Higher Hölder regularity for the fractional p-Laplace equation in the subquadratic case

Abstract

We study the fractional p-Laplace equation

$$\begin{aligned} (-\Delta _p)^s u = 0 \end{aligned}$$

for $0<s<1$ and in the subquadratic case $1<p<2$. We provide Hölder estimates with an explicit Hölder exponent. The inhomogeneous equation is also treated and there the exponent obtained is almost sharp for a certain range of parameters. Our results complement the previous results for the superquadratic case when $p\ge 2$. The arguments are based on a careful Moser-type iteration and a perturbation argument.

1 Introduction

We study the local regularity of solutions of the nonlinear and nonlocal equation

$$\begin{aligned} (-\Delta _p)^{s}u=f, \end{aligned}$$

(1.1)

where

$$\begin{aligned} (-\Delta _p)^{s}u(x)=\text {P.V.}\int _{{\mathbb {R}}^N} \frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+ps}}\,dy, \end{aligned}$$

is the fractional p-Laplace operator, where P.V. denotes the principal value. It arises as the first variation of the Sobolev–Slobodeckiĭ seminorm for $W^{s,p}({\mathbb {R}}^N)$, that is, as the first variation of the functional

$$\begin{aligned} u\mapsto \iint _{{\mathbb {R}}^N\times {\mathbb {R}}^N} \frac{|u(x)-u(x)|^p}{|x-y|^{N+s\,p}}\,dx\,dy. \end{aligned}$$

This operator has generated vast activities in recent years. The main contribution of our work is to provide Hölder regularity of weak solutions of equation (1.1), with an explicit Hölder exponent. This is done in Theorems 1.1 and 1.2. Our results complement the existing results for the superquadratic case, $p\ge 2$, obtained in [4]. To the best of our knowledge, this is the first result with an explicit Hölder exponent in the subquadratic case, $1<p<2,$ even for the homogeneous equation. We seize the moment to mention that we can verify that whenever

$$\begin{aligned} 1-\frac{1}{p}\ge s\ge \frac{N}{q}, \end{aligned}$$

the Hölder exponent $\Theta$ obtained in the inhomogeneous setting for $f\in L^q$, that is

$$\begin{aligned} {\Theta }=\frac{1}{p-1}\left( s\,p-\frac{N}{q}\right) , \end{aligned}$$

is the sharp one, see Sect. 1.2 below. Hence, our result is sharp under this assumptions.

1.1 Main results

We now present the main results of the paper. For the details regarding the notation used in the two theorems below, such as $\textrm{Tail}_{p-1,s\,p}$, we refer to Sect. 2.

Theorem 1.1

(Almost $sp/(p-1)$-regularity) Let $\Omega \subset {\mathbb {R}}^N$ be a bounded and open set and assume that $1<p<2$ and $0<s<1$. Suppose $u\in W^{s,p}_{\textrm{loc}}(\Omega )\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)$ is a local weak solution of

$$\begin{aligned} (-\Delta _p)^s u=0\qquad \text{ in } \Omega . \end{aligned}$$

Then $u\in C^{\Gamma -\varepsilon }_{\textrm{loc}}(\Omega )$ for every $\varepsilon \in (0,\Gamma )$, where

$$\begin{aligned} \Gamma = \min (sp/(p-1),1). \end{aligned}$$

In particular, for every $\varepsilon \in (0,\Gamma )$ and for every ball $B_{2R}(x_0)\Subset \Omega$, there exist constants $\sigma =\sigma (N,s,p,\varepsilon )\in (0,1)$ and $C=C(N,s,p,\varepsilon )>0$ such that

$$\begin{aligned} {[}u]_{C^{\Gamma -\varepsilon }(B_{\sigma R}(x_0))}\le \frac{C}{R^{\Gamma -\varepsilon }}\,\left( \Vert u\Vert _{L^\infty (B_{R}(x_0))} +\textrm{Tail}_{p-1,s\,p}(u;x_0,R)\right) . \end{aligned}$$

Theorem 1.2

Let $\Omega \subset {\mathbb {R}}^N$ be a bounded and open set and assume that $1<p<2$ and $0<s<1$. Suppose $u\in W^{s,p}_{\textrm{loc}}(\Omega )\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)$ is a local weak solution of

$$\begin{aligned} (-\Delta _p)^s u=f\qquad \text{ in } \Omega , \end{aligned}$$

where $f\in L^q_\text {loc}(\Omega )$ with

$$\begin{aligned} \left\{ \begin{array}{ll} q>\frac{N}{sp},&{} \quad \text{ if } s\,p\le N,\\ q\ge 1,&{} \quad \text{ if } s\,p>N. \end{array} \right. \end{aligned}$$

Let

$$\begin{aligned} \Theta = \min \Big (1,\frac{sp-N/q}{p-1}\Big ). \end{aligned}$$

Then $u\in C^{\Theta -\varepsilon }_{\textrm{loc}}(\Omega )$ for every $\varepsilon \in (0,\Theta )$.

In particular, for every $\varepsilon \in (0,\Theta )$ and for every ball $B_{4R}(x_0)\Subset \Omega$, there exists a constant $C=C(N,s,p,q,\varepsilon )>0$ such that

$$\begin{aligned} {[}u]_{C^{\Theta -\varepsilon }(B_{ R/8}(x_0))}\le & {} \frac{C}{R^{\Theta -\varepsilon }}\,\left( \Vert u\Vert _{L^\infty (B_{2R}(x_0))} +\textrm{Tail}_{p-1,s\,p}(u;x_0,2R)\right. \\{} & {} \left. +\left( R^{sp-N/q}\Vert f\Vert _{L^{q} (B_{2R}(x_0))}\right) ^\frac{1}{p-1}\right) . \end{aligned}$$

Remark 1.3

It is worth mentioning that in the case when $q<\infty$ and $\Theta =(sp-N/q)/(p-1)$, a careful inspection of the proof of Theorem 1.2 reveals that we obtain the stronger result that $u\in C^{\Theta }_{\textrm{loc}}(\Omega )$, with a similar estimate.

1.2 Comments on the results

We now discuss the sharpness of our results, in particular Theorem 1.2. Choose N, p, q such that

$$\begin{aligned} 1-\frac{1}{p}\ge \frac{N}{q} \end{aligned}$$

and pick $s\in [N/q,1-1/p]$. Then it follows that $sp <N$, $q >N/sp$, $sp\le p-1$ and that $(sp-N/q)/(p-1)\ge s$.

Define for some $\varepsilon >0$ the function

$$\begin{aligned} u(x)=|x|^{\gamma +\varepsilon }, \quad \gamma = (sp-N/q)/(p-1). \end{aligned}$$

By the assumptions, $u\in C^{s+\varepsilon }_{\text {loc}}({\mathbb {R}}^N)\cap W^{s,p}_{\text {loc}}({\mathbb {R}}^N)\cap L_{sp}^{p-1}({\mathbb {R}}^N)$. In addition, by homogeneity and radial symmetry it follows that

$$\begin{aligned} (-\Delta _p)^s u = f, \quad f(x)=C(s,p,\gamma ,\varepsilon )|x|^{(\gamma +\varepsilon -s)(p-1)-s} \end{aligned}$$

where $f\in L^q_{\text {loc}}({\mathbb {R}}^N)$ if and only if $\varepsilon >0$. It is clear that $u\not \in C^{\alpha }(B_1)$ for any $\alpha >\gamma +\varepsilon$. Therefore, the result is sharp in this region of parameters. Now we comment on the assumptions on q and p in Theorem 1.2. We believe that they are sharp and they do perfectly match the sharp assumptions in the local limit. Indeed, in the local case, that should correspond to the limiting case $s=1$, the assumptions become $q>N/p$ when $p\le N$ and $q\ge 1$ when $p>N$. These are the proper conditions for the inhomogeneous p-Laplace equation, see [25, 26].

1.3 Known results

The first appearance of equations similar to the fractional p-Laplacian that we are aware of is in [16]. There existence, uniqueness, and the convergence to the $p-$Laplace equation as s goes to 1, are proved in the viscosity setting. The starting point of the regularity theory was [11], where the local Hölder regularity was proved, using a nonlocal De Giorgi-type method. See also [10], for a related Harnack inequality. The paper [5] contains several useful regularity estimates for the inhomogeneous equation.

The literature on related Hölder regularity results is vast and we only mention a fraction. A local regularity result using viscosity methods was obtained in [21]. In [9], nonlocal analogues of the De Giorgi classes are introduced and used to prove regularity results in a very general setting. We also seize the opportunity to mention that fractional De Giorgi classes has been used in the context of local equations in [23].

The regularity up to the boundary has been studied in [14, 15]. Basic Hölder regularity up to the boundary is proved for general p and for $p\ge 2$ finer regularity results up to the boundary are obtained.

In terms of regularity for the inhomogeneous equation, we mention the papers [4, 12, 18]. In [18], the authors study the regularity for equations of the type (1.1) with a right hand side f belonging to a Lorentz space. Sharp results for when u is continuous are obtained. The paper [4] is the counterpart of the present paper in the superquadratic case. In [12], these results are improved and the authors obtain sharp Hölder regularity results when $p\ge 2$ and when the right hand side belongs to a Marcinkiewicz space.

We stress that for the subquadratic case $1<p<2$, none of the above mentioned papers include an explicit Hölder exponent.

In addition to Hölder regularity, there has been quite some development of results in terms of higher Sobolev differentiability. In the linear case $p=2$, see [1, 2, 8, 19] and [20], where the results are valid for more general kernels. For a general p, this has been studied in [3, 7, 12, 24].

We finally mention that the corresponding results for the p-Laplacian are well known. See for instance [25, 26].

1.4 Plan of the paper

In Sect. 2, we discuss notation, definitions and certain results in function spaces. The most important part of the paper is Sect. 3, where we prove Theorem 1.1, using a Moser-type argument that results in an improved differentiability that can be iterated in an unusual way. Following this, in Sect. 4, we treat the inhomogeneous equation, by means of a perturbation argument, using the regularity obtained for the homogeneous equation. Finally, in the Appendix, we include a list of pointwise inequalities that are used throughout the paper.

2 Preliminaries

In this section we present some auxiliary results needed in the rest of the paper.

2.1 Notation

Throughout the paper, we shall use the following notation: $B_r(x_0)$ denotes the ball of radius r with center at $x_0$. When $x_0=0$, we write $B_r(0):=B_r$. For a function u, we denote the positive and the negative part of u as $u_{\pm }=\max \{\pm u,0\}$. The conjugate exponent $\frac{l}{l-1}$ of $l>1$ will be denoted by $l'$. We write c or C to denote a positive constant which may vary from line to line or even in the same line. The dependencies on parameters are written in the parentheses.

For $1<q<\infty$, we define the function $J_q:{\mathbb {R}}\rightarrow {\mathbb {R}}$ by

$$\begin{aligned} J_q(t)=|t|^{q-2}t \end{aligned}$$

(2.1)

and for $0<s<1$ and $1<p<\infty$ we use the notation

$$\begin{aligned} d\mu =\frac{dx dy}{|x-y|^{N+ps}}. \end{aligned}$$

(2.2)

Moreover, for $0\le \delta \le 1$, we use the notation

$$\begin{aligned} {[}u]_{C^\delta (\Omega )}:=\sup _{x\ne y\in \Omega }\frac{|u(x)-u(y)|}{|x-y|^{\delta }}. \end{aligned}$$

We also define

$$\begin{aligned} \psi _h(x)=\psi (x+h),\qquad \delta _h \psi (x)=\psi _h(x)-\psi (x)\\ \end{aligned}$$

and

$$\begin{aligned} \delta ^2_h \psi (x)=\delta _h(\delta _h \psi (x))=\psi _{2\,h}(x)+\psi (x)-2\,\psi _h(x) \end{aligned}$$

for functions $\psi :{\mathbb {R}}^N\rightarrow {\mathbb {R}}$ and $h\in {\mathbb {R}}^N$. Note that the following discrete product rule holds:

$$\begin{aligned} \delta _h(\phi \psi )=\psi _h\delta _h\phi +\phi \delta _h\psi . \end{aligned}$$

(2.3)

2.2 Function spaces

It will be necessary to introduce two Besov-type spaces. For this reason, let $1\le q<\infty$ and $\psi \in L^q({\mathbb {R}}^N)$. For $0<\beta \le 1$, define

$$\begin{aligned} {[}\psi ]_{{\mathcal {N}}^{\beta ,q}_\infty ({\mathbb {R}}^N)}:=\sup _{|h|>0} \left\| \frac{\delta _h \psi }{|h|^{\beta }}\right\| _{L^q({\mathbb {R}}^N)}, \end{aligned}$$

and for $0<\beta <2$, define

$$\begin{aligned} {[}\psi ]_{{\mathcal {B}}^{\beta ,q}_\infty ({\mathbb {R}}^N)}:=\sup _{|h|>0} \left\| \frac{\delta _h^2 \psi }{|h|^{\beta }}\right\| _{L^q({\mathbb {R}}^N)}. \end{aligned}$$

The Besov-type spaces ${\mathcal {N}}^{\beta ,q}_\infty$ and ${\mathcal {B}}^{\beta ,q}_\infty$ are defined by

$$\begin{aligned} {\mathcal {N}}^{\beta ,q}_\infty ({\mathbb {R}}^N)=\left\{ \psi \in L^q({\mathbb {R}}^N)\,:\, [\psi ]_{{\mathcal {N}}^{\beta ,q}_\infty ({\mathbb {R}}^N)}<+\infty \right\} ,\qquad 0<\beta \le 1, \end{aligned}$$

and

$$\begin{aligned} {\mathcal {B}}^{\beta ,q}_\infty ({\mathbb {R}}^N)=\left\{ \psi \in L^q({\mathbb {R}}^N)\,:\, [\psi ]_{{\mathcal {B}}^{\beta ,q}_\infty ({\mathbb {R}}^N)}<+\infty \right\} ,\qquad 0<\beta <2. \end{aligned}$$

The Sobolev-Slobodeckiĭ space is defined as

$$\begin{aligned} W^{\beta ,q}({\mathbb {R}}^N)=\left\{ \psi \in L^q({\mathbb {R}}^N)\,:\, [\psi ]_{W^{\beta ,q}({\mathbb {R}}^N)}<+\infty \right\} ,\qquad 0<\beta <1, \end{aligned}$$

where the seminorm $[\,\cdot \,]_{W^{\beta ,q}({\mathbb {R}}^N)}$ is given by

$$\begin{aligned} {[}\psi ]_{W^{\beta ,q}({\mathbb {R}}^N)}=\left( \iint _{{\mathbb {R}}^N\times {\mathbb {R}}^N} \frac{|\psi (x)-\psi (y)|^q}{|x-y|^{N+\beta \,q}}\,dx\,dy\right) ^\frac{1}{q}. \end{aligned}$$

The above spaces are endowed with the norms

$$\begin{aligned} \Vert \psi \Vert _{{\mathcal {N}}^{\beta ,q}_\infty ({\mathbb {R}}^N)}= & {} \Vert \psi \Vert _{L^q({\mathbb {R}}^N)}+[\psi ]_{{\mathcal {N}}^{\beta ,q}_\infty ({\mathbb {R}}^N)},\\ \Vert \psi \Vert _{{\mathcal {B}}^{\beta ,q}_\infty ({\mathbb {R}}^N)}= & {} \Vert \psi \Vert _{L^q({\mathbb {R}}^N)}+[\psi ]_{{\mathcal {B}}^{\beta ,q}_\infty ({\mathbb {R}}^N)}, \end{aligned}$$

and

$$\begin{aligned} \Vert \psi \Vert _{W^{\beta ,q}({\mathbb {R}}^N)}=\Vert \psi \Vert _{L^q ({\mathbb {R}}^N)}+[\psi ]_{W^{\beta ,q}({\mathbb {R}}^N)}. \end{aligned}$$

We also introduce the space $W^{\beta ,q}(\Omega )$ for a subset $\Omega \subset {\mathbb {R}}^N$,

$$\begin{aligned} W^{\beta ,q}(\Omega )=\left\{ \psi \in L^q(\Omega )\,:\, [\psi ]_{W^{\beta ,q}(\Omega )}<+\infty \right\} ,\qquad 0<\beta <1, \end{aligned}$$

where naturally

$$\begin{aligned} {[}\psi ]_{W^{\beta ,q}(\Omega )}=\left( \iint _{\Omega \times \Omega } \frac{|\psi (x)-\psi (y)|^q}{|x-y|^{N+\beta \,q}}\,dx\,dy\right) ^\frac{1}{q}. \end{aligned}$$

It will also be convenient to use the following abuse of notation for the Sobolev exponent $p_s^*$ related to the space $W^{s,p}$: if $sp<N$ then

$$\begin{aligned} p_s^*= \frac{Np}{N-sp}, \quad (p_s^*)' = \frac{Np}{Np-N+sp} \end{aligned}$$

and if $sp>N$ then

$$\begin{aligned} p_s^* =\infty ,\quad (p_s^*)'=1. \end{aligned}$$

2.3 Embedding inequalities

The following result can be found for example in [6, Lemma 2.3].

Lemma 2.1

The following embedding

$$\begin{aligned} {\mathcal {B}}_\infty ^{\beta ,q}({\mathbb {R}}^N)\hookrightarrow {\mathcal {N}}_\infty ^{\beta ,q}({\mathbb {R}}^N) \end{aligned}$$

is continuous, provided $0<\beta <1$ and $1\le q<\infty$. Moreover,

$$\begin{aligned} {[}\psi ]_{{\mathcal {N}}_\infty ^{\beta ,q}({\mathbb {R}}^N)} \le \frac{C}{1-\beta }[\psi ]_{{\mathcal {B}}_\infty ^{\beta ,q}({\mathbb {R}}^N)}, \end{aligned}$$

for every $\psi \in {\mathcal {B}}_\infty ^{\beta ,q}({\mathbb {R}}^N)$, for some constant $C=C(N,q)>0$.

We have the following embedding result from [4, Theorem 2.8].

Theorem 2.2

Let $\psi \in {\mathcal {N}}_\infty ^{\beta ,q}({\mathbb {R}}^N)$, where $0<\beta <1$ and $1\le q<\infty$ such that $\beta q>N$. Then for every $0<\alpha <\beta -\frac{N}{q}$, we have $\psi \in C^\alpha _{\textrm{loc}}({\mathbb {R}}^N)$. More precisely,

$$\begin{aligned} \sup _{x\ne y}\frac{|\psi (x)-\psi (y)|}{|x-y|^\alpha }\le C\Big ([\psi ]_{{\mathcal {N}}_\infty ^{\beta ,q} ({\mathbb {R}}^N)}\Big )^\frac{\alpha q+N}{\beta q}\Big (\Vert \psi \Vert _{L^q({\mathbb {R}}^N)}\Big )^\frac{(\beta -\alpha )q-N}{\beta q}, \end{aligned}$$

for some positive constant $C=C(N,q,\alpha ,\beta )$ which blows up as $\alpha \nearrow \beta -\frac{N}{q}$.

The following result follows from [4, Proposition 2.7].

Lemma 2.3

Let $\Omega \subset {\mathbb {R}}^N$ be an open and bounded set. Assume that $1<p<\infty$ and $0<s<1$. Then

$$\begin{aligned} \Vert u\Vert ^p_{L^p(\Omega )}\le C|\Omega |^\frac{sp}{N}\int _{{\mathbb {R}}^N}\int _{{\mathbb {R}}^N} \frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}}\,dx dy, \end{aligned}$$

holds for every $u\in W^{s,p}({\mathbb {R}}^N)$ such that $u=0$ almost everywhere in ${\mathbb {R}}^N\setminus \Omega$, for some positive constant $C=C(N,p,s)$.

2.4 Tail spaces and weak solutions

For a priori estimates, the so-called tail spaces that takes into account the global behavior are expedient. The tail space is defined as

$$\begin{aligned} L^{q}_{\alpha }({\mathbb {R}}^N)=\left\{ u\in L^{q}_{\textrm{loc}}({\mathbb {R}}^N)\,:\, \int _{{\mathbb {R}}^N} \frac{|u|^q}{1+|x|^{N+\alpha }}\,dx<+\infty \right\} ,\qquad q>0 \text{ and } \alpha >0, \end{aligned}$$

and the global behavior of a function $u\in L^q_{\alpha }({\mathbb {R}}^N)$ is measured by the quantity

$$\begin{aligned} \textrm{Tail}_{q,\alpha }(u;x_0,R)=\left[ R^\alpha \,\int _{{\mathbb {R}}^N\setminus B_R(x_0)} \frac{|u|^q}{|x-x_0|^{N+\alpha }}\,dx\right] ^\frac{1}{q}. \end{aligned}$$

Here $x_0\in {\mathbb {R}}^N$, $R>0,\,\beta >0$.

Definition 2.4

(Local weak solution) Suppose $\Omega \subset {\mathbb {R}}^N$ is an open and bounded set. Assume that $1<p<2$ and $0<s<1$. Let $f\in L_\text {loc}^q(\Omega )$ with $q\ge (p_s^{*})'$ if $sp\ne N$ and $q>1$ if $sp=N$. We define $u\in W_{\textrm{loc}}^{s,p}(\Omega )\cap L^{p-1}_{sp}({\mathbb {R}}^N)$ to be a local weak solution of $(-\Delta _p)^s u=f$ in $\Omega$, if

$$\begin{aligned} \int _{{\mathbb {R}}^N}\int _{{\mathbb {R}}^N}J_p(u(x) -u(y))(\phi (x)-\phi (y))\,d\mu =\int _\Omega f \phi \,dx, \end{aligned}$$

(2.4)

for every compactly supported $\phi \in W^{s,p}(\Omega )$, where $J_p(t)=|t|^{p-2}t$ and $d\mu =\frac{dx dy}{|x-y|^{N+ps}}$ are defined in (2.1) and (2.2) respectively.

Now we define the notion of weak solution for the Dirichlet problem associated with $(-\Delta _p)^s$. To this end, for given open and bounded sets $\Omega \Subset \Omega '\subset {\mathbb {R}}^N$ and $g\in L_{sp}^{p-1}({\mathbb {R}}^N)$, we define

$$\begin{aligned} X_g^{s,p}(\Omega ,\Omega '):=\{w\in W^{s,p}(\Omega ')\cap L_{p-1}^{sp}({\mathbb {R}}^N):w=g\text { a.e. in }{\mathbb {R}}^N\setminus \Omega \}. \end{aligned}$$

Definition 2.5

(Dirichlet problem) Suppose $\Omega \Subset \Omega '\subset {\mathbb {R}}^N$ are two open and bounded sets. Assume that $1<p<2$ and $0<s<1$. Let $f\in L^q(\Omega )$ with $q\ge (p_s^{*})'$ if $sp\ne N$ and $q>1$ if $sp=N$ and $g\in L^{p-1}_{sp}({\mathbb {R}}^N)$. We define $u\in X_g^{s,p}(\Omega ,\Omega ')$ to be a weak solution of the boundary value problem

$$\begin{aligned} (-\Delta _p)^s u=f\text { in }\Omega ,\quad u=g\text { a.e. in }{\mathbb {R}}^N\setminus \Omega , \end{aligned}$$

(2.5)

if for every $\phi \in X_0^{s,p}(\Omega ,\Omega ')$, Eq. (2.4) holds.

By Proposition 2.12 in [4], there exists a unique weak solution of the Dirichlet problem (2.5) in the sense above, given $g\in W^{s,p}(\Omega ')\cap L_{sp}^{p-1}({\mathbb {R}}^N)$.

3 The homogeneous equation

In this section, we treat the regularity for the homogeneous equation. This is done through an iteration scheme built on improved Besov-type regularity and improved Hölder regularity.

3.1 Improved Besov-type regularity

The starting point is the following improved Besov-type regularity.

Proposition 3.1

Let $1<p <2$ and $0<s<1$. Assume that $u\in W^{s,p}_{\textrm{loc}}(B_2)\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)$ is a local weak solution of $(-\Delta _p)^s u=0$ in $B_2$. Suppose that

$$\begin{aligned}{} & {} \Vert u\Vert _{L^\infty (B_1)}\le 1, \nonumber \\{} & {} \textrm{Tail}_{p-1,s\,p}(u;0,1)^{p-1}=\int _{{\mathbb {R}}^N\setminus B_1} \frac{|u(y)|^{p-1}}{|y|^{N+s\,p}}\,dy\le 1 \qquad \text{ and } \quad [u]_{C^\gamma (B_1)}\, \le 1,\nonumber \\ \end{aligned}$$

(3.1)

for some $\gamma \in [0,1)$. Moreover, suppose that for some $\alpha \in [0,1)$, $1\le q<\infty$ and $0<h_0<\frac{1}{10}$, we have

$$\begin{aligned} \sup _{0<|h|< h_0}\left\| \frac{\delta ^2_h u }{|h|^{\alpha }}\right\| _{L^{q}(B_{R+4h_0})} <+\infty . \end{aligned}$$

(3.2)

Then for R such that $4\,h_0<R\le 1-5\,h_0$, we have

$$\begin{aligned} \begin{aligned} \sup _{0<|h|< h_0}\left\| \frac{\delta ^2_h u}{|h|^{\frac{sp-\gamma (p -2)+\alpha q}{q+1}}}\right\| _{L^{q+1}(B_{R-4\,h_0})}^{q+1}\le C\,\left( \sup _{0<|h|< h_0}\left\| \frac{\delta ^2_h u }{|h|^\alpha } \right\| _{L^{q}(B_{R+4\,h_0})}^{q}+1\right) , \end{aligned}\nonumber \\ \end{aligned}$$

(3.3)

for some positive constant $C=C(N,s,p,q,h_0,\alpha ,\gamma )$ and $C\nearrow +\infty$ as $h_0\searrow 0$.

Proof

We divide the proof into three steps.

Step 1: Discrete differentiation of the equation. We set $r=R-4h_0$ and recall $\quad d\mu =\frac{dx dy}{|x-y|^{N+ps}}.$ Take $\varphi \in W^{s,p}(B_R)$ vanishing outside $B_{\frac{R+r}{2}}$. Let $h\in {\mathbb {R}}^N\setminus \{0\}$ be such that $|h|<h_0$. Testing (2.4) with $\varphi _{-h}$ and performing a change of variable yields

$$\begin{aligned} \frac{1}{h} \int _{{\mathbb {R}}^N} \int _{{\mathbb {R}}^N} \Big (J_p(u_h(x)-u_h(y))-J_p(u(x)-u(y))\Big )\,\Big (\varphi (x)-\varphi (y)\Big )\,d\mu =0.\nonumber \\ \end{aligned}$$

(3.4)

In what follows, suppose $\eta \in C^\infty _0(B_R)$ is such that

$$\begin{aligned} 0\le \eta \le 1,\qquad \eta \equiv 0 \text{ in } {\mathbb {R}}^N\setminus B_{\frac{R+r}{2}},\qquad |\nabla \eta |\le \frac{C}{R-r}=\frac{C}{4\,h_0}. \end{aligned}$$

Testing (3.4) with

$$\begin{aligned} \varphi =J_{q+1}\left( \frac{\delta _h u}{|h|^{\theta }}\right) \,\eta ^2, \end{aligned}$$

we get

$$\begin{aligned} \begin{aligned}&\iint _{{\mathbb {R}}^N\times {\mathbb {R}}^N} \frac{\Big (J_p(u_h(x)-u_h(y))-J_p(u(x)-u(y))\Big )}{|h|^{1+\theta \,q}}\\&\quad \times \Big (J_{q+1}(u_h(x)-u(x))\,\eta (x)^2-J_{q+1}(u_h(y)-u(y))\,\eta (y)^2\Big )\,d\mu =0. \end{aligned} \end{aligned}$$

(3.5)

We split the above double integral into three pieces:

$$\begin{aligned} {\mathcal {I}}_1&:=\iint _{B_R\times B_R}\frac{\Big (J_p(u_h(x) -u_h(y))-J_p(u(x)-u(y))\Big )}{|h|^{1+\theta \,q}}\\&\qquad \times \Big (J_{q+1}(u_h(x)-u(x))\,\eta (x)^2-J_{q+1}(u_h(y)-u(y))\,\eta (y)^2\Big )d\mu ,\\ {\mathcal {I}}_2&:=\iint _{B_\frac{R+r}{2}\times ({\mathbb {R}}^N\setminus B_R)} \frac{\Big (J_p(u_h(x)-u_h(y))-J_p(u(x)-u(y))\Big )}{|h|^{1 +\theta \,q}}\\&\qquad J_{q+1}(u_h(x)-u(x))\,\eta (x)^2\,d\mu ,\\ \end{aligned}$$

and

$$\begin{aligned} {\mathcal {I}}_3&:=-\iint _{({\mathbb {R}}^N\setminus B_R)\times B_\frac{R+r}{2}}\frac{\Big (J_p(u_h(x)-u_h(y))-J_p(u(x)-u(y))\Big )}{|h|^{1 +\theta \,q}}\\&\qquad J_{q+1}(u_h(y)-u(y))\,\eta (y)^2\,d\mu , \end{aligned}$$

where we used that $\eta$ vanishes identically outside $B_{(R+r)/2}$. Thus the Eq. (3.5) can be written as

$$\begin{aligned} {\mathcal {I}}_1=-{\mathcal {I}}_2-{\mathcal {I}}_3. \end{aligned}$$

(3.6)

We estimate ${\mathcal {I}}_j$ for $j=1,2,3$ separately.

Estimate of ${\mathcal {I}}_1$. We observe that

$$\begin{aligned} \begin{aligned}&J_{q+1}(u_h(x)-u(x))\,\eta (x)^2-J_{q+1}(u_h(y)-u(y))\,\eta (y)^2\\&\quad =\frac{\Big (J_{q+1}(u_h(x)-u(x))-J_{q+1}(u_h(y) -u(y))\Big )}{2}\,\Big (\eta (x)^2+\eta (y)^2\Big )\\&\qquad +\frac{\Big (J_{q+1}(u_h(x)-u(x))+J_{q+1}(u_h(y) -u(y))\Big )}{2}\,(\eta (x)^2-\eta (y)^2). \end{aligned} \end{aligned}$$

Therefore, we have

$$\begin{aligned} \begin{aligned} {I}&=\Big (J_p(u_h(x)-u_h(y))-J_p(u(x)-u(y))\Big )\\&\quad \Big (J_{{q+1}}(u_h(x)-u(x))\,\eta (x)^2-J_{{q+1}}(u_h(y)-u(y))\,\eta (y)^2\Big )\\&\ge \Big (J_p(u_h(x)-u_h(y))-J_p(u(x)-u(y))\Big )\\&\quad \times \Big (J_{q+1}(u_h(x)-u(x))-J_{q+1}(u_h(y)-u(y)) \Big )\,\left( \frac{\eta (x)^2+\eta (y)^2}{2}\right) \\&\quad -\Big |J_p(u_h(x)-u_h(y))-J_p(u(x)-u(y))\Big |\\&\quad \times {\Big |J_{q+1}(u_h(x)-u(x))+J_{q+1}(u_h(y)-u(y)) \Big |}\,\left| \frac{\eta (x)^2-\eta (y)^2}{2}\right| \\&:=J_1-J_2. \end{aligned} \end{aligned}$$

(3.7)

Estimate of $J_1$: We will now estimate the positive term. With the notation

$$\begin{aligned} a=u_h(x), \quad b=u_h(y),\quad c=u(x)\quad \text { and } \quad d=u(y), \end{aligned}$$

we have by Lemma A.1 together with the fact that u is locally $\gamma$-Hölder continuous (recall (3.1))

$$\begin{aligned} \begin{aligned} J_1&=\Big (J_p(a-b)-J_p(c-d)\Big )\Big (J_{q+1}(a-c)-J_{q+1}(b -d)\Big )\Big (\frac{\eta (x)^2+\eta (y)^2}{2}\Big )\\&\ge C\left| |a-c|^\frac{q-1}{2}(a-c)-|b-d|^\frac{q-1}{2}(b -d)\right| ^2 \left( |a-b|^{p-2}+|c-d|^{p-2}\right) \\&\quad \Big (\frac{\eta (x)^2+\eta (y)^2}{2}\Big )\\&\ge C\left| |a-c|^\frac{q-1}{2}(a-c)-|b-d|^\frac{q-1}{2}(b -d)\right| ^2\\&\quad |x-y|^{\gamma (p-2)}\Big (\frac{\eta (x)^2+\eta (y)^2}{2}\Big )\\&=C\left| |\delta _h u(x)|^\frac{q-1}{2}(\delta _h u(x)) -|\delta _h u (y)|^\frac{q-1}{2}(\delta _h u(y))\right| ^2\\&\quad (\eta (x)^2 +\eta (y)^2) |x-y|^{\gamma (p-2)}, \quad C=C(p,q). \end{aligned} \end{aligned}$$

(3.8)

Estimate of $J_2$: We will absorb a part of the term

$$\begin{aligned}{} & {} \Big (J_p(u_h(x)-u_h(y))-J_p(u(x)-u(y))\Big )\times \Big (J_{q+1}(\delta _ h u(x)))+J_{q+1}(\delta _h u(y))\Big )\\{} & {} \quad \left( \eta (x)^2-\eta (y)^2\right) \end{aligned}$$

into the positive term

$$\begin{aligned}{} & {} \Big (J_p(u_h(x)-u_h(y))-J_p(u(x)-u(y))\Big )\times \Big ((J_{q+1}(\delta _hu(x))-J_{q+1}(\delta _hu(y))) \Big )\\{} & {} \quad \left( \eta (x)^2+\eta (y)^2\right) . \end{aligned}$$

We write, noticing that $\delta _h J_p(u(x)-u(y))$ and $\delta _h u(x)-\delta _h u(y)$ have the same sign

$$\begin{aligned}{} & {} \delta _h J_p(u(x)-u(y))\nonumber \\{} & {} \quad =\delta _h J_p(u(x)-u(y))\left( \frac{\big (J_{q+1}(\delta _hu(x))-J_{q+1}(\delta _hu(y))\big )\delta _h J_p(u(x)-u(y))}{\big (J_{q+1}(\delta _hu(x))-J_{q+1}(\delta _h u(y))\big )\delta _h J_p(u(x)-u(y))}\right) ^\frac{p-1}{p}\nonumber \\{} & {} \quad =\left[ \delta _h J_p(u(x)-u(y))\big (J_{q+1}(\delta _hu(x))-J_{q+1}(\delta _hu(y))\big )\right] ^\frac{p-1}{p}\delta _h J_p(u(x)-u(y))\nonumber \\{} & {} \qquad \times \left( \frac{(\delta _h u(x)-\delta _h u(y))}{\big (J_{q+1}(\delta _h u(x))-J_{q+1}(\delta _hu(y))\big )(\delta _h u(x)-\delta _h u(y))\delta _h J_p(u(x)-u(y))}\right) ^\frac{p-1}{p} \nonumber \\{} & {} \quad =\left[ \delta _h J_p(u(x)-u(y))\big (J_{q+1}(\delta _hu(x))-J_{q+1}(\delta _hu(y))\big )\right] ^\frac{p-1}{p}\nonumber \\{} & {} \qquad \times \left( \frac{1}{\big (J_{{q}+1}(\delta _hu(x))-J_{q+1}(\delta _hu(y))\big )(\delta _h u(x)-\delta _h u(y))}\right) ^\frac{p-1}{p} \nonumber \\{} & {} \qquad \times \delta _h J_p(u(x)-u(y))\left( \frac{\delta _h u(x)-\delta _h u(y)}{\delta _h J_p(u(x)-u(y))}\right) ^\frac{p-1}{p}. \end{aligned}$$

(3.9)

We observe that

$$\begin{aligned} \begin{aligned}&\Big |\delta _h J_p(u(x)-u(y))\left( \frac{\delta _h u(x) -\delta _h u(y)}{\delta _h J_p(u(x)-u(y))}\right) ^\frac{p-1}{p}\Big |\\&\quad \le |\delta _h u(x)-\delta _h u(y)|^\frac{p-1}{p}||\delta _h J_p(u(x)-u(y))|^\frac{1}{p}\\&\quad \le {C}|\delta _h u(x)-\delta _h u(y)|^{(p-1)/p+(p-1)/p}\\&\quad ={C}|\delta _h u(x)-\delta _h u(y)|^\frac{2(p-1)}{p}, \quad {C=C(p)} \end{aligned} \end{aligned}$$

(3.10)

which follows from the $(p-1)$-Hölder regularity for $J_p$. Using Lemma A.1 and (3.10) in (3.9) yields

$$\begin{aligned} \begin{aligned} |\delta _h J_p(u(x)-u(y))|&\le {C} \left[ \delta _h J_p(u(x)-u(y)) \big (J_{q+1}(\delta _hu(x))-J_{q+1}(\delta _hu(y))\big )\right] ^\frac{p-1}{p}\\&\quad \times \left( \frac{1}{(|\delta _h u(x)|^{q-1}+|\delta _h u(y)|^{q-1}) (\delta _h u(x)-\delta _h u(y))^2}\right) ^\frac{p-1}{p}\\&\quad \times |\delta _h u(x)-\delta _h u(y)|^\frac{2(p-1)}{p}\\&={C}\left[ \delta _h J_p(u(x)-u(y))\big (J_{q+1}(\delta _hu(x)) -J_{q+1}(\delta _hu(y))\big )\right] ^\frac{p-1}{p}\\&\quad \times \left( \frac{1}{|\delta _h u(x)|^{q-1}+|\delta _h u(y) |^{q-1}}\right) ^\frac{p-1}{p},\quad { C=C(p,q).} \end{aligned}\nonumber \\ \end{aligned}$$

(3.11)

Therefore, by the above estimate (3.11) and using Young’s inequality with p and $p/(p-1)$, we have

$$\begin{aligned}{} & {} |\delta _h J_p(u(x)-u(y))|\times \Big |J_{q+1}(\delta _ h u(x))+J_{q+1}(\delta _h u(y))\Big |\,\left| \eta (x)^2-\eta (y)^2\right| \nonumber \\{} & {} \quad =|\delta _h J_p(u(x)-u(y))|\times \Big |J_{q+1}(\delta _ h u(x))+J_{q+1}(\delta _h u(y))\Big |\,\left| \eta (x)-\eta (y)\right| \left( \eta (x)+\eta (y)\right) \nonumber \\{} & {} \quad \le {C}\left[ \delta _h J_p(u(x)-u(y))\big (J_{q+1}(\delta _hu(x))-J_{q+1}(\delta _hu(y))\big )\right] ^\frac{p-1}{p}\left( \eta (x)+\eta (y)\right) \nonumber \\{} & {} \qquad \times \Big |J_{q+1}(\delta _ h u(x))+J_{q+1}(\delta _h u(y))\Big |\left( \frac{1}{(|\delta _h u(x)|^{q-1}+|\delta _h u(y)|^{q-1})}\right) ^\frac{p-1}{p}|\eta (x)-\eta (y)|\nonumber \\{} & {} \quad \le \frac{1}{2} \delta _h J_p(u(x)-u(y))\big (J_{q+1}(\delta _hu(x))-J_{q+1}(\delta _hu(y))\big )\left( \eta (x)^\frac{p}{p-1}+\eta (y)^\frac{p}{p-1}\right) \nonumber \\{} & {} \qquad +{C} \Big |J_{q+1}(\delta _ h u(x))+J_{q+1}(\delta _h u(y))\Big |^p\left( \frac{1}{|\delta _h u(x)|^{q-1}+|\delta _h u(y)|^{q-1}}\right) ^{p-1}|\eta (x)-\eta (y)|^p\nonumber \\{} & {} \quad \le \frac{1}{2} \delta _h J_p(u(x)-u(y))\left( (J_{q+1}(\delta _hu(x))-J_{q+1}(\delta _hu(y)))\right) \left( \eta ^2(x)+\eta ^2(y)\right) \nonumber \\{} & {} \qquad +{C} \Big (|\delta _h u(x)|^{q+p-1}+|\delta _h u(y)|^{q+p-1}\Big )|\eta (x)-\eta (y)|^p,\quad { C=C(p,q)} \end{aligned}$$

(3.12)

where we used that $\eta ^\frac{p}{p-1}\le \eta ^2$ since $p/(p-1)\ge 2$. Thus from (3.12), we conclude that

$$\begin{aligned} \begin{aligned} J_2\le \frac{1}{2} J_1+ {C}\left( |\delta _h u(x)|^{q+p-1}+|\delta _h u(y)|^{q+p-1}\right) |\eta (x)-\eta (y)|^p,\quad C=C(p,q). \end{aligned}\nonumber \\ \end{aligned}$$

(3.13)

Therefore, using (3.8) and (3.13) in (3.7), we obtain

$$\begin{aligned} \begin{aligned} I&\ge \frac{1}{2} J_1-{C(p)}\left( |\delta _h u(x)|^{q+p-1}+|\delta _h u(y)|^{q+p-1}\right) |\eta (x)-\eta (y)|^p\\&\ge C\left| |\delta _h u(x)|^\frac{q-1}{2}(\delta _h u(x))-|\delta _h u (y)|^\frac{q-1}{2}(\delta _h u(y))\right| ^2(\eta (x)^2+\eta (y)^2) |x-y|^{\gamma (p-2)}\\&\quad -C(p,q) \left( |\delta _h u(x)|^{q+p-1}+|\delta _h u(y)|^{q+p-1}\right) |\eta (x)-\eta (y)|^p,\quad C=C(p,q). \end{aligned}\nonumber \\ \end{aligned}$$

(3.14)

Thus (3.14) gives

$$\begin{aligned} \begin{aligned}&{C}\iint _{B_R\times B_R} \frac{1}{|h|^{1+\theta q}}\left| |\delta _h u(x)|^\frac{q-1}{2}(\delta _h u(x))-|\delta _h u (y)|^\frac{q-1}{2}(\delta _h u(y))\right| ^2\\&\quad \qquad (\eta (x)^2+\eta (y)^2) |x-y|^{\gamma (p-2)}d\mu \\&\qquad \le {\mathcal {I}}_1+C\int _{B_R} \frac{|\delta _h u(x)|^{p+q-1}}{|h|^{1+\theta q}} dx,\quad C=C(N,h_0,p,q,s). \end{aligned} \end{aligned}$$

(3.15)

Here we have also used that the factor $|\eta (x)-\eta (y)|^p$ cancels out the singularity of the kernel in the last term above. We now observe that with

$$\begin{aligned} A=\frac{|\delta _h u(x)|^\frac{q-1}{2}\,\delta _h u(x)}{|h|^\frac{1+\theta \,q}{2}}\qquad \text{ and } \qquad B=\frac{|\delta _h u(y)|^\frac{q-1}{2}\,\delta _h u(y)}{|h|^\frac{1+\theta \,q}{2}} \end{aligned}$$

the convexity of $\tau \mapsto \tau ^2$ implies

$$\begin{aligned} \begin{aligned} \left| A\,\eta (x)-B\,\eta (y)\right| ^2&=\left| (A-B)\,\frac{\eta (x) +\eta (y)}{2}+(A+B)\,\frac{\eta (x)-\eta (y)}{2}\right| ^2\\&\le \frac{1}{2}\,|A-B|^2\,\left| \eta (x)+\eta (y)\right| ^2\\&\quad +\frac{1}{2}\, |A+B|^2\,\left| \eta (x)-\eta (y)\right| ^2\\&\le |A-B|^2\, (\eta (x)^2+\eta (y)^2)\\&\quad + (|A|^2+|B|^2)\, |\eta (x)-\eta (y)|^2. \end{aligned} \end{aligned}$$

Taking the above inequality into account and using (3.15), we get the following lower bound for ${\mathcal {I}}_1$, with $2\sigma = sp-\gamma (p-2)$:

$$\begin{aligned} \begin{aligned} {\mathcal {I}}_1&\ge c \left[ \frac{|\delta _h u|^\frac{q-1}{2}\,\delta _h u}{|h|^\frac{1+\theta \,q}{2}}\,\eta \right] ^2_{W^{\sigma ,2}(B_R)} -C\int _{B_R} \frac{|\delta _h u(x)|^{p+q-1}}{|h|^{1+\theta \,q}} dx\\&\quad -C\,\iint _{B_R\times B_R}\, \left( \frac{|\delta _h u(x)|^{(q+1) }}{|h|^{1+\theta \,q}}+\frac{|\delta _h u(y)|^{(q+1) }}{|h|^{1+\theta \,q}}\right) \, |\eta (x)-\eta (y)|^2\,d\mu \\&\ge {c}\left[ \frac{|\delta _h u|^\frac{q-1}{2}\,\delta _h u}{|h|^\frac{1+\theta \,q}{2}}\,\eta \right] ^2_{W^{\sigma ,2}(B_R)}-C\int _{B_R} \frac{|\delta _h u(x)|^{p+q-1}}{|h|^{1+\theta \,q}} dx\\&\quad -C\,\int _{B_R}\, \frac{|\delta _h u(x)|^{q+1}}{|h|^{1+\theta \,q}}dx, \end{aligned} \end{aligned}$$

where we again used that $\eta$ is Lipschitz. Here $c=c(p,q)>0$ and $C=C(N,h_0,p,q,s)>0$. Note that $\sigma \in (0,1)$ since $\gamma \in [0,1)$ and $s\in (0,1)$. By recalling that ${\mathcal {I}}_1+{\mathcal {I}}_2+{\mathcal {I}}_3=0$ from (3.6) and using the estimate for ${\mathcal {I}}_1$, we arrive at

$$\begin{aligned} \left[ \frac{|\delta _h u|^\frac{q-1}{2}\,\delta _h u}{|h|^\frac{1+\theta \,q}{2}}\,\eta \right] ^2_{W^{\sigma ,2}(B_R)}\le C\,\Big (\int _{B_R} \frac{|\delta _h u(x)|^{p+q-1}}{|h|^{1+\theta \,q}} +\frac{|\delta _h u(x)|^{q+1}}{|h|^{1+\theta \,q}}+|{\mathcal {I}}_2|+|{\mathcal {I}}_3|\Big ),\nonumber \\ \end{aligned}$$

(3.16)

for $C=C(N,h_0,p,q,s)>0$.

Step 2: Estimates of the nonlocal terms ${\mathcal {I}}_2$ and ${\mathcal {I}}_3$: Both nonlocal terms ${\mathcal {I}}_2$ and ${\mathcal {I}}_3$ can be treated in the same way. We only estimate ${\mathcal {I}}_2$ for simplicity. Using (3.1), since $|u|\le 1$ in $B_1$, for every $x\in B_{(R+r)/2}$ and $y\in {\mathbb {R}}^N{\setminus } B_R$, we have

$$\begin{aligned} \begin{aligned}&\Big |\big (J_p(u_h(x)-u_h(y))-J_p(u(x)-u(y))\big )\, J_{q+1}(\delta _h u(x))\Big |\\&\quad \le C\left( 1+|u_h(y)|^{p-1}+|u(y)|^{p-1}\right) \,|\delta _h u(x)|^{q}, \end{aligned} \end{aligned}$$

where $C=C(p)>0$. For $x\in B_{(R+r)/2}$ we have $B_{(R-r)/2}(x)\subset B_{R}$ and thus

$$\begin{aligned} \int _{{\mathbb {R}}^N\setminus B_{R}}\frac{1}{|x-y|^{N+s\,p}}\, dy\le \int _{{\mathbb {R}}^N\setminus B_\frac{R-r}{2}(x)} \frac{1}{|x-y|^{N+s\,p}}\, dy\le C(N,h_0,p,s), \end{aligned}$$

by recalling that $R-r=4\,h_0$. By using [4, Lemma 2.2 and Lemma 3.3], we get for $x \in B_{(R+r)/2}$

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}^N\setminus B_{R}} \frac{|u(y)|^{p-1}}{|x-y|^{N+s\,p}}\, dy&\le \left( \frac{2\,R}{R-r}\right) ^{N+s\,p}\,\int _{{\mathbb {R}}^N\setminus B_R} \frac{|u(y)|^{p-1}}{|y|^{N+s\,p}} \, dy\\&\le \left( \frac{2\,R}{R-r}\right) ^{N+s\,p}\,\int _{{\mathbb {R}}^N\setminus B_1} \frac{|u(y)|^{p-1}}{|y|^{N+s\,p}} \, dy\\&\quad +\left( \frac{2\,R}{R-r}\right) ^{N+s\,p}\,R^{-N}\,\int _{B_1} |u|^{p-1}\,dy\\&\le C(N,h_0,p,s). \end{aligned} \end{aligned}$$

In the last estimate we have used the bounds assumed on u in (3.1) and $4\,h_0< R \le 1$. The term involving $u_h$ can be estimated similarly. Recall also that $\eta =0$ outside $B_{(R+r)/2}$. Hence, we have

$$\begin{aligned} |{\mathcal {I}}_2|\le C\int _{B_R}\frac{|\delta _h u(x)|^{q}}{|h|^{1+\theta \,q}} dx. \end{aligned}$$

(3.17)

Similarly, we get

$$\begin{aligned} |{\mathcal {I}}_3|\le C\int _{B_R}\frac{|\delta _h u(x)|^{q}}{|h|^{1+\theta \,q}} dx. \end{aligned}$$

(3.18)

The combination of (3.16), (3.17) and (3.18) now implies

$$\begin{aligned} \begin{aligned} \left[ \frac{|\delta _h u|^\frac{q-1}{2}\,\delta _h u}{|h|^\frac{1+\theta \,q}{2}}\,\eta \right] ^2_{W^{\sigma ,2}(B_R)}\le C\,\Big (\int _{B_R} \frac{|\delta _h u(x)|^{p+q-1}}{|h|^{1+\theta \,q}} +\frac{|\delta _h u(x)|^{q+1 }}{|h|^{1+\theta \,q}}+\frac{|\delta _h u(x)|^{q}}{|h|^{1+\theta \,q}}dx\Big ), \end{aligned}\nonumber \\ \end{aligned}$$

(3.19)

where $C=C(N,h_0,p,s,q)>0.$

Step 3: Transformation to double differences. For $\xi ,h\in {\mathbb {R}}^N\setminus \{0\}$ such that $|h|,|\xi |<h_0$, we let

$$\begin{aligned} A=u(x+h+\xi )-u(x+\xi ),\qquad B=u(x+h)-u(x),\qquad \gamma =\frac{q+1}{2}. \end{aligned}$$

Lemma A.5 implies

$$\begin{aligned} |\delta _h \delta _\xi u|^\frac{q+1}{2}\le C \delta _\xi |\delta _h u|^\frac{q-1}{2}\delta _h u,\quad C=C(q). \end{aligned}$$

Therefore,

$$\begin{aligned} \left\| \frac{|\delta _\xi \delta _h u|^\frac{q+1}{2}}{|\xi |^\sigma \,|h|^\frac{1+\theta \,q}{2}}\right\| ^{2}_{L^{2}(B_r)}\le & {} C \left\| \frac{\delta _\xi \left( |\delta _h u|^\frac{q-1}{2} \,\delta _h u\right) }{|\xi |^\sigma \,|h|^\frac{1+\theta \,q}{2}}\right\| ^2_{L^2(B_r)}\nonumber \\\le & {} C\,\left\| \eta \,\frac{\delta _\xi }{|\xi |^\sigma } \left( \frac{|\delta _h u|^\frac{q-1}{2}\,(\delta _h u)}{|h|^\frac{1+\theta \,q}{2}}\right) \right\| ^2_{L^2({\mathbb {R}}^N)}, \end{aligned}$$

(3.20)

where $C=C(q)>0$. Here we used that $\eta \equiv 1$ on $B_r$. By a discrete version of Leibniz rule (see (2.3)),

$$\begin{aligned} \eta \,\delta _\xi \left( |\delta _h u|^\frac{q-1}{2}\,(\delta _h u)\right) = \delta _\xi \left( \eta \,|\delta _h u|^\frac{q-1}{2}\,(\delta _h u)\right) -\left( |\delta _h u|^\frac{q-1}{2}\,(\delta _h u)\right) _\xi \,\delta _{\xi }\eta .\nonumber \\ \end{aligned}$$

(3.21)

Inserting (3.21) into (3.20) yields

$$\begin{aligned} \left\| \frac{|\delta _\xi \delta _h u|^\frac{q+1}{2}}{|\xi |^\sigma \,|h|^\frac{1+\theta \,q}{2}}\right\| ^2_{L^{2}(B_r)}\le & {} C\, \left\| \frac{\delta _\xi }{|\xi |^\sigma }\left( \frac{|\delta _h u|^\frac{q-1}{2}\,(\delta _h u)\,\eta }{|h|^\frac{1+\theta \,q}{2}} \right) \right\| ^2_{L^2({\mathbb {R}}^N)}\nonumber \\{} & {} + C\, \left\| \frac{\delta _\xi \eta }{|\xi |^\sigma }\frac{\left( |\delta _h u |^\frac{q-1}{2}\,(\delta _h u)\right) _\xi }{|h|^\frac{1+\theta \,q}{2}} \right\| ^2_{L^2({\mathbb {R}}^N)}, \end{aligned}$$

(3.22)

where $C=C(q)>0$. For the first term in (3.22), we apply [3, Proposition 2.6] with the choice

$$\begin{aligned} \psi =\frac{|\delta _h u|^\frac{q-1}{2}\,(\delta _h u)\,\eta }{|h|^\frac{1+\theta \,q}{2}}, \end{aligned}$$

and get

$$\begin{aligned} \sup _{|\xi |>0}\left\| \frac{\delta _\xi }{|\xi |^\sigma }\frac{|\delta _h u|^\frac{q-1}{2}\,(\delta _h u)\,\eta }{|h|^\frac{1+\theta \,q}{2}}\right\| ^2_{L^2({\mathbb {R}}^N)}\le C\,(1-\sigma )\left[ \frac{|\delta _h u|^\frac{q-1}{2}\,(\delta _h u)\,\eta }{|h|^\frac{1+\theta \,q}{2}}\right] ^2_{W^{\sigma ,2}(B_R)},\nonumber \\ \end{aligned}$$

(3.23)

where $C=C(N,h_0,\sigma )>0$. Here we also used that $\frac{R+r}{2} + 2h_0 = R$.

As for the second term in (3.22), we observe that for every $0<|\xi |<h_0$

$$\begin{aligned} \begin{aligned} \left\| \frac{\delta _\xi \eta }{|\xi |^\sigma }\frac{\left( |\delta _h u |^\frac{q-1}{2}\,(\delta _h u)\right) _\xi }{|h|^\frac{1+\theta \,q}{2}} \right\| ^2_{L^2({\mathbb {R}}^N)}&\le C\,\left\| \frac{\left( |\delta _h u|^\frac{q-1}{2}\,(\delta _h u) \right) _\xi }{|h|^\frac{1+\theta \,q}{2}}\right\| ^2_{L^2(B_{\frac{R+r}{2}+h_0})}\\&\le C\,\int _{B_{\frac{R+r}{2}+2h_0}}\frac{|\delta _h u|^{q+1}}{|h|^{1+\theta \,q}} dx\\&\le C\,\int _{B_{R}}\frac{|\delta _h u|^{q+1}}{|h|^{1+\theta \,q}} dx, \end{aligned} \end{aligned}$$

(3.24)

where $C=C(N,h_0,s)>0$. Here we have used the estimate of $\nabla \eta$.

From (3.22), (3.23), and (3.24) we get for any $0< |\xi | < h_0$

$$\begin{aligned} \left\| \frac{\delta _\xi \delta _h u}{|\xi |^\frac{2\sigma }{q+1}|h|^\frac{1+\theta \,q}{(q+1)}}\right\| ^{q+1}_{L^{q+1}(B_r)}\le C\,\left[ \frac{|\delta _h u|^\frac{q-1}{2}\,(\delta _h u)}{|h|^\frac{1+\theta \,q}{2}}\eta \right] ^2_{W^{\sigma ,2}(B_R)}+C\,\int _{B_{R}}\frac{|\delta _h u|^{q+1}}{|h|^{1+\theta \,q}} dx,\nonumber \\ \end{aligned}$$

(3.25)

with $C=C(N,h_0,s,q,\sigma )>0$. We then choose $\xi =h$ and take the supremum over h for $0<|h|< h_0$. Then (3.25) together with (3.19) imply

$$\begin{aligned}{} & {} \sup _{0<|h|< h_0}\int _{B_r}\left| \frac{\delta ^2_h u}{|h|^{\frac{2\sigma }{q+1}+\frac{1+\theta \,q}{(q+1)}}}\right| ^{q+1}\,dx\nonumber \\{} & {} \qquad \quad \le C\,\Big (\int _{B_R} \frac{|\delta _h u(x)|^{p+q-1}}{|h|^{1+\theta \,q}} +\frac{|\delta _h u(x)|^{q+1}}{|h|^{1+\theta \,q}}+\frac{|\delta _h u(x)|^{q}}{|h|^{1+\theta \,q}}dx\Big ), \end{aligned}$$

(3.26)

where $C=C(N,h_0,p,s,q,\sigma )>0$.

Now we choose $\theta =\alpha -1/q$ and observe that since $\Vert u\Vert _{L^\infty (B_1)}\le 1$ from (3.1), the assumption that $4h_0<R\le 1-5h_0$ and the fact that $q\le q+p-1\le q+1$, implies that the first and the second terms in the right hand side of (3.26) can be estimated by the third one. Recalling also that $2\sigma = sp-\gamma (p-2)$, this yields

$$\begin{aligned} \sup _{0<|h|< h_0}\int _{B_r}\left| \frac{\delta ^2_h u}{|h|^{\frac{sp-\gamma (p-2)+\alpha q}{q+1}}}\right| ^{q+1}\,dx\le C\,\int _{B_{R+h_0}} \frac{|\delta _h u(x)|^{q}}{|h|^{\alpha \,q}}dx, \end{aligned}$$

(3.27)

where $C=C(N,h_0,p,s,q,\gamma )>0$. Since $\alpha <1$, taking into account (3.2) and using the second estimate of [4, Lemma 2.6] we replace the first order difference quotient in the right-hand side of (3.27) with a second order difference quotient. Then (3.27) transforms into the desired inequality (3.3), upon recalling the relations between R, r and $h_0$. $\square$

3.2 Improved Hölder regularity

We can now iterate the improved Besov-type regularity to obtain an improved Hölder regularity.

Proposition 3.2

Assume $1<p<2$, $0<s<1$ and $\gamma \in [0,1)$. Let $u\in W^{s,p}_{\textrm{loc}}(B_2)\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)$ be a local weak solution of $(-\Delta _p)^s u=0$ in $B_2$. Suppose that

$$\begin{aligned}{} & {} \Vert u\Vert _{L^\infty (B_1)}\le 1, \\{} & {} \textrm{Tail}_{p-1,s\,p}(u;0,1)^{p-1}=\int _{{\mathbb {R}}^N\setminus B_1} \frac{|u(y)|^{p-1}}{|y|^{N+s\,p}}\,dy\le 1\quad \text{ and } \quad [u]_{C^\gamma (B_1)}\le 1. \end{aligned}$$

Let $\tau =\min (sp-\gamma (p-2),1)$. Then for any $\varepsilon \in (0,\tau )$, we have

$$\begin{aligned} {[}u]_{C^{\tau -\varepsilon }(B_\frac{1}{2})}\le C(s,p,\varepsilon ,N,\gamma ). \end{aligned}$$

Proof

Take $0<\varepsilon <\tau$ and choose q so that

$$\begin{aligned} \tau -\frac{\varepsilon }{2}-\frac{N}{q}>\tau -\varepsilon >0. \end{aligned}$$

Then we define the sequence of exponents

$$\begin{aligned} \alpha _0=0, \quad \alpha _i=\frac{sp-\gamma (p-2)+\alpha _{i-1} q}{q+1},\qquad i=0,\dots ,i_\infty , \end{aligned}$$

where we choose $i_\infty \ge 1$ such that

$$\begin{aligned} \alpha _{i_\infty -1}<\tau -\varepsilon /2\le \alpha _{i_\infty }. \end{aligned}$$

Note that this is possible since the sequence of exponents $\alpha _i$ are increasing towards $sp-\gamma (p-2)$. Define also

$$\begin{aligned} h_0=\frac{1}{64\,i_\infty },\qquad R_i=\frac{7}{8}-4\,(2i+1)\,h_0=\frac{7}{8}-\frac{2i+1}{16\,i_\infty },\qquad \text{ for } i=0,\dots ,i_\infty . \end{aligned}$$

We note that

$$\begin{aligned} R_0+4\,h_0=\frac{7}{8}\qquad \text{ and } \qquad R_{i_\infty -1}-4\,h_0=\frac{3}{4}. \end{aligned}$$

By applying Proposition 3.1 and with $R=R_i$ and observing that $R_i-4\,h_0=R_{i+1}+4\,h_0$, we obtain the iterative scheme of inequalities

$$\begin{aligned} \left\{ \begin{array}{ll} \sup \limits _{0<|h|< h_0}\left\| \dfrac{\delta ^2_h u}{|h|^{\alpha _1}} \right\| _{L^{q}(B_{R_1+4h_0})}\le C\,\sup \limits _{0<|h|< h_0} \left( \left\| \delta ^2_h u \right\| _{L^q(B_{7/8})}+1\right) &{}\\ \sup \limits _{|h|\le h_0}\left\| \dfrac{\delta ^2_h u}{|h|^{\alpha _{i+1}}} \right\| _{L^{q}(B_{R_{i+1}+4h_0})}\le C\,\sup \limits _{0<|h|< h_0} \left( \left\| \dfrac{\delta ^2_h u }{|h|^{\alpha _i}}\right\| _{L^{q} (B_{R_i+4h_0})}+1\right) ,&{}\quad \text{ for } i=1,\ldots ,i_\infty -2, \end{array} \right. \end{aligned}$$

and finally

$$\begin{aligned} \sup _{0<|h|< {h_0}}\left\| \frac{\delta ^2_h u}{|h|^{\alpha _{i_\infty }}}\right\| _{L^{q}(B_{3/4})}= & {} \sup _{0<|h|< h_0}\left\| \frac{\delta ^2_h u}{|h|^{\alpha _{i_\infty }}}\right\| _{L^{q}(B_{R_{i_\infty -1}-4h_0})}\\\le & {} C\sup _{0<|h|< h_0}\left( \left\| \frac{\delta ^2_h u }{|h|^{\alpha _{i_\infty -1}}}\right\| _{L^{q}(B_{R_{i_\infty -1}+4h_0})}+1\right) . \end{aligned}$$

Here $C=C(N,\varepsilon ,p,s,\gamma )>0$. Also,

$$\begin{aligned} \sup \limits _{0<|h|< h_0}\left\| \delta ^2_h u \right\| _{L^q(B_{7/8})}&\le {3}\Vert u\Vert _{L^{\infty }(B_{1})}\le {3}. \end{aligned}$$

Hence, the iterative scheme of inequalities leads us to

$$\begin{aligned} \sup _{0<|h|< {h_0}}\left\| \frac{\delta ^2_h u}{|h|^{\alpha _{i_\infty }}}\right\| _{L^{q}(B_{3/4})}\le C(N,\varepsilon ,p,s,\gamma ). \end{aligned}$$

(3.28)

Using $\alpha _{i_\infty }\ge \tau -\varepsilon /2$ in (3.28) implies

$$\begin{aligned} \sup _{0<|h|< {h_0}}\left\| \frac{\delta ^2_h u}{|h|^{\tau -\varepsilon /2}}\right\| _{L^{q}(B_{3/4})}\le C(N,\varepsilon ,p,s,\gamma ). \end{aligned}$$

(3.29)

Take now $\chi \in C_0^\infty (B_{5/8})$ such that

$$\begin{aligned} 0\le \chi \le 1, \qquad |\nabla \chi |\le C,\qquad |D^2 \chi |\le C \text { in }B_\frac{5}{8}\qquad \text { and }\qquad \chi =1 \text { in }B_{\frac{1}{2}}. \end{aligned}$$

In particular we have for all $|h| > 0$

$$\begin{aligned} \frac{|\delta _h\chi |}{|h|}\le C,\qquad \frac{|\delta ^2_h\chi |}{{|h|^2}}\le C. \end{aligned}$$

We also recall that

$$\begin{aligned} \delta ^2_h (u\,\chi )=\chi _{2h}\,\delta ^2_h u+2\,\delta _h u\, \delta _h \chi _h+u\,\delta ^2_h\chi . \end{aligned}$$

Hence, for $0<|h|< h_0$, using the above properties of $\chi$ and (3.29), we have

$$\begin{aligned} {[}u\,\chi ]_{{\mathcal {B}}^{\tau -\varepsilon /2,q}_\infty ({\mathbb {R}}^N)}&=\left\| \frac{\delta ^2_h (u\,\chi )}{|h|^{\tau -\varepsilon /2}} \right\| _{L^{q}({\mathbb {R}}^N)}\nonumber \\&\le C\,\left( \left\| \frac{\chi _{2h}\, \delta ^2_h u}{|h|^{\tau -\varepsilon /2}}\right\| _{L^{q}({\mathbb {R}}^N)} +\left\| \frac{\delta _h u\,\delta _h\chi }{|h|^{\tau -\varepsilon /2}} \right\| _{L^{q}({\mathbb {R}}^N)}+\left\| \frac{u\,\delta ^2_h\chi }{|h|^{\tau -\varepsilon /2}}\right\| _{L^{q}({\mathbb {R}}^N)}\right) \nonumber \\&\le C\,\left( \left\| \frac{\delta ^2_h u}{|h|^{\tau -\varepsilon /2}} \right\| _{L^{q}(B_{5/8+2\,h_0})}+\Vert \delta _h u\Vert _{L^{q} (B_{5/8+h_0})}+\Vert u\Vert _{L^{q}(B_{5/8+2h_0})}\right) \\&\le C\,\left( \left\| \frac{\delta ^2_h u}{|h|^{\tau -\varepsilon /2}} \right\| _{L^{q}(B_{3/4})}+\Vert u\Vert _{L^{q}(B_{3/4})}\right) \le C(N,\varepsilon ,p,s,\gamma ). \nonumber \end{aligned}$$

(3.30)

Thus by (3.30) and Lemma 2.1, we have

$$\begin{aligned} {[}u\,\chi ]_{{\mathcal {N}}_\infty ^{\tau -\frac{\varepsilon }{2},q}({\mathbb {R}}^N)}\le C(N,\varepsilon ,q)\,[u\,\chi ]_{{\mathcal {B}}_\infty ^{\tau -\varepsilon /2,q}({\mathbb {R}}^N)}\le C(N,\varepsilon ,p,s,\gamma ). \end{aligned}$$

Finally, thanks to the choice of q we have

$$\begin{aligned} \tau -\varepsilon <\tau -\frac{\varepsilon }{2}-\frac{N}{q}. \end{aligned}$$

We may therefore apply Theorem 2.2 with $\beta =\tau -\frac{\varepsilon }{2}$ and $\alpha =\tau -\varepsilon$ to obtain

$$\begin{aligned} {[}u]_{C^{\tau -\varepsilon }(B_{1/2})}= & {} [u\,\chi ]_{C^{\tau -\varepsilon }(B_{1/2})}\le C\left( [u\,\chi ]_{{\mathcal {N}}_\infty ^{\tau -\frac{\varepsilon }{2},q} ({\mathbb {R}}^N)}\right) ^{\frac{(\tau -\varepsilon )\,q+N}{(\tau -\frac{\varepsilon }{2})\,q}} \,\left( \Vert u\,\chi \Vert _{L^q({\mathbb {R}}^N)}\right) ^\frac{\frac{q\varepsilon }{2} -N}{(\tau -\frac{\varepsilon }{2})\,q}\\\le & {} C(N,\varepsilon ,p,s,\gamma ). \end{aligned}$$

This concludes the proof. $\square$

3.3 Final Hölder regularity

We first prove a normalized version of Theorem 1.1. This is accomplished by iterating the previously obtained improved Hölder regularity.

Theorem 3.3

(Almost $sp/(p-1)$-regularity) Let $1<p<2$ and $0<s<1$. Suppose $u\in W^{s,p}_{\textrm{loc}}(B_2)\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)$ is a local weak solution of

$$\begin{aligned} (-\Delta _p)^s u=0\qquad \text{ in } B_2 \end{aligned}$$

such that

$$\begin{aligned} \Vert u\Vert _{L^\infty (B_1)}\le 1\qquad \text{ and } \qquad \textrm{Tail}_{p-1,s\,p}(u;0,1)^{p-1}=\int _{{\mathbb {R}}^N\setminus B_1} \frac{|u(y)|^{p-1}}{|y|^{N+s\,p}}\, dy\le 1. \end{aligned}$$

Then for any $\varepsilon \in (0,\Gamma )$, there is $\sigma (\varepsilon ,N,s,p)>0$ such that $u\in C^{\Gamma -\varepsilon }(B_\sigma )$, where

$$\begin{aligned} \Gamma = \min (sp/(p-1),1). \end{aligned}$$

Moreover,

$$\begin{aligned} {[}u]_{C^{\Gamma -\varepsilon }(B_\sigma )}\le C(s,p,\varepsilon ,N). \end{aligned}$$

Proof

The idea is to apply Proposition 3.2 iteratively. Take $\varepsilon \in (0,\Gamma )$ and define

$$\begin{aligned} \gamma _0=0,\qquad \gamma _{i+1}=sp-\gamma _i(p-2)-\frac{\varepsilon (p-1)}{2}. \end{aligned}$$

Then $\gamma _i$ is an increasing sequence and $\gamma _i\rightarrow sp/(p-1)-\varepsilon /2$, as $i\rightarrow \infty$. Define also $v_i(x)=u(2^{-i} x)$ and

$$\begin{aligned} M_i= & {} \Vert u\Vert _{L^\infty (B_{2^{-i}})}+ \textrm{Tail}_{p-1,s\,p}(u;0,2^{-i})+2^{-i\gamma _i}[u]_{C^{\gamma _i}(B_{2^{-i}})}\\\le & {} C(i)(1+[u]_{C^{\gamma _i}(B_{2^{-i}})}). \end{aligned}$$

It is clear that there is $i_\infty =i_\infty (\varepsilon )\in {\mathbb {N}}$ such that $\gamma _{i_\infty }\ge \Gamma -\varepsilon$ and $\gamma _{i_\infty -1}<1$. Now we apply Proposition 3.2 to $v_i/M_i$ successively with $\gamma =\gamma _i$ and $\varepsilon$ replaced by $\frac{\varepsilon (p-1)}{2}$ and obtain

$$\begin{aligned} \begin{aligned} 2^{\gamma _1}[v_1]_{C^{\gamma _1}(B_1)}&=[v_0]_{C^{\gamma _1}(B_\frac{1}{2})}\le C(s,p,\varepsilon ,N)\\ 2^{\gamma _{i+1}}[v_{i+1}]_{C^{\gamma _{i+1}}(B_1)}&=[v_i]_{C^{\gamma _{i+1}}(B_\frac{1}{2})}\le M_i C(s,p,\varepsilon ,N)\\&\le C(s,p,\varepsilon ,N)(1+[v_i]_{C^{\gamma _i}(B_1)})\\&\cdots \\ 2^{\Gamma -\varepsilon }[v_{i_{\infty }}]_{C^{\Gamma -\varepsilon }(B_\frac{1}{2})}&\le [v_{i_{\infty }-1}]_{C^{\min (\gamma _{i_{\infty }},1 -\frac{\varepsilon (p-1)}{2})}(B_\frac{1}{2})}\\&\le C(s,p,\varepsilon ,N)(1 +[v_{i_{\infty -1}}]_{C^{\gamma _{i_{\infty -1}}}(B_{1})}). \end{aligned} \end{aligned}$$

Note that at every iteration step we get the estimate multiplied by a constant $C(s,p,\varepsilon ,N)$. Hence, by scaling back we obtain

$$\begin{aligned} {[}u]_{C^{\Gamma -\varepsilon }(B_{2^{-i_\infty -1}})}=2^{i_\infty (\Gamma -\varepsilon )}[v_{i_{\infty }}]_{C^{\Gamma -\varepsilon }(B_\frac{1}{2})}\le C(s,p,\varepsilon ,N). \end{aligned}$$

This is the desired result with $\sigma = 2^{-i_\infty -1}$. $\square$

The proof of the main Hölder regularity now easily follows. We spell out the details.

Proof of Theorem 1.1

By Theorem 1.1 in [11], $u\in L^{\infty }_{\textrm{loc}}(B_{2R}(x_0))$, so the assumption on the boundedness makes sense. Assume for simplicity that $x_0=0$ and let

$$\begin{aligned} u_R(x):=\frac{1}{{\mathcal {M}}_R}\,u(R\,x),\qquad \text{ for } x\in B_2, \end{aligned}$$

where

$$\begin{aligned} {\mathcal {M}}_R=\Vert u\Vert _{L^\infty (B_{R})}+\textrm{Tail}_{p-1,s\,p}(u;0,R)>0. \end{aligned}$$

Then $u_R$ is a local weak solution of $(-\Delta _p)^s u=0$ in $B_2$ and satisfies

$$\begin{aligned} \Vert u_R\Vert _{L^\infty (B_1)}\le 1,\qquad \int _{{\mathbb {R}}^N\setminus B_1}\frac{|u_R(y)|^{p-1}}{|y|^{N+s\,p}}\, dy\le 1. \end{aligned}$$

By Theorem 3.3, $u_R$ satisfies the estimate

$$\begin{aligned} {[}u_R]_{C^{\Gamma -\varepsilon }(B_{\sigma })}\le C. \end{aligned}$$

By scaling back, we obtain the desired estimate. $\square$

Remark 3.4

We note that as usual, once a local estimate of the spirit of Theorem 1.1 is obtained, one may obtain a similar estimate for any ball strictly contained in $\Omega$ by a standard covering argument. See for instance Remark 4.3 in [4] for a proof of such a fact.

4 The inhomogeneous equation

In this section, we treat the regularity for the inhomogeneous equation by approximation.

4.1 Basic regularity for the inhomogeneous equation

For our purpose, we need a uniform Hölder estimate for some exponent $\alpha \in (0,1)$. The argument used to prove this is inspired by [4, 18].

We begin with a Caccioppoli estimate for solutions to the inhomogeneous equation.

Lemma 4.1

Let $1<p<2$ and $0<s<1$. Suppose $\Omega \subset {\mathbb {R}}^N$ is an open and bounded set such that $B_r(x_0)\Subset B_R(x_0)\subset \Omega$. For $f\in L^q_{\textrm{loc}}(\Omega )$, with

$$\begin{aligned} q\ge (p^*_s)'\quad \text{ if } s\,p\not =N\qquad \text{ or } \qquad q>1\quad \text{ if } s\,p=N, \end{aligned}$$

we consider a local weak solution $u\in W^{s,p}_{\textrm{loc}}(\Omega )\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)$ of the equation

$$\begin{aligned} (-\Delta _p)^s u=f,\qquad \text{ in } \Omega . \end{aligned}$$

Then

$$\begin{aligned} \begin{aligned} {[}u]^p_{W^{s,p}(B_{r}(x_0))}&\le C{\Big (\frac{R}{R-r}\Big )^{N+sp+p}}R^{N-sp}\\&\quad \Big \{\Vert u\Vert ^p_{L^\infty (B_R (x_0))}+(\textrm{Tail}_{p-1,sp}(u;x_0,R))^{p-1}\Vert u\Vert _{L^\infty (B_R(x_0))}\Big \}\\&\quad + CR^{\frac{N}{q'}} \Vert u\Vert _{L^\infty (B_R(x_0))}\Vert f\Vert _{L^q(B_R(x_0))} \end{aligned} \end{aligned}$$

for some positive constant $C=C(N,s,p)$.

Proof

We only perform the proof for $u_+$. Proceeding exactly as in the proof of Corollary 3.6 in [5], we take a smooth function $\phi$ such that $\phi =1$ in $B_r(x_0)$, $0\le \phi \le 1$ in $B_\frac{R+r}{2}(x_0)$, $\phi =0$ outside $B_{(R+r)/2}(x_0)$ and $|\nabla \phi |\le C/(R-r)$. Testing the equation with $\phi ^p u_+$ as in the proof of Corollary 3.6 in [5] we obtain

$$\begin{aligned} \begin{aligned}&\int _{B_R(x_0)}\int _{B_R(x_0)}|u_+(x)\phi (x)-u_+(y)\phi (y)|^p\,d\mu \le C\Big (\frac{R}{R-r}\Big )^{N+sp+p}R^{-sp}\\&\qquad \left( \Vert u_+\Vert _{L^p (B_R(x_0))}^p+(\text {Tail}_{p-1,sp}(u_+;x_0,R))^{p-1} \Vert u_+\Vert _{L^1(B_R(x_0))}\right) \\&\qquad +\int _{B_R(x_0)} fu_+ \, dx \end{aligned} \end{aligned}$$

(4.1)

for some positive constant $C=C(N,s,p)$. We note that since $\phi =1$ in $B_r(x_0)$, the left hand side of (4.1) can be bounded from below by $[u_+]_{W^{s,p}(B_r(x_0))}^p$. In addition, the first two terms can be estimated as

$$\begin{aligned} \begin{aligned}&C\Big (\frac{R}{R-r}\Big )^{N+sp+p}R^{-sp}\left( \Vert u_+\Vert _{L^p (B_R(x_0))}^p+(\textrm{Tail}_{p-1,sp}(u_+;x_0,R))^{p-1} \Vert u_+\Vert _{L^1(B_R(x_0))}\right) \\&\quad \le {C{\Big (\frac{R}{R-r}\Big )^{N+sp+p}}R^{N-sp} \Big \{\Vert u\Vert ^p_{L^\infty (B_R(x_0))}+(\textrm{Tail}_{p-1, sp}(u;x_0,R))^{p-1}\Vert u\Vert _{L^\infty (B_R(x_0))}\Big \}}, \end{aligned} \end{aligned}$$

which matches the first two terms in the statement of the lemma. It remains to estimate the term involving f. By Hölder’s inequality, we have

$$\begin{aligned} \begin{aligned} \int _{B_R(x_0)} |f u_+| \,dx&\le \Vert u_+\Vert _{L^{q'}(B_R(x_0))} \Vert f\Vert _{L^q(B_R(x_0))}\\&\le CR^\frac{N}{q'} \Vert u_+\Vert _{L^{\infty }(B_R(x_0))} \Vert f\Vert _{L^q(B_R(x_0))}\\&\le CR^{\frac{N}{q'}}\Vert u\Vert _{L^\infty (B_R(x_0))} \Vert f\Vert _{L^q(B_R(x_0))}. \end{aligned} \end{aligned}$$

This completes the proof. $\square$

The following results provides stability for the inhomogeneous equation.

Lemma 4.2

Let $1<p<2$, $0<s<1$ and $\Omega \subset {\mathbb {R}}^N$ be an open and bounded set. Suppose that $u\in W^{s,p}_{\textrm{loc}}(\Omega )\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)$ is a local weak solution of the equation

$$\begin{aligned} (-\Delta _p)^s u=f,\qquad \text{ in } \Omega , \end{aligned}$$

where $f\in L^q_{\textrm{loc}}(\Omega )$, with

$$\begin{aligned} q\ge (p^*_s)'\quad \text{ if } s\,p\not =N\qquad \text{ or } \qquad q>1\quad \text{ if } s\,p=N. \end{aligned}$$

Let $B=B_{\sigma r}\Subset B'=B_r\Subset \Omega$ be a pair of concentric balls and take $v\in X^{s,p}_u(B,B')$ to be the unique weak solution of

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta _p)^s\,v=0,&{}\quad \text{ in } B,\\ v=u,&{}\quad \text{ in } {\mathbb {R}}^N\setminus B. \end{array} \right. \end{aligned}$$

For any $\varepsilon \in (0,1/2)$ we have

$$\begin{aligned} {[}u-v]^p_{W^{s,p}({\mathbb {R}}^N)}\le C\varepsilon ^\frac{p-2}{p-1} \,|B|^{\frac{p'}{q'}-\frac{p}{p-1}\,\frac{N-s\,p}{N\,p}} \,\left( \int _B |f|^{q}\,dx\right) ^\frac{p'}{q} +\varepsilon [u]^p_{W^{s,p}(B')},\nonumber \\ \end{aligned}$$

(4.2)

and

(4.3)

whenever $s\,p\not =N$ and for a constant $C=C(N,p,s,\sigma )>0$.

If instead $s\,p=N$, a similar estimate holds with $N\,p/(N-s\,p)$ replaced by an arbitrary exponent $m<\infty$ and the constant C depending on m as well.

Proof

We only perform the proof in the case $sp<N$. We first observe that the existence of v is guaranteed by Theorem 2.12 in [4], since $u\in W^{s,p}(B')\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)$. By using the weak formulations of the equations solved by u and v with the test function $w=u-v$, we get

$$\begin{aligned} \iint _{{\mathbb {R}}^N\times {\mathbb {R}}^N} \frac{\Big (J_p(u(x)-u(y))-J_p(v(x)-v(y))\Big )\,\big (w(x) -w(y)\big )}{|x-y|^{N+s\,p}}\,dx\,dy={\int _{B}} f w\,dx. \end{aligned}$$

Let $a=u(x), b=u(y), c=v(x)$ and $d=v(y)$. By Lemma B.4 in [5],

$$\begin{aligned} \begin{aligned}&\Big (J_p(a-b)-J_p(c-d)\Big )\Big ((a-c)-(b-d)\Big )\\&\qquad \ge (p-1)|(a-c)-(b-d)|^2 \left( |a-b|^2+|c-d|^2\right) ^\frac{p-2}{2}. \end{aligned} \end{aligned}$$

(4.4)

By (4.4) and Hölder’s inequality together with some trivial manipulations, we obtain

$$\begin{aligned} \begin{aligned}&\iint _{B'\times B'} |(a-c)-(b-d)|^p d\mu \\&\quad \le {C(p)}\left( \iint _{B'\times B'} \Big (J_p(a-b) -J_p(c-d)\Big )\Big ((a-c)-(b-d)\Big )\,{d\mu }\right) ^\frac{p}{2}\\&\qquad \times \left( \iint _{B'\times B'}(|a-b|^{2} +|c-d|^{2})^\frac{p}{2} d\mu \right) ^\frac{2-p}{2},\\&\quad \le {C(p)}\left( \iint _{B'\times B'} \Big (J_p(a-b) -J_p(c-d)\Big )\Big ((a-c)-(b-d)\Big ){\,d\mu }\right) ^\frac{p}{2}\\&\qquad \times \left( \iint _{B'\times B'}(|a-b|^{p}+|(a-c) -(b-d)|^{p}) d\mu \right) ^\frac{2-p}{2}. \end{aligned} \end{aligned}$$

(4.5)

Recalling the above choices of a, b, c, d and using Hölder’s inequality together with the localized Sobolev inequality (cf. Proposition 2.3 in [5]), from (4.5) we have

$$\begin{aligned} \begin{aligned} {[}w]^p_{W^{s,p}(B')}&\le C\,\left( \int _{B} |fw|\,dx\right) ^\frac{p}{2} \left( [w]^p_{W^{s,p}(B')}+[u]^p_{W^{s,p}(B')}\right) ^\frac{2-p}{2}\\&\le C\,\left\{ \Vert f\Vert _{L^q(B)}\,\Vert w\Vert _{L^{q'}(B)}\right\} ^{\frac{p}{2}} \left( [w]^p_{W^{s,p}(B')}+[u]^p_{W^{s,p}(B')}\right) ^\frac{2-p}{2}\\&\le C\left\{ |B|^{\frac{1}{q'}-\frac{1}{p^*_s}}\,\Vert f\Vert _{L^q(B)}\, \Vert w\Vert _{L^{p^*_s}(B)}\right\} ^\frac{p}{2}\left( [w]^p_{W^{s,p}(B')} +[u]^p_{W^{s,p}(B')}\right) ^\frac{2-p}{2}\\&\le C\,\left\{ \frac{|B'|}{|B|}\frac{|B|^\frac{1}{N}}{|B'|^\frac{1}{N} -|B|^\frac{1}{N}}\left( \frac{|B'|^\frac{1}{N}}{|B'|^\frac{1}{N} -|B|^\frac{1}{N}}\right) ^{{sp}}+1\right\} ^{\frac{1}{2}}\\&\quad \times \left\{ |B|^{\frac{1}{q'}-\frac{1}{p^*_s}}\,\Vert f\Vert _{L^q(B)} \,[w]_{W^{s,p}(B')}\right\} ^\frac{p}{2}\left( [w]^p_{W^{s,p}(B')} +[u]^p_{W^{s,p}(B')}\right) ^\frac{2-p}{2}\\&\le C\left\{ |B|^{\frac{1}{q'}-\frac{1}{p^*_s}}\,\Vert f\Vert _{L^q(B)} \,[w]_{W^{s,p}(B')}\right\} ^\frac{p}{2}\left( [w]^p_{W^{s,p}(B')} +[u]^p_{W^{s,p}(B')}\right) ^\frac{2-p}{2}, \end{aligned} \end{aligned}$$

where $C=C(N,p,s,\sigma )$. By Young’s inequality with exponent 2 this implies

$$\begin{aligned} {[}w]^p_{W^{s,p}(B')}\le C \left\{ |B|^{\frac{1}{q'}-\frac{1}{p^*_s}}\,\Vert f\Vert _{L^q(B)} \right\} ^p\left( [w]^p_{W^{s,p}(B')}+[u]^p_{W^{s,p}(B')}\right) ^{2-p}, \end{aligned}$$

with $C=C(N,p,s,\sigma )$. Using Young’s inequality with exponents $1/(p-1)$ and $1/(2-p)$ we obtain

$$\begin{aligned} {[}w]^p_{W^{s,p}(B')}\le C\varepsilon ^\frac{p-2}{p-1} \left\{ |B|^{\frac{1}{q'}-\frac{1}{p^*_s}}\,\Vert f\Vert _{L^q(B)} \right\} ^{p'}+\varepsilon [u]^p_{W^{s,p}(B')}, \end{aligned}$$

where $C=C(N,p,s,\sigma )$.

Using the above estimate and arguing as in the proof of Proposition 2.3 in [5], we have

$$\begin{aligned} \begin{aligned} {[}w]^{p}_{W^{s,p}({\mathbb {R}}^N)}&=[w]^{p}_{W^{s,p}(B')} +2\int _{B'}\int _{{\mathbb {R}}^N\setminus B'} |w(x)|^p \,d\mu \\&\le C\left( 1+\frac{|B'|}{|B|}\frac{|B|^\frac{1}{N}}{|B'|^\frac{1}{N} -|B|^\frac{1}{N}}\left( \frac{|B'|^\frac{1}{N}}{|B'|^\frac{1}{N} -|B|^\frac{1}{N}}\right) ^{{sp}}\right) [w]^p_{W^{s,p}(B')}\\&\le C{\varepsilon ^\frac{p-2}{p-1}}\left\{ |B|^{\frac{1}{q'} -\frac{1}{p^*_s}}\,\Vert f\Vert _{L^q(B)}\right\} ^{p'}+\varepsilon [u]^p_{W^{s,p}(B')}, \end{aligned} \end{aligned}$$

for some constant $C=C(N,p,s,\sigma )>0$, which in turn gives (4.2).

Estimate (4.3) now follows by applying Poincaré’s inequality in (4.2), see Lemma 2.3. $\square$

In the lemma below we obtain a Campanato estimate. Here we use the notation

to denote the average of u in $B_r(x_0)$.

Lemma 4.3

(Decay transfer) Let $1<p<2$, $0<s<1$ and $\Omega \subset {\mathbb {R}}^N$ be an open and bounded set. Suppose that $u\in W^{s,p}_{\textrm{loc}}(\Omega )\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)$ is a local weak solution of the equation

$$\begin{aligned} (-\Delta _p)^s u=f,\qquad \text{ in } \Omega , \end{aligned}$$

where $f\in L^{q}_{\textrm{loc}}(\Omega )$ with

$$\begin{aligned} q\ge (p^*_s)'\quad \text{ if } s\,p\not =N\qquad \text{ or } \qquad q>1\quad \text{ if } s\,p=N. \end{aligned}$$

If $B_{4R}(x_0)\Subset \Omega$ such that $0<R\le 1$, then there is $\alpha \in (0,1)$ such that for any $\varepsilon \in (0,\frac{1}{2})$ we have

for every $0<r\le R$. Here

$$\begin{aligned} \gamma :=\left\{ \begin{array}{ll} s\,p\,p'+N\,\left( \dfrac{p'}{q'}-\dfrac{1}{p-1}-1\right) , &{}\quad \text{ if } s\,p\not =N,\\ N\,p'\,\left( \dfrac{1}{q'}-\dfrac{1}{m}\right) , &{}\quad \text{ for } \text{ an } \text{ arbitrary } q'< m<\infty , \text{ if } s\,p=N, \end{array} \right. \nonumber \\ \end{aligned}$$

(4.6)

and $C=C(N,s,p,q,m)>0$.

Proof

The proof is the same as the proof of Lemma 3.5 in [4], except for the last term that appears when applying (4.3) in the present case $p< 2$. We present some details in the case $s\,p<N$. In order to estimate this extra term, we use that the boundedness of u together with Lemma 4.1 applied to the balls $B_{7R/2}(x_0)$ and $B_{4R}(x_0)$ gives

$$\begin{aligned} \begin{aligned}&R^{sp-N}[u]_{W^{s,p}(B_{7R/2}(x_0))}^p \\&\quad \le C\Big \{\Vert u\Vert ^{p}_{L^\infty (B_{4R}(x_0))}+\textrm{Tail}_{p-1,sp}(u;x_0,4R)^{p-1} \Vert u\Vert _{L^\infty (B_{4R}(x_0))}\Big \}\\&\qquad +CR^{\frac{N}{q'}+sp-N}\Vert f\Vert _{L^q(B_{4R}(x_0))} \Vert u\Vert _{L^\infty (B_{4R}(x_0))}\\&\quad \le C\Big \{\Vert u\Vert _{L^\infty (B_{4R}(x_0))}^p+\textrm{Tail}_{p-1,sp} (u;x_0,4R)^p\Big \}+C\Vert f\Vert _{L^q(B_{4R}(x_0))}^{p'}, \end{aligned} \end{aligned}$$

for some constant $C=C(N,s,p)$, where we also used Young’s inequality and that $\frac{N}{q'}+sp-N>0$, $\gamma >0$ and $0<r\le R\le 1$. $\square$

We are now ready to prove Hölder regularity.

Theorem 4.4

Let $1<p<2$ and $0<s<1$. Suppose that $u\in W^{s,p}_{\textrm{loc}}(\Omega )\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)$ is a local weak solution of the equation

$$\begin{aligned} (-\Delta _p)^s u=f,\qquad \text{ in } \Omega , \end{aligned}$$

for $f\in L^q_{\textrm{loc}}(\Omega )$ with

$$\begin{aligned} \left\{ \begin{array}{ll} q>{\frac{N}{s\,p}},&{} \quad \text{ if } s\,p\le N,\\ q\ge 1,&{} \quad \text{ if } s\,p>N. \end{array} \right. \end{aligned}$$

Then $u\in C^{\beta }_{\textrm{loc}}(\Omega )$, where

$$\begin{aligned} \beta =\frac{\alpha \gamma (p-1)}{\gamma (p-1)+(\alpha p+N)}, \end{aligned}$$

with $\gamma$ as in (4.6) and $\alpha$ as in Lemma 4.3.

More precisely, for every ball $B_{R_0}(z)\Subset \Omega$ we have the estimate

$$\begin{aligned} \begin{aligned} {[}u]_{C^{\beta }(B_{R_0}(z))}^p&\le C\,\left[ 1+\Vert u\Vert _{L^\infty (B_{R_1}(z))}^p+(\textrm{Tail}_{p-1,sp}(u;z,R_1))^{p}+ \Vert f\Vert ^{p'}_{L^q(B_{R_1}(z))}\right] , \end{aligned} \end{aligned}$$

where

$$\begin{aligned} R_1=R_0+\frac{\textrm{dist}(B_{R_0}(z),\partial \Omega )}{2}. \end{aligned}$$

Here, the constant C depends only $N,p,s,q,R_0$ and $\textrm{dist}(B_{R_0}(z),\partial \Omega )$.

Proof

The proof is almost identical with the proof of Theorem 3.6 of [4]. The only difference is that there is a parameter $\varepsilon$ and an additional term when applying Lemma 4.3. We take a ball $B_{R_0}(z)\Subset \Omega$ and set

$$\begin{aligned} \textrm{d}=\textrm{dist}(B_{R_0}(z),\partial \Omega )>0\qquad \text{ and } \qquad R_1=\frac{\textrm{d}}{2}+R_0. \end{aligned}$$

Choose a point $x_0\in B_{R_0}(z)$ and consider the ball $B_{4R}(x_0)$ with $R<\min \{1,\textrm{d}/8\}$. If¹$s\,p\not =N$, applying Lemma 4.3 and obtain

for every $0<r\le R<\min \{1,\textrm{d}/8\}$.

As in the proof of Theorem 3.6 in [4] it is straightforward to estimate these terms and obtain

where $C=C(N,s,p,q)>0$.

To simplify the notation, let

$$\begin{aligned} A=\textrm{Tail}_{p-1,sp}(u,z,R_1)^p+\Vert u\Vert _{L^\infty (B_{R_1}(z))}^p +\Vert f\Vert _{L^q(B_{R_1}(z))}^{p'}+1. \end{aligned}$$

Then the above estimate reads

We will now see that for a specific choice of R and $\delta$ in terms of r, this implies that

decays in a power fashion. Indeed, let

$$\begin{aligned} \varepsilon = \left( \frac{r}{R}\right) ^{\alpha p+N},\quad R=r^{\sigma }, \end{aligned}$$

where

$$\begin{aligned} \sigma =\frac{\alpha p+N}{\alpha p+N+\gamma (p-1)}\in (0,1). \end{aligned}$$

Then

for $x_0\in B_{R_0}(z)$ and $r<\min \{1,(\textrm{d}/8)^\frac{1}{\sigma }\}$ where $\beta =\frac{\alpha \gamma (p-1)}{\gamma (p-1)+(\alpha p+N)}$. This shows that u belongs to the Campanato space²${\mathcal {L}}^{p,N+\beta \,p}(B_{R_0}(z))$, which is isomorphic to $C^{\beta }(\overline{B_{R_0}(z)})$. The proof is complete. $\square$

4.2 Final Hölder regularity

In order to prove Theorem 1.2, we first establish the following stability result.

Lemma 4.5

(Stability in $L^\infty$) Let $1<p<2$, $0<s<1$. Suppose $\Omega \subset {\mathbb {R}}^N$ is an open and bounded set and $f\in L^q_{\textrm{loc}}(\Omega )$ with

$$\begin{aligned} \left\{ \begin{array}{ll} q>\big ({p^*_s}\big )',&{}\quad \text{ if } s\,p< N,\\ q>1,&{} \quad \text{ if } s\,p= N,\\ q\ge 1,&{} \quad \text{ if } s\,p>N. \end{array} \right. \end{aligned}$$

Consider a local weak solution $u\in W^{s,p}_{\textrm{loc}}(\Omega )\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)$ of the equation

$$\begin{aligned} (-\Delta _p)^s u=f,\qquad \text{ in } \Omega . \end{aligned}$$

Let $B_{4}\Subset \Omega$ and assume that

$$\begin{aligned} \Vert u\Vert _{L^\infty (B_2)}+\int _{{\mathbb {R}}^N\setminus B_2} \frac{|u(x)|^{p-1}}{|x|^{N+s\,p}}\, dx\le M\qquad \text{ and } \qquad \Vert f\Vert _{L^q(B_2)}\le \eta . \end{aligned}$$

Suppose that $h\in X^{s,p}_u({B_\frac{3}{2},B_{4}})$ weakly solves

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta _p)^s h = 0,&{} \quad \text{ in } {B_\frac{3}{2}},\\ h=u, &{}\quad \text{ in } {\mathbb {R}}^N\setminus {B_\frac{3}{2}}. \end{array} \right. \end{aligned}$$

Then there is $\tau _{M}(\eta )$ such that

$$\begin{aligned} \Vert u-h\Vert _{L^\infty (B_{\frac{5}{4}})}\le \tau _{M}(\eta ) \end{aligned}$$

(4.7)

and $\tau _{M}(\eta )$ converges to 0 as $\eta$ goes to 0.

Proof

The existence of a bound of the form (4.7) is a consequence of the triangle inequality and the local $L^\infty$ estimate for the equation (Theorem 3.8 in [5]). We will now prove that $\tau _{M}(\eta )\rightarrow 0$ as $\eta \rightarrow 0$.

We assume towards a contradiction that there exist two sequences $\{f_n\}_{n\in {\mathbb {N}}}\subset L^{q}(B_2)$ and $\{u_n\}_{n\in {\mathbb {N}}}$ such that

$$\begin{aligned} \Vert u_n\Vert _{L^\infty (B_2)}+\int _{{\mathbb {R}}^N\setminus B_2} \frac{|u_n|^{p-1}}{|x|^{N+s\,p}}\, dx\le M,\qquad \Vert f_n\Vert _{L^q(B_2)}\rightarrow 0, \end{aligned}$$

but

$$\begin{aligned} \liminf _{n\rightarrow \infty } \Vert u_n-h_n\Vert _{L^\infty (B_{\frac{5}{4}})}>0. \end{aligned}$$

We note that by Lemma 4.1, any u satisfying the assumptions of the lemma also satisfies the bound

$$\begin{aligned} {[}u]_{W^{s,p}(B_\frac{5}{3})}\le C(M,N,s,p). \end{aligned}$$

Therefore, (4.2) implies that for every $\varepsilon \in (0,1/2)$, we have

$$\begin{aligned} \limsup _{n\rightarrow \infty }[u_n-h_n]^p_{W^{s,p}({{\mathbb {R}}^N})}\le & {} C\varepsilon ^{\frac{p-2}{p-1}}\limsup _{n\rightarrow \infty }\,\left( \int _{B_{\frac{3}{2}}} |f_n|^{q}\,dx\right) ^\frac{p'}{q}\\{} & {} +\varepsilon \lim _{n\rightarrow \infty }[u_n]^p_{W^{s,p}(B_\frac{5}{3})}\le C\varepsilon , \end{aligned}$$

where $C=C(M,N,p,s)>0$ is a constant. Since this holds for any $\varepsilon \in (0,1/2)$, we conclude that

$$\begin{aligned} \lim _{n\rightarrow \infty }[u_n-h_n]^p_{W^{s,p}({{\mathbb {R}}^N})}=0. \end{aligned}$$

(4.8)

This, together with the fractional Sobolev inequality and Theorem 1.1 in [11] implies that $h_n$ is locally uniformly bounded in $B_{3/2}$. Theorem 3.1 in [4] or Theorem 1.1 implies that $h_n$ is uniformly bounded in $C^{\beta }(B_{5/4})$ and Theorem 4.4 implies that $u_n$ is uniformly bounded in $C^{\beta }(B_{5/4})$ for some $\beta >0$. Therefore, by the Ascoli–Arzelà theorem, we may conclude that $u_n-h_n$ converges uniformly in $\overline{B_{5/4}}$, up to a subsequence. By (4.8) we get that

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert u_n-h_n\Vert _{L^\infty (B_{5/4})}=0, \end{aligned}$$

which gives the desired contradiction. $\square$

The following proposition is a rescaled version of Theorem 1.2.

Proposition 4.6

Let $1<p<2$, $0<s<1$. Take q such that

$$\begin{aligned} \left\{ \begin{array}{ll} q>\big ({p^*_s}\big )',&{}\quad \text{ if } s\,p< N,\\ q>1,&{} \quad \text{ if } s\,p= N,\\ q\ge 1,&{} \quad \text{ if } s\,p>N, \end{array} \right. \end{aligned}$$

and define

$$\begin{aligned} \Theta = \min \Big (1,\frac{sp-N/q}{p-1}\Big ). \end{aligned}$$

For every $0<\varepsilon <\Theta$ there exists $\eta =$ $\eta (N,p,q,s,\varepsilon )>0$ such that if $f\in L^q_{\textrm{loc}}(B_4(x_0))$ and

$$\begin{aligned} \Vert f\Vert _{L^q({B_2(x_0))}}\le \eta , \end{aligned}$$

then every local weak solution $u\in W^{s,p}_{\textrm{loc}}(B_4(x_0))\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)$ of the equation

$$\begin{aligned} (-\Delta _p)^s u=f,\qquad \text{ in } B_4(x_0), \end{aligned}$$

such that

$$\begin{aligned} \Vert u\Vert _{L^\infty ({B_2(x_0))}}\le 1,\qquad \int _{{\mathbb {R}}^N\setminus {B_2(x_0)}}\frac{|u|^{p-1}}{|x|^{N+s\,p}}\, dx\le 1 \end{aligned}$$

(4.9)

belongs to $C^{\Theta -\varepsilon }(\overline{B_{1/8}(x_0)})$ with the estimate

$$\begin{aligned} {[}u]_{C^{\Theta -\varepsilon }(\overline{B_{1/8}}(x_0))}\le C(N,p,q,s,\varepsilon ), \end{aligned}$$

for some constant $C(N,p,q,s,\varepsilon )>0$.

Proof

Without loss of generality, we may assume that $x_0=0$. We divide the proof in two parts.

Part 1: Regularity at the origin. We claim that for any $0<\varepsilon <\Theta$ and every $0<r<1/2$, there exists $\eta =\eta (N,p,q,s,\varepsilon )>0$ and a constant $C=C(N,p,q,s,\varepsilon )>0$ such that if f and u are as above, then we have

$$\begin{aligned} \sup _{x\in B_r} |u(x)-u(0)|\le C\,r^{\Theta -\varepsilon }. \end{aligned}$$

Without loss of generality, we assume $u(0)=0$. Let us fix $0<\varepsilon <\Theta$. Then we remark that it is enough to prove that there exists $\lambda <1/2$ and $\eta >0$ (depending on N, p, q, s and $\varepsilon$) such that if f and u are as above, then

$$\begin{aligned} \sup _{B_{{2\lambda ^k}}}|u|\le \lambda ^{k\,(\Theta -\varepsilon )},\qquad \int _{{\mathbb {R}}^N\setminus {B_2}}\left| \frac{u(\lambda ^k\, x)}{\lambda ^{k\,(\Theta -\varepsilon )}}\right| ^{p-1}\,|x|^{-N-s\,p}\, dx\le 1, \end{aligned}$$

(4.10)

for every $k\in {\mathbb {N}}$. Indeed, if this is true, then for every $0<r<1/2$, there exists $k\in {\mathbb {N}}$ such that $2\lambda ^{k+1}< r\le 2\lambda ^k$. Using the first property from (4.10), we deduce that

$$\begin{aligned} \sup _{B_r} |u|\le \sup _{B_{2\lambda ^k}} |u|\le \lambda ^{k\,(\Theta -\varepsilon )}=\frac{1}{\lambda ^{\Theta -\varepsilon }}\,\lambda ^{(k+1)\,(\Theta -\varepsilon )}\le C\,r^{\Theta -\varepsilon }, \end{aligned}$$

where $C=\frac{1}{(2\lambda )^{\Theta -\epsilon }}$ as desired.

We prove (4.10) by an induction argument. First, we note that (4.10) holds true for $k=0$, using the assumptions in (4.9). Suppose (4.10) is valid up to k. We prove that this is also valid for $k+1$ assuming that

$$\begin{aligned} \Vert f\Vert _{L^q({B_2})}\le \eta , \end{aligned}$$

for small enough $\eta$, which is independent of k. We define

$$\begin{aligned} w_k(x)=\frac{u(\lambda ^k x)}{\lambda ^{k\,(\Theta -\varepsilon )}}. \end{aligned}$$

We observe that by the hypotheses, it follows that

$$\begin{aligned} \Vert w_k\Vert _{L^\infty ({B_2})}\le 1 \qquad \text{ and } \qquad \int _{{\mathbb {R}}^N\setminus {B_2}}\frac{|w_k|^{p-1}}{|x|^{N+s\,p}}\,dx\le 1. \end{aligned}$$

(4.11)

Furthermore,

$$\begin{aligned} (-\Delta _p)^s w_k (x) = \lambda ^{k\,[sp\,-(\Theta -\varepsilon )(p-1)]}\,f(\lambda ^k\, x)=:f_k(x). \end{aligned}$$

We notice that

$$\begin{aligned} \Vert f_k\Vert _{L^{q}({B_2})}=\lambda ^{k(sp\,-(\Theta -\varepsilon ) (p-1))}\lambda ^{-\frac{N}{q}\,k}\,\left( \int _{{B_{2\lambda ^k}}} |f|^{q}\,dx\right) ^{\frac{1}{q}}\le \Vert f\Vert _{L^{q}({B_2})}\le \eta , \end{aligned}$$

where we have used the hypotheses on f, the definition of $\Theta$, and again the fact that $\lambda <1/2$. By Proposition 2.12 in [4], we consider $h_k\in X_{w_k}^{s,p}(B_\frac{3}{2},B_4)$ to be the weak solution of

$$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta _p)^s h = 0,&{} \quad \text{ in } B_\frac{3}{2},\\ h=w_k,&{} \quad \text{ in } {\mathbb {R}}^N\setminus B_\frac{3}{2}. \end{array} \right. \end{aligned}$$

From Lemma 4.5, we obtain

$$\begin{aligned} \Vert w_k-h_k\Vert _{L^\infty (B_{\frac{5}{4}})}<\tau _\eta , \end{aligned}$$

where $\tau _\eta \rightarrow 0$ as $\eta \rightarrow 0$ and $\tau _\eta$ is independent of k. Therefore,

$$\begin{aligned} \begin{aligned} |w_k(x)|&\le |w_k(x)-h_k(x)|+|h_k(x)-h_k(0)|+|h_k(0)-w_k(0)|\\&\le 2\tau _\eta +[h_k]_{C^{\Theta -\varepsilon /2}(B_{1})}\,|x|^{\Theta -\frac{\varepsilon }{2}},\qquad \qquad \text{ for } x\in B_{1}. \end{aligned} \end{aligned}$$

(4.12)

To obtain the above estimate, we have also used the fact that $h_k$ belongs to $C^{\Theta -\varepsilon /2}(\overline{B_{1}})$, which follows from Theorem 1.1 and Remark 3.4, with the estimate

$$\begin{aligned}{} & {} {[}h_k]_{C^{{\Theta -\varepsilon /2}}(B_{1})}\le C\, \left( \Vert h_k\Vert _{L^\infty ({B_{\frac{5}{4}}})}+\textrm{Tail}_{p-1,sp} \big (h_k;0,{\frac{5}{4}}\big )\right) \le C_1,\\{} & {} C_1=C_1(N,p,q,s,\varepsilon ). \end{aligned}$$

We obtained the above estimate by observing that the quantities in the right-hand side are uniformly bounded, independently of k. To this end, Lemma 4.5 and (4.11) along with the triangle inequality gives that

$$\begin{aligned} \Vert h_k\Vert _{L^\infty (B_{\frac{5}{4}})}\le \Vert h_k-w_k\Vert _{L^\infty (B_{\frac{5}{4}})}+\Vert w_k\Vert _{L^\infty (B_{\frac{5}{4}})}\le \tau _\eta +1. \end{aligned}$$

For the tail term, by the triangle inequality, the hypothesis on $w_k$ and (4.3) combined with Lemma 4.1, we obtain

$$\begin{aligned} \textrm{Tail}_{p-1,sp}\big (h_k;0,{\frac{5}{4}}\big )\le & {} C \Big (\int _{{\mathbb {R}}^N\setminus B_{\frac{5}{4}}} \frac{|h_k-w_k|^{p-1}}{|x|^{N+ps}}\,dx\Big )^\frac{1}{p-1}\\{} & {} +C\Big (\int _{{\mathbb {R}}^N\setminus B_{\frac{5}{4}} } \frac{|w_k|^{p-1}}{|x|^{N+ps}}\,dx\Big )^\frac{1}{p-1}\\\le & {} C\Big (\int _{B_\frac{3}{2}\setminus B_{\frac{5}{4}}} \frac{|h_k-w_k|^{p-1}}{|x|^{N+ps}}\,dx\Big )^\frac{1}{p-1}\\{} & {} +C\Big (\int _{{\mathbb {R}}^N\setminus B_{2}}\frac{|w_k|^{p-1}}{|x|^{N+ps}}\,dx\Big )^\frac{1}{p-1}\\{} & {} +C\Big (\int _{B_2\setminus B_{\frac{5}{4}}} \frac{|w_k|^{p-1}}{|x|^{N+ps}}\,dx\Big )^\frac{1}{p-1}\\\le & {} C(1+\eta ), \end{aligned}$$

with $C=C(N,s,p,q)$. We also made use of (4.11) and that $h_k=w_k$ outside $B_{3/2}$, by construction. Therefore, the estimate (4.12) is uniform in k. Let

$$\begin{aligned} w_{k+1}(x)=\frac{u(\lambda ^{k+1}\, x)}{\lambda ^{(k+1)\,(\Theta -\varepsilon )}}=\frac{w_k(\lambda \, x)}{\lambda ^{\Theta -\varepsilon }}. \end{aligned}$$

We can transfer estimate (4.12) to $w_{k+1}$ by choosing $\eta$ so that $2\tau _\eta <\lambda ^\Theta$ and $\lambda$ small enough. Indeed, we observe that

$$\begin{aligned} \begin{aligned} |w_{k+1}(x)|\le 2\tau _\eta \,\lambda ^{\varepsilon -\Theta } +C_1\,\lambda ^{\varepsilon /2}|x|^{\Theta -\varepsilon /2}\le (1 +C_1\,|x|^{\Theta -\varepsilon /2})\,\lambda ^{\varepsilon /2}, \qquad x\in B_\frac{1}{\lambda }. \end{aligned} \end{aligned}$$

In particular, the above estimate gives that $\Vert w_{k+1}\Vert _{L^\infty ({B_2})}\le 1$ for $\lambda$ satisfying

$$\begin{aligned} \lambda <\min \left\{ {\frac{1}{4}},(1+C_12^{\Theta -\varepsilon /2})^{-\frac{2}{\varepsilon }}\right\} . \end{aligned}$$

(4.13)

This information, rescaled back to u, gives precisely the first part of (4.10) for $k+1$. To obtain the second part of (4.10), we use the upper bound for $|w_{k+1}|$ and the fact that $\Theta <\frac{sp}{p-1}$, which gives

$$\begin{aligned} \begin{aligned} \int _{B_{\frac{1}{\lambda }}\setminus B_{2}} \frac{|w_{k+1}|^{p-1}}{|x|^{N+s\,p\,}}\,dx&\le \lambda ^{\varepsilon \, (p-1)/2} \int _{B_{\frac{1}{\lambda }}\setminus B_{2}} \frac{(1 +C_1\,|x|^{\Theta -\varepsilon /2})^{p-1}}{|x|^{N+s\,p}}\,dx\\&\le (1+C_1)^{p-1}\,\lambda ^{\varepsilon \, (p-1)/2} \int _{B_{\frac{1}{\lambda }}\setminus B_{2}} \frac{1}{|x|^{N+ sp+(\varepsilon /2-\Theta )\,(p-1)}}\,dx\\&\le \frac{C_2}{s\,p-(\Theta -\varepsilon /2)\,(p-1)} \,\lambda ^{\varepsilon \,(p-1)/2}. \end{aligned}\nonumber \\ \end{aligned}$$

(4.14)

Since $|w_k|\le 1$ in $B_2$, a change of variable gives

$$\begin{aligned} \int _{B_{\frac{2}{\lambda }}\setminus B_{\frac{1}{\lambda }}} \frac{|w_{k+1}|^{p-1}}{|x|^{N+s\,p\,}}\,dx=\lambda ^{(\varepsilon -\Theta )\,(p-1)+s\,p}\,\int _{B_2\setminus B_1} \frac{|w_k(x)|^{p-1}}{|x|^{N+s\,p}}\,dx\le {C_3\,\lambda ^{\varepsilon \,(p-1)/2}}.\nonumber \\ \end{aligned}$$

(4.15)

In addition, by the integral bound on $w_k$ in (4.11) and using that $\text {Tail}_{p-1,sp}(w_k;0,2)\le 1$, we get

$$\begin{aligned} \int _{{\mathbb {R}}^N\setminus B_{\frac{2}{\lambda }}} \frac{|w_{k+1}(x)|^{p-1}}{|x|^{N+s\,p}}\,dx= \lambda ^{(\varepsilon -\Theta )\,(p-1)+s\,p}\,\int _{{\mathbb {R}}^N\setminus B_2}\frac{|w_k(x)|^{p-1}}{|x|^{N+s\,p}}\,dx\le \lambda ^{\varepsilon \, (p-1)/2}.\nonumber \\ \end{aligned}$$

(4.16)

The condition $\lambda <1/2$ and the fact that

$$\begin{aligned} (\varepsilon -\Theta )\,(p-1)+s\,p\ge \varepsilon \,\frac{p-1}{2} \end{aligned}$$

is used in both estimates. Here the constants $C_2$ and $C_3$ depend on N, p, q, s and $\varepsilon$ only. By (4.14), (4.15) and (4.16), we get that the second part of (4.10) holds, provided that

$$\begin{aligned} \left( \frac{C_2}{\varepsilon \,(p-1)}+C_3+1\right) \,\lambda ^{\varepsilon \,(p-1)/2}\le 1. \end{aligned}$$

Recalling (4.13), we finally obtain that (4.10) holds true at step $k+1$ as well, when $\lambda$ and $\eta$ (depending on N, p, q, s and $\varepsilon$) are chosen so that

$$\begin{aligned} \lambda<\min \left\{ \frac{1}{2},( 1+C_12^{\Theta -\varepsilon /2})^{-\frac{2}{\varepsilon }}, \left( \frac{C_2}{\varepsilon \,(p -1)}+C_3+1\right) ^\frac{2}{\varepsilon \,(p-1)}\right\} \qquad \text{ and } \qquad \tau _\eta <\frac{\lambda ^\Theta }{2}. \end{aligned}$$

The induction is complete.

Part 2: We prove the desired regularity in the whole ball $B_{1/8}$. To this end, we take $0<\varepsilon <\Theta$ and choose the associated $\eta$, obtained in Part 1. Take $z_0\in B_{1}$, let $L=2^{N+1}\,(1+|B_2|)$ and define

$$\begin{aligned} v(x):=L^{-\frac{1}{p-1}}\,u\left( \frac{x}{2}+z_0\right) ,\qquad x\in {\mathbb {R}}^N. \end{aligned}$$

We observe that $v\in W^{s,p}_{\textrm{loc}}(B_4)\cap L^{p-1}_{s\,p}({\mathbb {R}}^N)$ and that v is a weak solution in $B_4$ of

$$\begin{aligned} (-\Delta _p)^s v(x)=\frac{2^{-sp}}{L}\,f \left( \frac{x}{2}+z_0\right) =:{{\widetilde{f}}}(x), \end{aligned}$$

with

$$\begin{aligned} \left\| {{\widetilde{f}}}\right\| _{L^{q}(B_2)} =\frac{2^{N/q-sp}}{L}\,\Vert f\Vert _{L^{q}(B_{1}(z_0))}\le \frac{2^{N/q-sp}}{L}\,\eta <\eta . \end{aligned}$$

Moreover, by construction, we have

$$\begin{aligned} \Vert v\Vert _{L^\infty (B_2)}\le 1. \end{aligned}$$

Observing that $B_{1}(z_0)\subset B_2$ along with the definition of L and the hypotheses in (4.9), we get

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}^N\setminus B_2}\frac{|v(x)|^{p-1}}{|x|^{N+s\,p}}\, dx&=\frac{2^{-s\,p}}{L}\,\int _{{\mathbb {R}}^N\setminus B_{1}(z_0)}\frac{|u(y)|^{p-1}}{|y-z_0|^{N+s\,p}}\,dy\\&\le \frac{1}{L}\,\left( \frac{1}{2}\right) ^{s\,p}\, \left( \frac{2}{2-|z_0|}\right) ^{N+s\,p}\,\int _{{\mathbb {R}}^N \setminus B_2}\frac{|u(y)|^{p-1}}{|y|^{N+s\,p}}\,dy\\&\quad +\frac{2^{N}}{L}\,\Vert u\Vert ^{p-1}_{L^{p-1}(B_2)}\\&\le \frac{2^{N}}{L}\,\int _{{\mathbb {R}}^N\setminus B_2} \frac{|u(y)|^{p-1}}{|y|^{N+s\,p}}dy+\frac{2^N\,|B_2|}{L} \,\Vert u\Vert ^{p-1}_{L^\infty (B_2)}\le 1. \end{aligned} \end{aligned}$$

In the above estimate, we have also used Lemma 2.3 in [4] with the balls $B_{1}(z_0)\subset B_2$. Therefore applying Part 1 to v, we obtain

$$\begin{aligned} \sup _{x\in B_r}|v(x)-v(0)|\le C\,r^{\Theta -\varepsilon },\quad 0<r<\frac{1}{2}, \end{aligned}$$

which in terms of u is same as

$$\begin{aligned} \sup _{x\in B_r(z_0)}|u(x)-u(z_0)|\le C\,L^\frac{1}{p-1}\,r^{\Theta -\varepsilon },\qquad 0<r<\frac{1}{4}. \end{aligned}$$

(4.17)

We remark that the above estimate holds for any $z_0\in B_{1}$. We choose any pair $x,y\in B_{1/8}$ such that $|x-y|= r$. Then $r<1/4$. Setting $z=(x+y)/2$, we apply (4.17) with $z_0=z$ and obtain

$$\begin{aligned} \begin{aligned} |u(x)-u(y)|&\le |u(x)-u(z)|+|u(y)-u(z)|\le 2\sup _{w\in B_r(z)}|u(w)-u(z)|\\&\le 2\,C\,L^\frac{1}{p-1}\,r^{\Theta -\varepsilon }=2\,C\, L^\frac{1}{p-1}\,|x-y|^{\Theta -\varepsilon }, \end{aligned} \end{aligned}$$

which is the desired result. $\square$

We are now ready to give the proof of the final Hölder regularity result.

Proof of Theorem 1.2

Without loss of generality, we may assume $x_0=0$. We modify u in such a way that it fits into the setting of Proposition 4.6. Let

$$\begin{aligned} {\mathcal {A}}_R=\Vert u\Vert _{L^\infty (B_{2R})} +\left( R^{s\,p}\,\int _{{\mathbb {R}}^N\setminus B_{2R}}\frac{|u(y)|^{p-1}}{|y|^{N+s\,p}}\, dy\right) ^\frac{1}{p-1}+\left( \frac{R^{sp-N/q} \Vert f\Vert _{L^{q}(B_{2R})}}{\eta }\right) ^\frac{1}{p-1}, \end{aligned}$$

where we have chosen $\varepsilon \in (0,\Theta )$ and $\eta$ as in Proposition 4.6. Note that u is locally bounded by Theorem 3.8 in [5]. By scaling arguments, it is enough to prove that the rescaled function

$$\begin{aligned} u_R(x):=\frac{1}{{\mathcal {A}}_R}\,u(R\,x), \qquad \text{ for } x\in B_4, \end{aligned}$$

satisfies the estimate

$$\begin{aligned} {[}u_R]_{C^{\Theta -\varepsilon }(B_{1/8})}\le C. \end{aligned}$$

It is straightforward to see that the choice of ${\mathcal {A}}_R$ implies

$$\begin{aligned} \Vert u_R\Vert _{L^\infty (B_2)}\le 1,\qquad \int _{{\mathbb {R}}^N\setminus B_2}\frac{|u_R|^{p-1}}{|x|^{N+s\,p}}\, dx\le 1. \end{aligned}$$

Also, $u_R$ is a local weak solution of

$$\begin{aligned} (-\Delta _p)^s u_R\, (x) = \frac{R^{sp}}{{\mathcal {A}}_R^{p -1}}\,f(R\,x):= f_R(x),\qquad x\in B_4, \end{aligned}$$

with $\Vert f_R\Vert _{L^{q}(B_{2})}\le \eta$. Therefore, applying Proposition 4.6 to $u_R$, we obtain

$$\begin{aligned} {[}u_R]_{C^{\Theta -\varepsilon }(B_{1/8})}\le C. \end{aligned}$$

After scaling back, this concludes the proof. $\square$

Useful Inequalities

The following inequality follows from from [22, (I), Page 95].

Lemma A.1

Let $1<p<\infty$ and $a,b,c,d\in {\mathbb {R}}$. Then we have

$$\begin{aligned} (J_{\beta +1}(a)-J_{\beta +1}(b))(a-b)\ge {2^{-1}}(|a|^{\beta -1}+|b|^{\beta -1})|a-b|^2. \end{aligned}$$

Lemma A.2

Let $p\ge 2$ and $q>1$. For every $a,b\in {\mathbb {R}}$, we have

$$\begin{aligned} J_{q}(a-b)\,\Big (J_p(a)-J_p(b)\Big )\ge (p-1)\,\left( \frac{q}{p-2+q}\right) ^q\, \left| |a|^\frac{p-2}{q} a-|b|^\frac{p-2}{q}{b}\right| ^q. \end{aligned}$$

This is Lemma A.1 in [4].

Lemma A.3

Let $1<p<2$ and $a,c \in {\mathbb {R}}^n$. Then

$$\begin{aligned} (p-1)\frac{|a-c|^2}{(|a|+|c|)^{2-p}} \le \left( |a|^{p-2}a-|c|^{p-2}c\right) \cdot (a-c). \end{aligned}$$

This is inequality (2.3) in [17].

Lemma A.4

Let $\gamma \ge 2$, $1<p<2$ and $a,b,c,d \in {\mathbb {R}}$. Then

$$\begin{aligned} \begin{aligned}&\big (J_\gamma (a-b)-J_\gamma (c-d)\big )\big (J_p(a-c)-J_p(b-d)\big ) \ge 4\frac{(\gamma -1)(p-1)}{\gamma ^2}\\&\quad \left| |a -b|^{\frac{\gamma -2}{2}}(a-b)-|c-d|^{\frac{\gamma -2}{2}} (c-d)\right| ^2\big (|a-c|+|b-d|\big )^{p-2}. \end{aligned} \end{aligned}$$

Proof

The proof is just a combination of Lemmas A.2 and A.3. $\square$

The following inequality can be found in [4, Lemma A.3].

Lemma A.5

Let $\gamma \ge 1$. Then for every $A,B\in {\mathbb {R}}$, we have

$$\begin{aligned} \left| |A|^{\gamma -1}\,A-|B|^{\gamma -1}\,B\right| \ge \frac{1}{C}\, |A-B|^\gamma , \end{aligned}$$

for some constant $C=C(\gamma )>0$.