $\Omega$, and $\varphi$ is a $C^{2}$ function that touches $u$ from above at a point $x\in\Omega$, then $\mathcal{I}u\left(x\right)$ is defined in the classical sense and $\mathcal{I}u\left(x\right)\geq f\left(x\right)$.

Another important property of $\mathcal{I}$ is the continuity of $\mathcal{I}\varphi$ in $\Omega$ if $\varphi\in C^{1,1}\left(\Omega\right)$.

Lemma 2.10**.**

Let $v$ be a bounded function in $\mathbb{R}^{n}$ and $C^{1,1}$ in some open set $\Omega$. Then $\mathcal{I}v$ is continuous in $\Omega$.

The next lemma allows us to conclude that the difference between a subsolution of the maximal operator $\mathcal{M}^{+}$ and a supersolution of the minimal operator $\mathcal{M}^{-}$ is a subsolution of the maximal operator.

Lemma 2.11**.**

Let $\Omega$ be a bounded open set and $u$ and $v$ be two bounded functions in $\mathbb{R}^{n}$ such that

$u$* is upper-semicontinuous and $v$ is lower-semicontinuous in $\overline{\Omega}$;* 2. 2.

$\mathcal{I}u\geq f$* and $\mathcal{I}v\leq g$ in the viscosity sense in $\Omega$ for two continuous functions $f$ and $g$.*

Then

[TABLE]

in the viscosity sense.

3 Hölder Regularity: smooth solutions

As in [25] we will use the De Giorgi’s approach to achieve the $C^{\gamma}$-regularity for $\Delta^{\beta,s}$-harmonic smooth functions. We begin with a Growth lemma.

Lemma 3.1** (Growth lemma).**

If $u$ is a function that satisfies:

$(-\Delta)^{\beta,s}u\leq 0$* in $B_{1}$;* 2. 2.

$u\leq 1$* in $B_{1}$;* 3. 3.

$u(x)\leq 2|2x|^{\tau}-1$* for all $x\in\mathbb{R}^{n}\setminus B_{1}$;* 4. 4.

$|\left\{x\in B_{1}:u(x)\leq 0\right\}|>\delta$.

Then, there exists a constant $\mu=\mu(n,s,\beta,\delta)>0$ such that $u\leq 1-\mu$ in $B_{1/2}$.

Proof.

Consider $\mu=\kappa(\eta(1/2)-\eta(3/4))$. Suppose, for the purpose of contradiction, that there exists $x_{0}\in B_{\frac{1}{2}}$ such that

[TABLE]

Thus, since $\eta$ is decreasing in any ray from the origin and $u\leq 1$ in $B_{1}$, we have

[TABLE]

where $v(x)=u(x)+\kappa\eta(x)$. Then, we conclude that

[TABLE]

for some $x_{1}\in B_{\frac{3}{4}}$. If we define

[TABLE]

we can write

[TABLE]

where we denote

[TABLE]

Since $v$ has a maximum at $x_{1}$ and $v(x_{1})\geq 1$ we estimate

[TABLE]

Using the conditions 2 and 3 we find

[TABLE]

Moreover, since $(\mathbb{R}^{n}\setminus\mathcal{B})\subset(\mathbb{R}^{n}\setminus B_{1/4})$ we obtain

[TABLE]

From condition 1 we have

[TABLE]

and using the condition 4 we obtain

[TABLE]

which contradicts (2.10). ∎

Using the anisotropic scaling $T_{\max,r}$ and Lemma 3.1 we get the following scaled version.

Lemma 3.2** (Growth lemma-anisotropic).**

If $u$ is a function that satisfies:

$(-\Delta)^{\beta,s}u\leq 0$* in $E^{\max}_{r,1}(x_{0})$;* 2. 2.

$u\leq C$* in $E^{\max}_{r,1}(x_{0})$;* 3. 3.

$u(x)\leq C\left(2|2T^{-1}_{\max,r}(x-x_{0})|^{\tau}-1\right)$* for all $x\in\mathbb{R}^{n}\setminus E^{\max}_{r,1}(x_{0})$;* 4. 4.

$\dfrac{|\left\{x\in E^{\max}_{r,1}(x_{0}):u(x)\leq 0\right\}|}{r^{\frac{b_{\max}}{2}c}}>\delta$.

Then, there exists a constant $\mu=\mu(n,s,\beta,\delta)>0$ such that $u\leq C(1-\mu)$ in $E^{\max}_{\frac{r}{2},1}$.

Proof.

Define

[TABLE]

for all $x\in\mathbb{R}^{n}$. Since $T_{\max,r}(B_{1})=E^{\max}_{r,1}$ we conclude that $v$ satisfies 2 and 3. Furthermore, we find

[TABLE]

By Lemma 3.1 there exists a constant $\mu=\mu(n,s,\beta)>0$ such that $v\leq 1-\mu$ in $B_{1/2}$. Thus, we find $u\leq C(1-\mu)$ in $E^{\max}_{r,1/2}$. Finally, by Lemma 2.1 we have $E^{\max}_{\frac{r}{2},1}\subset E^{\max}_{r,1/2}$ and the Lemma 3.2 is concluded. ∎

Theorem 3.3**.**

If $u$ is a bounded function that satisfies $(-\Delta)^{\beta,s}u=0$ in $E^{\max}_{2r,1}$, then for $\delta=\frac{|B_{1}|}{2}$ there exist constants $\gamma=\gamma(n,s,\beta)\in(0,1)$ and $C=C(n,s,\beta)>0$ such that

[TABLE]

In particular, $u\in C_{loc}^{\frac{\gamma b_{\min}}{2}}(E^{\max}_{r,1})$.

Proof.

By considering the anisotropic scaling $v(x)=u(T_{\max,r}x)/2\|u\|_{\infty}$ we can suppose that $\text{osc}_{\mathbb{R}^{n}}u=1$ and $r=1$. As in [25], given $x_{0}\in B_{1}$ we will construct a nondecreasing sequence $c_{k}$ and a nonincreasing sequence $d_{k}$ such that $d_{k}-c_{k}=2^{-k\alpha}$

[TABLE]

where $r_{k}=r_{0}^{k}$ for any integer number $k$ and $0<\alpha<1$ will be chosen appropriately. Now we consider two cases:

Case 1: $k\leq 0$.

Since $\text{osc}_{\mathbb{R}^{n}}u=1,$ we can write

[TABLE]

for $k\leq 0$ and for all $\alpha\in(0,1)$.

Case 2: $k\geq 1$.

Suppose that we already have $c_{j}$ and $d_{j}$ for $j=1,\dots,k$. We will find $c_{k+1}$ and $d_{k+1}$ satisfying (3.8). In fact, if

[TABLE]

then by (3.8) we find

[TABLE]

Now define

[TABLE]

for all $x\in E^{\max}_{r_{k},1}(x_{0})$. Clearly, we have

[TABLE]

and

[TABLE]

Next, we will analysis two cases:

(i) Assume that

[TABLE]

Taking into account that

[TABLE]

we obtain

[TABLE]

Thus, there exists $j\in\mathbb{N}$ such that

[TABLE]

Hence, we find

[TABLE]

and from Lemma 2.1

[TABLE]

Thus, by inductive hypothesis we estimate

[TABLE]

and since $c_{k}$ is a nondecreasing sequence we obtain

[TABLE]

for all $x\in\mathbb{R}^{n}\setminus E^{\max}_{r_{k},1}(x_{0})$. If we take $\alpha\in\left(0,\tau\right]$ we get

[TABLE]

Then, we can apply the Lemma 3.2 to obtain $v\leq 1-\mu$ in $E^{\max}_{r_{k}/2,1}(x_{0})=E^{\max}_{r_{k+1},1}(x_{0})$. We then scale back to $u$ to find

[TABLE]

Now we define $c_{k+1}=c_{k}$ and $d_{k}=c_{k}+r_{k+1}^{\alpha}$. Clearly, $c_{k+1}\leq u$ in $E^{\max}_{r_{k+1},1}(x_{0})$. Finally, if we choose $\alpha=\min\left\{\tau,\frac{\ln(1-\mu/2)}{\ln 2}\right\}$ we obtain

[TABLE]

(ii) In the case

[TABLE]

we consider $v=-u$ to obtain

[TABLE]

Now we define $d_{k+1}=d_{k}$ and $c_{k+1}=d_{k}-\left(\dfrac{2-\mu}{2}\right)r_{k}^{\alpha}$.

Finally, given $x_{0}\in B_{1}$ and $y\in\mathbb{R}^{n}$ we can choose an integer $k$ such that $x_{0}-y\in(E^{\max}_{r_{k-1},1}\setminus E^{\max}_{r_{k},1})$. Thus, by Lemma 2.1 we can conclude

[TABLE]

where $C=C(n,\alpha,b_{\min},b_{\max})>1$ and $\gamma=\frac{2\alpha}{b_{\max}}$. ∎

Corollary 3.4** (Liouville property).**

Let $u$ be a bounded function that satisfies $(-\Delta)^{\beta,s}u=0$ in $\mathbb{R}^{n}$. Then, $u$ is constant.

Proof.

Given $x,y\in\mathbb{R}^{n}$, choose $R>0$ such that $x,y\in E^{\max}_{R,1}$. By Theorem 3.3 we have

[TABLE]

Taking $R>0$ large enough, we get $u(x)=u(y)$. Hence, $u$ is constant. ∎

4 Hölder Regularity: viscosity solutions

In this section, we obtain the ingredients necessary to reach the interior $C^{\gamma}$ and $C^{1,\gamma}$ regularity for viscosity solutions of $\mathcal{I}u=0$.

4.1 Nonlocal anisotropic ABP estimate

In this subsection we get an ABP estimate for integro-differential equations like anisotropic fractional Laplacian.

Let $u$ be a non positive function outside the ball $B_{1}$. We define the concave envelope of $u$ by

[TABLE]

Lemma 4.1**.**

Let $u\leq 0$ in $\mathbb{R}^{n}\setminus B_{1}$ and $\Gamma$ be its concave envelope. Suppose $f\in L^{\infty}$ and $\mathcal{M}^{+}u\left(x\right)\geq-f\left(x\right)$ in $B_{1}$. Let $\rho_{0}=\rho_{0}\left(n\right)>0$,

[TABLE]

where $\mathfrak{C}=\mathfrak{C}(b_{\min},b_{\max})$ is a natural number such that

[TABLE]

with $l=2^{-\mathfrak{C}\left[\frac{b_{\min}}{2}\right]}$ for all $r>0$ and $q_{\min,s}=\frac{4}{b_{\min}}-s$. Given $M>0$, we define

[TABLE]

Then there exists a constant $C_{0}>0$, depending only on $n$, $\lambda$, $b_{\min}$ and $b_{\max}$, such that, for any $x\in\left\{u=\Gamma\right\}$ and any $M>0$, there is a $k$ such that

[TABLE]

Proof.

Notice that $u$ is touched by the plane

[TABLE]

from above at $x$. From Lemma 2.9, $\mathcal{M}^{+}u\left(x\right)$ is defined classically and we get

[TABLE]

We will show that

[TABLE]

In fact, if both $x-y\in B_{3}$ and $x+y\in B_{3}$ then we conclude that $\delta\left(y\right)\leq 0$, since $u\left(x\right)=\Gamma\left(x\right)=p\left(x\right)$, for some plane $p$ that remains above $u$ in the whole ball $B_{3}$. Moreover, if either $x-y\notin B_{3}$ or $x+y\notin B_{3}$, then both $x-y$ and $x+y$ are not in $B_{1}$, and thus $u\left(x+y\right)\leq 0$ and $u\left(x-y\right)\leq 0$. Therefore, in any case the inequality (4.3) is proved. Combining (4.2) and (4.3), we find

[TABLE]

where $r_{0}=\rho_{0}2^{-\frac{1}{q_{\min,s}}}$. Since $x\in\left\{u=\Gamma\right\}$, we would like to emphasize that $y\in W_{k}\left(x\right)$ implies $-y\in W_{k}\left(x\right)$. Hence, we find

[TABLE]

Using (4.4), we estimate

[TABLE]

Moreover, we have

[TABLE]

where $l=2^{-\mathfrak{C}\frac{b_{\min}}{2}}$. Therefore, we find

[TABLE]

Let us assume by contradiction that (4.1) is not valid. Then, from (4.5), (4.6) and (4.1), we obtain

[TABLE]

Then, we get

[TABLE]

Finally, since $\frac{q_{\min,s}}{1-2^{-q_{\min,s}}}$ is bounded away from zero, for all $s\in\left(0,\dfrac{4}{b_{\max}}\right)$, we find

[TABLE]

which is a contradiction if $C_{0}$ is chosen large enough. ∎

As in [8], the following result is a direct consequence of the arguments used in the proof of [9, Lemma 8.4].

Lemma 4.2**.**

Let $\Gamma$ be a concave function in $B_{1}$ and $v\in\mathbb{R}^{n}$. Assume that, for a small $\varepsilon>0$,

[TABLE]

where $T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is a linear map. Then

[TABLE]

in the whole ball $B_{\frac{1}{2}}$.

Proof.

Let $y\in B_{\frac{1}{2}}$. There exist $B_{\frac{1}{2}}\left(y_{1}\right)\subset B_{1}\setminus B_{1/2}$ and $B_{\frac{1}{2}}\left(y_{2}\right)\subset B_{1}\setminus B_{1/2}$ such that

[TABLE]

where $L:B_{\frac{1}{2}}\left(y_{1}\right)\rightarrow B_{\frac{1}{2}}\left(y_{2}\right)$ is the linear map

[TABLE]

Geometrically, the balls $B_{\frac{1}{2}}\left(y_{1}\right)$ and $B_{\frac{1}{2}}\left(y_{2}\right)$ are symmetrical with respect to $y$. Then, if $\varepsilon>0$ is sufficiently small, there will be two points $z_{1}\in B_{\frac{1}{2}}\left(y_{1}\right)$ and $z_{2}\in B_{\frac{1}{2}}\left(y_{2}\right)$ such that

$y=\dfrac{z_{1}+z_{2}}{2}$; 2. 2.

$\Gamma\left(z_{1}\right)\geq\Gamma\left(0\right)+\langle T\left(z_{1}\right),v\rangle-h$; 3. 3.

$\Gamma\left(z_{2}\right)\geq\Gamma\left(0\right)+\langle T\left(z_{2}\right),v\rangle-h$.

Hence, since $T$ and $\langle\cdot,v\rangle$ are linear maps and $\Gamma$ is a concave function, we obtain

[TABLE]

∎

As in [8] , we use Lemma 4.2 to prove the version of Lemma 8.4 in [9] for our problem.

Lemma 4.3**.**

Let $r>0$ and $\Gamma$ be a concave function in $E_{r,1}$. There exists $\varepsilon_{0}>0$ such that if

[TABLE]

for $0<\varepsilon\leq\varepsilon_{0}$, then

[TABLE]

in the whole set $E_{r,\frac{1}{2}}$.

Proof.

Consider

[TABLE]

and

[TABLE]

Notice that

[TABLE]

where $\tilde{\Gamma}\left(x\right):=\Gamma\left(T_{\beta,r}\left(x\right)\right)$. Moreover,

[TABLE]

Then, taking into account that $\tilde{\Gamma}$ is concave, the lemma follows from Lemma 4.2. ∎

Corollary 4.4**.**

Let $\varepsilon_{0}>0$ be as in Lemma 4.3. Given $0<\varepsilon\leq\varepsilon_{0}$, there exists a constant $C\left(n,\lambda,b_{\min},b_{\max},\varepsilon\right)>0$ such that for any function $u$ satisfying the same hypothesis as in Lemma 4.1, there exist $r\in\left(0,\rho_{0}2^{-\frac{1}{q_{\min,s}}}\right)$ and $k=k\left(x\right)$ such that

[TABLE]

[TABLE]

and

[TABLE]

where $r=\rho_{0}2^{-\frac{1}{q_{\min,s}}}2^{-\mathfrak{C}\left[\frac{b_{\min}}{2}\right]k}$ and $l=2^{-\mathfrak{C}\left[\frac{b_{\min}}{2}\right]}$.

Proof.

Taking $M=\frac{C_{0}}{\varepsilon C^{-1}_{1}}f\left(x\right)$ in Lemma 4.1, we obtain (4.7) with $C_{2}:=\frac{C_{0}}{\varepsilon C^{-1}_{1}}$, where

[TABLE]

Consider the sets

[TABLE]

and

[TABLE]

Then, since

[TABLE]

for $y\in E_{r,1}$, we have $W_{1,r}\subset W_{2,r}\subset W_{r}\left(x\right)$. Thus, from (4.7) we obtain

[TABLE]

Moreover, we estimate

[TABLE]

Then, from Lemma 4.3 and the concavity of $\Gamma$, we find

[TABLE]

where

[TABLE]

Notice that

[TABLE]

Then, since $F$ is concave, we find

[TABLE]

Thus, we have

[TABLE]

and obtain

[TABLE]

Finally, taking $C=\max\left\{C_{2},C_{4}\right\}$, the lemma is proven. ∎

The following covering lemma is a fundamental tool in our analysis.

Lemma 4.5** (Covering Lemma, [5, Lemma 3]).**

Let $S$ be a bounded subset of $\mathbb{R}^{n}$ such that for each $x\in S$ there exists an $n$-dimensional rectangle $\mathcal{R}\left(x\right)$, centered at $x$, such that:

• •

the edges of $\mathcal{R}\left(x\right)$ are parallel to the coordinate axes;

• •

the length of the edge of $\mathcal{R}\left(x\right)$ corresponding to the $i$-th axis is given by $h_{i}\left(t\right)$, where $t=t\left(x\right)$, $h_{i}\left(t\right)$ is an increasing function of the parameter $t\geq 0$, continuous at $t=0$, and $h_{i}\left(0\right)=0$.

Then there exist points $\left\{x_{k}\right\}$ in $S$ such that

$S\subset\bigcup_{k=1}^{\infty}\mathcal{R}\left(x_{k}\right)$; 2. 2.

each $x\in S$ belongs to at most $C=C\left(n\right)>0$ different rectangles.

The Corollary 4.4 and the Covering Lemma 4.5 allow us to obtain a lower bound on the volume of the union of the level sets $E_{r,1}$ where $\Gamma$ and $u$ detach quadratically from the corresponding tangent planes to $\Gamma$ by the volume of the image of the gradient map, as in the standard ABP estimate.

Corollary 4.6**.**

For each $x\in\Sigma=\left\{u=\Gamma\right\}\cap B_{1}$, let $E_{r,1}\left(x\right)$ be the level set obtained in Corollary 4.4. Then, we have

[TABLE]

The nonlocal anisotropic version of the ABP estimate now reads as follows.

Theorem 4.7**.**

Let $u$ and $\Gamma$ be as in Lemma 4.1. There is a finite family of open rectangles $\left\{\mathcal{R}_{j}\right\}_{j\in\left\{1,\dots,m\right\}}$ with diameters $d_{j}$ such that the following hold:

Any two rectangles $\mathcal{R}_{i}$ and $\mathcal{R}_{j}$ in the family do not intersect. 2. 2.

$\left\{u=\Gamma\right\}\subset\bigcup_{j=1}^{m}\overline{\mathcal{R}}_{j}$. 3. 3.

$\left\{u=\Gamma\right\}\cap\overline{\mathcal{R}}_{j}\neq\emptyset$* for any $\mathcal{R}_{j}$.* 4. 4.

$d_{j}\leq\sqrt{\sum\limits_{i=1}^{n}\left(\rho_{0}2^{-\frac{1}{q_{\min,s}}}\right)^{\frac{4}{b_{i}}}}$. 5. 5.

$\left|\nabla\Gamma\left(\overline{\mathcal{R}}_{j}\right)\right|\leq C\left(\max_{\tilde{\mathcal{R}}_{j}}f^{+}\right)^{n}\left|\tilde{\mathcal{R}}_{j}\right|$. 6. 6.

$\left|\left\{y\in C\tilde{\mathcal{R}}_{j}:u\left(y\right)\geq\Gamma\left(y\right)-C\left(\max_{\tilde{\mathcal{R}}_{j}}f\right)\left(\tilde{d}_{j}\right)^{2}\right\}\right|\geq\varsigma\left|\tilde{\mathcal{R}}_{j}\right|$,

where $\tilde{d}_{j}$ is the diameter of the rectangle $\tilde{\mathcal{R}}_{j}$ corresponding to $\mathcal{R}_{j}$. The constants $\varsigma>0$ and $C>0$ depend only on $n$, $\lambda$, $\Lambda$, $b_{\min}$, $b_{\max}$, and $s$.

Proof.

We cover the ball $B_{1}$ with a tiling of rectangles of edges

[TABLE]

We discard all those that do not intersect $\left\{u=\Gamma\right\}$. Whenever a rectangle does not satisfy (5) and (6), we split its edges by $2^{n\mathfrak{C}}$ and discard those whose closure does not intersect $\left\{u=\Gamma\right\}$. Now we prove that all remaining rectangles satisfy (5) and (6) and that this process stops after a finite number of steps.

As in [8] we will argue by contradiction. Suppose the process is infinite. Then, there is a sequence of nested rectangles $\mathcal{R}_{j}$ such that the intersection of their closures will be a point $x_{0}$. Moreover, since

[TABLE]

and $\left\{u=\Gamma\right\}$ is closed, we have $x_{0}\in\left\{u=\Gamma\right\}$. Let $0<\varepsilon_{1}<\varepsilon_{0}$, where $\varepsilon_{0}$ is as in Corollary 4.4. Thus, there exist

[TABLE]

and $k_{0}=k_{0}\left(x_{0}\right)$ such that

[TABLE]

[TABLE]

and

[TABLE]

where

[TABLE]

Let $\mathcal{R}_{j}$ be the largest rectangle in the family containing $x_{0}$ such that

[TABLE]

Thus, from Lemma 2.1 we obtain

[TABLE]

for some $C=C(n,b_{\min},b_{\max})>1$. Furthermore, since $\Gamma$ is concave in $B_{2}$, we find

[TABLE]

in $B_{2}$. Thus, denoting

[TABLE]

using (4.10), (4.11), we obtain

[TABLE]

and

[TABLE]

Then $\mathcal{R}_{j}$ would not be split and the process must stop, which is a contradiction. ∎

Remark 4.8*.*

We emphasize that if $b_{\max}=b_{\min}=2$ we recover the ABP estimate obtained in [9]. Furthemore, for $b_{\max}=n+\sigma_{\max}$ and $b_{\min}=n+\sigma_{\min}$ with $\sigma_{\max},\sigma_{\min}\in(0,2)$ we have the ABP estimate reached in [8].

4.2 A barrier function

In order to locate the contact set of a solution $u$ of the maximal equation, as in Lemma 4.1, we build a barrier function which is a supersolution of the minimal equation outside a small ellipse and is positive outside a large ellipse.

Lemma 4.9**.**

Given $R>1,$ there exist $p>0$ and $s_{0}\in\left(0,\frac{4}{b_{\max}}\right)$ such that the function

[TABLE]

satisfies

[TABLE]

for $s_{0}<s$ and $1\leq|x|\leq R$, where $p=p\left(n,\lambda,\Lambda,b_{\min},b_{\max},R\right)$, $s_{0}=s_{0}\left(n,\lambda,\Lambda,b_{\min},b_{\max},R\right)$.

Proof.

Consider the following elementary inequalities:

[TABLE]

and

[TABLE]

where $0<a_{1}<a_{2}$ and $l>0$. Suppose without loss of generality that $b_{1}=b_{\max}$. Taking into account the inequalities (4.12) and (4.13), we estimate, for $|y|<\frac{1}{2}$,

[TABLE]

If $1\leq|x|\leq R$, there is a rotation $T_{x}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ such that $x=|x|Te_{1}$. Then, changing variables, we obtain

[TABLE]

Thus, we can estimate

[TABLE]

where $I_{1}$, $I_{2}$, $I_{3}$ and $I_{4}$ represent the three terms on the right-hand side of the above inequality.

Changing variables again, we get

[TABLE]

Moreover, without loss of generality, we can assume that

[TABLE]

From Lemma 2.1 there exists $r_{0}=r_{0}(n,b_{min},b_{max})\in(0,1)$ such that $E_{r_{0},1}\subset B_{1/4}$. Then, from (4.2) we estimate

[TABLE]

where $C_{3}=C_{3}\left(n,\lambda,\Lambda,b_{\min},b_{\max},R\right)>0$. Let $C=C\left(n,b_{\max},b_{\min}\right)$ be a positive constant such that $B_{1/4}\left(0\right)\subset E_{C,1}$. Then, for $|x|\geq 1$ we get

[TABLE]

where $C_{4}=C_{4}\left(n,\lambda,\Lambda,b_{\min},b_{\max},R\right)$. We have also

[TABLE]

where $r_{k}:=C2^{-k}$ and $C_{5}=C_{5}\left(n,\lambda,\Lambda,b_{\max},b_{min}\right)$. Moreover, we have

[TABLE]

and, if $r_{1}=r_{1}(r_{0})>0$ is such that $\Theta_{r_{1}}\subset E_{r_{0},1}$, we obtain

[TABLE]

for positive constants $C_{7}=C_{7}\left(n,\lambda,\Lambda,b_{\min},b_{\max},p\right)$ and $C_{8}=C_{8}\left(n,\lambda,\Lambda,b_{\min},b_{\max},p\right)$. Choosing $p=p\left(n,\lambda,\Lambda,b_{\min},b_{\max},R\right)>0$ such that

[TABLE]

and combining (4.14), (4.18) and (4.2), there is a positive constant $s_{0}=s_{0}\left(n,\lambda,\Lambda,b_{\min},b_{\max},R\right)<\frac{4}{b_{\max}}$ such that

[TABLE]

for a positive constant $C_{9}=C_{9}\left(n,\lambda,\Lambda,b_{\min},b_{\max},R\right)$. ∎

As in [8], from Lemma 4.9 we get the following results:

Corollary 4.10**.**

Given $s_{0}\in\left(0,\frac{4}{b_{\max}}\right)$, and $R>1$, there exist $\kappa>0$ and $p>0$ such that the function

[TABLE]

satisfies

[TABLE]

for $1\leq|x|\leq R$ and $s_{0}<s<\frac{4}{b_{\max}}$, where $p=p\left(n,\lambda,\Lambda,b_{\min},b_{\max},R\right)$ and $\kappa=\kappa\left(n,\lambda,\Lambda,b_{\min},b_{\max},s_{0},R\right)$.

Corollary 4.11**.**

Given $r>0$, $R>1$ and $s_{0}\in\left(0,\frac{4}{b_{\max}}\right)$, there exist $\kappa>0$ and $p>0$ such that the function

[TABLE]

satisfies

[TABLE]

for $s_{0}<s<\frac{4}{b_{max}}$ and $x\in E_{r,R}\setminus E_{r,1}$, where $p=p\left(n,\lambda,\Lambda,b_{\min},b_{\max},R\right)$ and $\kappa=\kappa\left(n,\lambda,\Lambda,b_{\min},b_{\max},s_{0},R\right)$.

Lemma 4.12**.**

Given $s_{0}\in\left(0,\frac{4}{b_{\max}}\right)$, there is a function $\Psi:\mathbb{R}^{n}\rightarrow\mathbb{R}$ satisfying

$\Psi$* is continuous in $\mathbb{R}^{n}$;* 2. 2.

$\Psi=0$* for $x\in\mathbb{R}^{n}\setminus E_{\frac{1}{4},3\sqrt{n}}$;* 3. 3.

$\Psi>3$* for $x\in\mathcal{R}_{\frac{1}{4},3}$;* 4. 4.

$\mathcal{M}^{-}\Psi\left(x\right)>-\varphi\left(x\right)$* for some positive function $\varphi\in C_{0}\left(E_{\frac{1}{4},1}\right)$ for $s_{0}<s<\frac{4}{b_{\max}}$.*

Proof.

Consider the function $\Psi:\mathbb{R}^{n}\rightarrow\mathbb{R}$ defined by

[TABLE]

where $q_{p,\beta}$ is a quadratic function with different coefficients in different directions so that $\Psi$ is $C^{1,1}$ across $E_{\frac{1}{4},1}$. Choose $\tilde{c}>0$ such that $\Psi>3$ in $\mathcal{R}_{\frac{1}{4},3}$. By Lemma 2.10, we get

[TABLE]

and, from Corollary 4.11, we find $\mathcal{M}^{-}\Psi\geq 0$ in $\mathbb{R}^{n}\setminus E_{\frac{1}{4},1}$. The lemma is proved. ∎

4.3 Harnack inequality and regularity

The next lemma is the fundamental tool towards the proof of the Harnack inequality. It bridges the gap between a pointwise estimate and an estimate in measure.

Lemma 4.13**.**

Let $0<s_{0}<\frac{4}{b_{\max}}$. If $s\in\left(s_{0},\frac{4}{b_{\max}}\right)$, then there exist constants $\varepsilon_{0}>0$, $0<\varsigma<1$, and $M>1$, depending only $s_{0},s$, $\lambda$, $\Lambda$, $b_{\min}$, $b_{\max}$, and $n$, such that if

$u\geq 0$* in $\mathbb{R}^{n}$;* 2. 2.

$u\left(0\right)\leq 1$; 3. 3.

$\mathcal{M}^{-}u\leq\varepsilon_{0}$* in $E_{\frac{3}{4}n^{\frac{b_{\max}}{4}},1}$,*

then

[TABLE]

Proof.

Let $v=\Psi-u$ and let $\Gamma$ be the concave envelope of $v$ in $E_{\frac{3}{4}n^{\frac{b_{\max}}{4}},3}$. We have

[TABLE]

Applying Theorem 4.7 to $v$ (anisotropically scaled), we obtain a family of rectangles $\mathcal{R}_{j}$ such that

[TABLE]

Thus, by Theorem 4.7 and condition (3) in Lemma 4.12, we obtain

[TABLE]

Furthermore, since $\Psi>3$ in $E_{\frac{3}{4}n^{\frac{b_{\max}}{4}},1}\supset\mathcal{R}_{\frac{1}{4},3}$ and $u\left(0\right)\leq 1$, we get

[TABLE]

If $\varepsilon_{0}>0$ is small enough, we have

[TABLE]

where we used that $\varphi$ is supported in $E_{\frac{1}{4},1}$. We also have that the diameter of $\tilde{\mathcal{R}}_{j}$ is bounded by $\left(\rho_{0}=\frac{1}{C}\right)^{\frac{2}{b_{\max}}}$. Then, if $\tilde{\mathcal{R}}_{j}\cap E_{\frac{1}{4},1}\neq\emptyset$ we have $C\tilde{\mathcal{R}}_{j}\subset B_{\frac{1}{2}}$. By Theorem 4.7, we get

[TABLE]

where we used that $Cd^{2}_{j}<C\rho_{0}^{\frac{4}{b_{\max}}}$. For each rectangles $\tilde{\mathcal{R}}_{j}$ that intersects $E_{\frac{1}{4},1}$ we consider $C\tilde{\mathcal{R}_{j}}$. The family $\left\{C\tilde{\mathcal{R}_{j}}\right\}$ is an open covering for $\bigcup_{i=1}^{m}\tilde{\mathcal{R}}_{j}$. We consider a subcover with finite overlapping (Lemma 4.5) that also covers $\bigcup_{i=1}^{m}\tilde{\mathcal{R}}_{j}$. Then, using (4.21) and (4.22) we obtain

[TABLE]

We recall that $B_{\frac{1}{2}}\subset Q_{1}$ and $\Gamma\geq 0$. Hence, if $M:=\sup\limits_{B_{\frac{1}{2}}}\Psi+C\rho_{0}^{\frac{4}{b_{\max}}}$, we have

[TABLE]

∎

The next lemma is fundamental to iterate Lemma 4.13 and to get the $L_{\varepsilon}$ decay of the distribution function $\lambda_{u}:=\left|\left\{u>t\right\}\cap B_{1}\right|$. Since our scaling is anisotropic, the following Calderón-Zygmund decomposition is performed with $n$-dimensional rectangles that satisfy the covering lemma of Caffarelli-Calderón (Lemma 4.5). We can then apply Lebesgue’s differentiation theorem having these $n$-dimensional rectangles as a differentiation basis, see Lemma 5.2 in [8].

Lemma 4.14**.**

Let $u$ be as in Lemma 4.13. Then

[TABLE]

where $M$ and $\varsigma$ are as in Lemma 4.13. Thus, there exist positive constants $d$ and $\varepsilon$, depending only $s_{0},s$, $\lambda$, $\Lambda$, $b_{\min}$, $b_{\max}$, and $n$ such that

[TABLE]

Using standard covering arguments we get the following theorem.

Theorem 4.15**.**

Let $u\geq 0$ in $\mathbb{R}^{n}$, $u\left(0\right)\leq 1$ and $\mathcal{M}^{-}u\leq\varepsilon_{0}$ in $B_{2}$. Suppose that $s_{0}<s<\frac{4}{b_{\max}}$ for some $s_{0}>0$. Then

[TABLE]

where $C=C\left(n,\lambda,\Lambda,b_{\max},b_{\min},s_{0},s\right)>0$ and $\varepsilon=\varepsilon\left(n,\lambda,\Lambda,b_{\max},b_{\min},s_{0},s\right)>0$.

Remark 4.16*.*

For each $l>0$, we will denote $E^{j}_{r,l}:=E_{r^{b_{j}},l}$. Let $u\geq 0$ in $\mathbb{R}^{n}$ and $\mathcal{M}^{-}u\leq C_{0}$ in $E^{j}_{r,2}$, with $0<r\leq 1$. We consider the anisotropic scaling

[TABLE]

where $T_{j,\beta,r}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is defined by

[TABLE]

We find $v\geq 0$ in $\mathbb{R}^{n}$, $v\left(0\right)\leq 1$ and $T_{j,\beta,r}\left(B_{2}\right)=E^{j}_{r,2}$. Moreover, changing variables, we estimate

[TABLE]

for all $x\in B_{2}$.

Then, using the anisotropic scaling $T_{j,\beta,r}$ and Theorem 4.15 we have the following scaled version.

Theorem 4.17** (Pointwise Estimate).**

Let $u\geq 0$ in $\mathbb{R}^{n}$ and $\mathcal{M}^{-}u\leq C_{0}$ in $E^{j}_{r,2}$. Suppose that $s_{0}<s<\frac{4}{b_{\max}}$ for some $s_{0}>0$. Then

[TABLE]

where $C=C\left(n,\lambda,\Lambda,b_{\min},b_{\max},s_{0},s\right)>0$ and $\varepsilon=\varepsilon\left(n,\lambda,\Lambda,b_{\min},b_{\max},s_{0},s\right)>0$.

We are now ready to prove the Harnack inequality.

Theorem 4.18** (Harnack Inequality).**

Let $u\geq 0$ in $\mathbb{R}^{n}$, $\mathcal{M}^{-}u\leq C_{0}$, and $\mathcal{M}^{+}u\geq-C_{0}$ in $B_{2}$. Suppose that $s_{0}<s<\frac{4}{b_{\max}}$, for some $s_{0}>0$. Then

[TABLE]

Proof.

Without loss of generality, we can suppose that $u\left(0\right)\leq 1$ and $C_{0}=1$. Let

[TABLE]

where $\varepsilon>0$ is as in Theorem 4.15. For each $\vartheta>0$, we define the function

[TABLE]

Let $t>0$ be such that $u\leq f_{t}$ in $B_{1}$. There is an $x_{0}\in B_{1}$ such that $u\left(x_{0}\right)=f_{t}\left(x_{0}\right)$. Let $d:=\left(1-|x_{0}|\right)$ be the distance from $x_{0}$ to $\partial B_{1}$.

We will estimate the portion of the ellipsoid $E^{\max}_{r,1}\left(x_{0}\right)$ covered by $\left\{u>\frac{u\left(x_{0}\right)}{2}\right\}$ and by $\left\{u<\frac{u\left(x_{0}\right)}{2}\right\}$. As in [9], we will prove that $t>0$ cannot be too large. Thus, since $\tau\leq\dfrac{C(n,b_{\min},b_{\max})}{\varepsilon}$, we conclude the proof of the theorem. By Theorem 4.15, we have

[TABLE]

where $r=\frac{d}{2}$. Thus, we get

[TABLE]

Now we will estimate $\left|\left\{u>\frac{u\left(x_{0}\right)}{2}\right\}\cap E^{\max}_{\theta r,1}\left(x_{0}\right)\right|$, where $0<\theta<1$. Since

[TABLE]

we have

[TABLE]

for $x\in B_{r\theta}\left(x_{0}\right)$. Hence, if $x\in B_{r\theta}\left(x_{0}\right)$, we get

[TABLE]

Then, since $\mathcal{M}^{+}u\geq-1$, the function

[TABLE]

satisfies

[TABLE]

We will consider the function $w:=v^{+}$. For $x\in\mathbb{R}^{n}$ we have

[TABLE]

and

[TABLE]

where $I_{1}$ and $I_{2}$ represent the two terms in the right-hand side above. Using the elementary equality

[TABLE]

and denoting $\delta_{w}:=\delta\left(w,x,y\right)$ and $\delta_{v}:=\delta\left(v,x,y\right)$, we obtain

[TABLE]

Thus, taking in account that

[TABLE]

we estimate

[TABLE]

Analogously, we get

[TABLE]

We also have

[TABLE]

Then, from (4.26) and (4.25), we obtain

[TABLE]

Hence, using (4.24), (4.27), and changing variables, we find

[TABLE]

Moreover, if $x\in B_{\frac{r\theta}{2}}\left(x_{0}\right)$, we have

[TABLE]

[TABLE]

If $\iota>0$ is the largest value such that $u\left(x\right)\geq\iota\left(1-|4x|^{2}\right)$, then there is a point $x_{1}\in B_{\frac{1}{4}}$ such that $u\left(x_{1}\right)=\left(1-|4x_{1}|^{2}\right)$. Moreover, since $u\left(0\right)\leq 1$, we get $\iota\leq 1$. Then, we have

[TABLE]

where the constant $C>0$ is independent of $s$. Moreover, since $\mathcal{M}^{-}u\left(x_{1}\right)\leq 1$, we find

[TABLE]

Recall that $u\left(x_{1}-y\right)\geq 0$ and $u\left(x_{1}\right)\leq 1$. Thus, we obtain

[TABLE]

Since $t>0$ is large enough, we can suppose that $u\left(x_{0}\right)>2$. Let

[TABLE]

and

[TABLE]

Then, we have the inequalities

[TABLE]

and

[TABLE]

Then, taking into account the obvious equalities

[TABLE]

and

[TABLE]

we estimate

[TABLE]

Thus, we have

[TABLE]

Applying Theorem 4.17 to $w$ in $E^{\max}_{\frac{r\theta}{2n},1}\left(x_{0}\right)\subset B_{\frac{r\theta}{2}}\left(x_{0}-x\right)$ and using that

[TABLE]

we get

[TABLE]

Thus, using (4.28) and the elementary inequalities

[TABLE]

[TABLE]

and

[TABLE]

for $\theta>0$ sufficiently small, and yet

[TABLE]

[TABLE]

we obtain

[TABLE]

Now we choose $\theta>0$ sufficiently small such that

[TABLE]

Having fixed $\theta>0$ (independently of $t$), we take $t>0$ sufficiently large such that

[TABLE]

Then, using (4.28), we find

[TABLE]

Hence, we have, for $t>0$ large,

[TABLE]

which is a contradiction to (4.23). ∎

As a consequence of the Harnack inequality we obtain the $C^{\gamma}$ regularity.

Theorem 4.19** ($C^{\gamma}$ estimates).**

Let $u$ be a bounded function such that

[TABLE]

If $0<s_{0}<s<\frac{4}{b_{\max}}$, then there is a positive constant $0<\gamma<1$, that depends only $n$, $\lambda$, $\Lambda$, $b_{\min}$, $b_{\max},s_{0}$, and $s$, such that $u\in C^{\gamma}\left(B_{1/2}\right)$ and

[TABLE]

for some constant $C>0$.

The next result is a consequence of the arguments used in [9, 8] and Theorem 4.19. As in [9, 8], if we suppose a modulus of continuity of $\mathcal{K}_{\alpha\beta}$ in measure, then as to make sure that faraway oscillations tend to cancel out, we obtain the interior $C^{1,\gamma}$ regularity for solutions of equation $\mathcal{I}u=0$.

Theorem 4.20** ($C^{1,\gamma}$ estimates).**

Suppose that $0<s_{0}<s<\frac{4}{b_{\max}}$. There exists a constant $\tau_{0}>0$, that depends only on $\lambda$, $\Lambda$, $n$, $b_{\min}$, $b_{\max},s_{0}$ and $s$, such that

[TABLE]

If $u$ is a bounded function satisfying $\mathcal{I}u=0$ in $B_{1}$, then there is a constant $0<\gamma<1$, that depends only $n$, $\lambda$, $\Lambda$, $b_{\min}$, $b_{\max},s_{0}$ and $s$, such that $u\in C^{1,\gamma}\left(B_{1/2}\right)$ and

[TABLE]

for some constant $C=C\left(n,\lambda,\Lambda,b_{\min},b_{\max},s_{0},s,C_{0}\right)>0$.

Remark 4.21*.*

As in [8], we can also obtain $C^{\gamma}$ and $C^{1,\gamma}$ estimates for truncated kernels, i.e., kernels that satisfy (1.4) only in a neighborhood of the origin. Let $\mathfrak{L}$ be the class of operators $L_{\alpha\beta}$ such that the corresponding kernels $\mathcal{K}_{\alpha\beta}$ have the form

[TABLE]

where

[TABLE]

and $\mathcal{K}_{\alpha\beta,2}\in L^{1}\left(\mathbb{R}^{n}\right)$ with $\|\mathcal{K}_{\alpha\beta,2}\|_{L^{1}\left(\mathbb{R}^{n}\right)}\leq c_{1}$, for some constant $c_{1}>0$. The class $\mathfrak{L}$ is larger than $\mathfrak{L}_{0}$ but the extremal operators $\mathcal{M}_{\mathfrak{L}}^{-}$ and $\mathcal{M}_{\mathfrak{L}}^{+}$ are controlled by $\mathcal{M}^{+}$ and $\mathcal{M}^{-}$ plus the $L^{\infty}$ norm of $u$ (see Lemma 14.1 and Corollary 14.2 in [9, 8]). Thus the interior $C^{\gamma}$ and $C^{1,\gamma}$ regularity follow.

Acknowledgments

EBS and RAL thank the Analysis research group of UFC for fostering a pleasant and productive scientific atmosphere. EBS supported by CAPES-Brazil.