1. Introduction and main results

Let $n\ge2$, $s\in(0,1)$, and $\Omega\subset{{{\mathbb R}}^n}$ be a bounded Lipschitz domain. In this paper, we study the global (higher-order) Sobolev regularity estimate for the weak solution to the fractional Dirichlet problem

(1.1)\begin{equation} \left\{\begin{aligned} (-\Delta)^su&=f & & \text{in}\ \ \Omega,\\ u&=0 & & \text{in}\ \ {{{\mathbb R}}^n}\setminus\Omega. \end{aligned} \right. \end{equation}

It is known that the fractional Laplacian $(-\Delta)^s$ is defined by, for any $u\in C^\infty_{\rm c}({{{\mathbb R}}^n})$ (the set of all infinitely differentiable functions on ${{{\mathbb R}}^n}$ with compact support) and $x\in{{{\mathbb R}}^n}$,

\begin{equation*} (-\Delta)^su(x):=c_{(n,s)}\,\mathrm{P.V.}\int_{{{{\mathbb R}}^n}} \frac{u(x)-u(y)}{|x-y|^{n+2s}}\,dy \end{equation*}

with

\begin{equation*}c_{(n,s)}:=2^{2s}\pi^{-\frac n2}\frac{{\Gamma}(\frac n2+s)}{{\Gamma}(1-s)}, \end{equation*}

where Γ is the usual Gamma function and $\mathrm{P.V.}$ denotes the integral taken in the sense of the Cauchy principal value (see, for instance, [Reference Bucur and Valdinoci15, Reference Carbotti, Dipierro and Valdinoci17, Reference Dyda22, Reference Fernández-Real and Ros-Oton23]).

Now, we recall the definition of weak solutions to the fractional Dirichlet problem (1.1).

Definition 1.1. Let $s\in(0,1)$ and $\Omega\subset{{{\mathbb R}}^n}$ be a domain. Define

\begin{equation*} \mathbb{X}^s(\Omega):=\left\{\phi\in C^s({{{\mathbb R}}^n}):\ \phi(x)=0\ \ \text{for all}\ \ x\in{{{\mathbb R}}^n}\setminus\Omega\ \ \text{and}\ \ (-\Delta)^s\phi\in L^{\infty}(\Omega)\right\}, \end{equation*}

where $C^s({{{\mathbb R}}^n})$ denotes the Hölder space on ${{{\mathbb R}}^n}$ with exponent s, defined by

\begin{equation*} C^s({{{\mathbb R}}^n}):= \left\{\phi\ \ \text{is bounded and continuous on}\ {{{\mathbb R}}^n} :\ [\phi]_{C^s({{{\mathbb R}}^n})} \lt \infty\right\} \end{equation*}

with

\begin{equation*} [\phi]_{C^s({{{\mathbb R}}^n})}:=\sup_{\overset{x,\,y\in{{{\mathbb R}}^n}}{x\neq y}}\frac{|\phi(x)-\phi(y)|}{|x-y|^s}. \end{equation*}

Then $u\in L^1({{{\mathbb R}}^n})$ is said to be a weak solution of the problem (1.1) if, for any $\phi\in\mathbb{X}^s(\Omega)$,

\begin{equation*} \int_{\Omega}u(x)(-\Delta)^s\phi(x)\,dx=\int_{\Omega}\phi(x)f(x)\,dx \end{equation*}

and $u\equiv0$ in ${{{\mathbb R}}^n}\setminus\Omega$.

It is known that when $\Omega\subset{{{\mathbb R}}^n}$ is a bounded Lipschitz domain, the problem (1.1) with any given $f\in L^1(\Omega)$ has the unique weak solution (see, for instance, [Reference Abdellaoui and Peral3, Reference Chen and Véron18]).

Remark 1.2. Let $s\in(0,1)$, $\Omega\subset{{{\mathbb R}}^n}$ be a bounded Lipschitz domain, and $y\in\Omega$. Denote by δ_y the Dirac mass centred at y. Recall that the Green function $G_s(\cdot, y)$ is defined as a weak solution to the problem

(1.2)\begin{equation} \left\{\begin{aligned} (-\Delta)^s_xG_s(x,y)&=\delta_y(x) & & \text{in}\ \ \Omega,\\ G_s(x,y)&=0 & & \text{in}\ \ {{{\mathbb R}}^n}\setminus\Omega, \end{aligned} \right. \end{equation}

where $(-\Delta)_x^s$ denotes the fractional Laplacian of order s with respect to x (see, for instance, [Reference Abdellaoui and Peral3, Reference Bogdan8–Reference Bogdan, Kulczycki and Nowak10, Reference Chen and Véron18]). Then the weak solution u to the problem (1.1) with any given $f\in L^1(\Omega)$ can be expressed by, for almost every $x\in\Omega$,

(1.3)\begin{equation} u(x)=\int_\Omega G_s(x,y)f(y)\,dy \end{equation}

(see, for instance, [Reference Abdellaoui and Peral3, Reference Chen and Véron18, Reference Kulczycki28, Reference Liu and Zhuo30]).

The fractional Laplacian arises naturally in the study of Lévy processes, and also appears in a variety of models, from Physics to Biology or Finance (see, for instance, [Reference Bucur and Valdinoci15, Reference Carbotti, Dipierro and Valdinoci17, Reference Fernández-Real and Ros-Oton23, Reference Pozrikidis33]). Applications of the fractional Laplacian can be envisioned in mathematical finance, quantum mechanics, and obstacle problems (see, for instance, [Reference Barrios, Figalli and Ros-Oton6, Reference Barrios, Figalli and Ros-Oton7, Reference Bucur and Valdinoci15, Reference Carbotti, Dipierro and Valdinoci17, Reference Fernández-Real and Ros-Oton23] and references therein). Over the past years, fractional partial differential equations have been extensively studied from the point of probability, potential theory, and PDEs (see, for instance, [Reference Abatangelo and Cozzi1, Reference Abdellaoui, Fernández, Leonori and Younes2, Reference Bogdan8, Reference Bogdan9, Reference Borthagaray and Nochetto11, Reference Dong and Liu20, Reference Dong and Ryu21, Reference Kim and Weidner26, Reference Ros-Oton and Serra34, Reference Ros-Oton and Serra35] and references therein).

Recently, there has been abundant literature focusing on the Hölder and the Sobolev type regularity estimates of solutions to the fractional Dirichlet problem (1.1) (see, for instance, [Reference Abdellaoui, Fernández, Leonori and Younes2, Reference Acosta and Borthagaray4, Reference Borthagaray and Nochetto11, Reference Cozzi19, Reference Dong and Ryu21, Reference Kim and Weidner26, Reference Ros-Oton and Serra34, Reference Ros-Oton and Serra36, Reference Silvestre37]). In particular, Silvestre [Reference Silvestre37] showed that, for any solution $u\in L^{\infty}({{{\mathbb R}}^n})$ to the equation $(-\Delta)^su=f$ in ${{{\mathbb R}}^n}$, if $f\in C^{\beta}({{{\mathbb R}}^n})$ with $\beta\in(0,1]$, then $u\in C^{\beta+2s}({{{\mathbb R}}^n})$, and, if $f\in L^{\infty}({{{\mathbb R}}^n})$, then $u\in C^{\beta}({{{\mathbb R}}^n})$ for any $\beta\in(0,2s)$. Based on the conclusion in [Reference Silvestre37], Ros-Oton and Serra [Reference Ros-Oton and Serra34] gave the optimal Hölder regularity estimate for the problem (1.1) when Ω is a bounded Lipschitz domain satisfying the uniform exterior ball condition. More generally, Ros-Oton and Serra [Reference Ros-Oton and Serra36] established sharp Hölder boundary regularity estimates in C ¹ and $C^{1,\kappa}$ domains for nonlocal Dirichlet problems of the form Lu = f in Ω, u = 0 in ${{{\mathbb R}}^n}\setminus\Omega$, where L is a nonlocal elliptic operator of order 2s with $s\in(0,1)$. Moreover, Kim and Weidner [Reference Kim and Weidner26] obtained the sharp global Hölder regularity for solutions to nonlocal elliptic equations with Hölder continuous coefficients in divergence form in $C^{1,\kappa}$ domains.

Furthermore, Cozzi [Reference Cozzi19] obtained the interior $W^{2s-\varepsilon,2}$ regularity for weak solutions of certain linear elliptic integro-differential equations via intermediate estimates in Nikol’skii spaces. Borthagaray and Nochetto [Reference Borthagaray and Nochetto11] studied global Besov regularity estimates for the weak solution of the problem (1.1) on any given bounded Lipschitz domain Ω, by using the variational structure of the problem (1.1) and the difference quotient technique. In particular, Borthagaray and Nochetto [Reference Borthagaray and Nochetto11] obtained the sharp regularity estimate for the weak solution of the problem (1.1) in the scale of the Sobolev space $W^{t,2}(\Omega)$ based on the Lebesgue space $L^2(\Omega)$. Precisely, Borthagaray and Nochetto [Reference Borthagaray and Nochetto11] proved that the weak solution of the problem (1.1) satisfies

\begin{equation*} \|u\|_{W^{s+\min\{s,1/2\}-\varepsilon,2}(\Omega)}\le C\|f\|_{L^2(\Omega)}, \end{equation*}

where $\varepsilon\in(0,s)$ is an arbitrary constant. Moreover, Borthagaray and Nochetto [Reference Borthagaray and Nochetto12] also studied the global Sobolev regularity estimate for the weak solution of the problem (1.1) in the scale of general Sobolev spaces $W^{t,p}(\Omega)$ with p ≠ 2, under the extra assumption that the solution of (1.1) satisfies the global s-order Hölder continuity. Additionally, Abdellaoui et al. [Reference Abdellaoui, Fernández, Leonori and Younes2] established the global Sobolev regularity estimate for the solution of the problem (1.1) in general Sobolev spaces $W^{t,p}(\Omega)$ with $t\in[s,\min\{1,2s\})$ when Ω is a bounded C ² domain, by using some pointwise upper bound estimates for the Green function G_s and its gradient. Recently, Dong and Ryu [Reference Dong and Ryu21] studied the weighted Sobolev regularity estimate for the weak solution of the problem (1.1) under the assumption that Ω is a bounded $C^{1,\kappa}$ domain with $\kappa\in(0,s)$.

In this paper, motivated by the recent work in [Reference Abdellaoui, Fernández, Leonori and Younes2, Reference Borthagaray and Nochetto11, Reference Borthagaray and Nochetto12, Reference Dong and Ryu21] on global Sobolev regularity estimates for the weak solution to the Dirichlet problem (1.1), we continue to study the global regularity estimate for the weak solution of the problem (1.1) on bounded Lipschitz domains in the scale of (higher-order) Sobolev spaces and try to improve some known Sobolev regularity estimates.

To state the main results of this paper, we first recall the definitions of the Sobolev space $W^{t,p}(\Omega)$ and the space $\mathrm{BMO}({{{\mathbb R}}^n})$ (see, for instance, [Reference Grafakos24]).

In the following, for any $x\in{\mathbb{R}^n}$ and $r\in(0,\infty)$, we define $B(x,r):=\{y\in{\mathbb{R}^n}:\ |y-x| \lt r\}$. Let $\Omega\subset{\mathbb{R}^n}$ be a domain. Denote by $L^1_{\mathrm{loc}}(\Omega)$ the set of all locally integrable functions on Ω.

Definition 1.4. Let $f\in L^1_{{\mathrm{loc}}}({{{\mathbb R}}^n})$. Then f is said to belong to the space BMO $({{{\mathbb R}}^n})$ if

\begin{equation*} \|f\|_{{{\mathrm{BMO}}}({{{\mathbb R}}^n})}:=\sup_{B\subset{{{\mathbb R}}^n}}\frac{1}{|B|}\int_{B}|f(x)-f_{B}|\,dx \lt \infty, \end{equation*}

where the supremum is taken over all balls B in ${{{\mathbb R}}^n}$ and $f_B:=\frac{1}{|B|}\int_{B}f(y)\,dy$.

Now, we give the first main result of this paper.

Theorem 1.5. Let $n\ge2$, $s\in(0,1)$, and $\Omega\subset{{{\mathbb R}}^n}$ be a bounded Lipschitz domain. Then there exists a constant $\varepsilon\in(0,s)$ depending only on n, s, and the Lipschitz constant of Ω such that, for any $t\in[\varepsilon,\min\{1+\varepsilon,2s\})$, the weak solution u to the problem (1.1) with $f\in L^q(\Omega)$ for some $q\in(\frac{n}{2s-t},\infty]$ satisfies $u\in W^{t,p}({{{\mathbb R}}^n})$ for all $ p\in[1,\frac{1}{t-\varepsilon})$, and

(1.4)\begin{equation} \|u\|_{W^{t,p}({{{\mathbb R}}^n})}\le C\|f\|_{L^q(\Omega)}, \end{equation}

where C is a positive constant depending only on n, s, t, p, q, and Ω.

Theorem 1.5 provides the global Sobolev regularity estimate for the solution of the problem (1.1) on any given bounded Lipschitz domain. In particular, when $s\in(1/2,1)$, we can take t > 1 in (1.4); in this case, (1.4) gives the higher-order Sobolev regularity estimate for the problem (1.1) on bounded Lipschitz domains.

We prove Theorem 1.5 by borrowing some ideas from the work of Borthagaray and Nochetto [Reference Borthagaray and Nochetto12, Theorems 3.6 and 3.7]. Precisely, using the global Hölder estimate for the solution of the problem (1.1) (see Proposition 3.1), we first prove (1.4) in the case of $t\in[\varepsilon,1)$. Then, by applying the famous Bourgain–Brezis–Mironescu formula (see, for instance, [Reference Bourgain, Brezis and Mironescu13, Reference Bourgain, Brezis and Mironescu14]) and the conclusion in the case of $t\in[\varepsilon,1)$, we further show that the estimate (1.4) holds when t = 1. Finally, by using the estimate (1.4) in the case of t = 1 and the local Hölder regularity estimate on $\nabla u$, and establishing the pointwise upper bound estimate for $\nabla u$ near the boundary of Ω, we obtain the estimate (1.4) in the case of $t\in(1,\min\{1+\varepsilon,2s\})$.

Moreover, to obtain the global Hölder estimate for the solution of the problem (1.1), we need some precise pointwise estimates for the Green function G_s near the boundary of Ω which can be obtained by using the boundary Harnack inequality for nonnegative s-harmonic functions established by Bogdan in [Reference Bogdan8, Reference Bogdan9] (see also Lemma 2.4).

Remark 1.6. Borthagaray and Nochetto [Reference Borthagaray and Nochetto12, Theorem 3.7] established the global Sobolev regularity estimate similar to (1.4) for the problem (1.1) with the larger range for t, under the assumption that the global s-order Hölder estimate

(1.5)\begin{equation} \|u\|_{C^s({{{\mathbb R}}^n})}\le C\|f\|_{L^\infty(\Omega)} \end{equation}

holds for the solution of the problem (1.1). We point out that the estimate (1.5) holds when Ω is a bounded $C^{1,\kappa}$ domain with some $\kappa\in(0,s)$ (see [Reference Kim and Weidner26, Theorem 1.6]) or a bounded Lipschitz domain satisfying the uniform exterior ball condition (see [Reference Ros-Oton and Serra34, Proposition 1.1]). Compared with [Reference Borthagaray and Nochetto12, Theorem 3.7], Theorem 1.5 establishes the global Sobolev regularity estimate for the problem (1.1) on bounded Lipschitz domains without any additional assumptions.

Let $t\in(0,1)$. Recall that, for any $u\in C^\infty_{\rm c}({{{\mathbb R}}^n})$, the Riesz gradient of order t of u is defined by setting, for any $x\in{{{\mathbb R}}^n}$,

\begin{equation*} \nabla^tu(x):=\mu_{(n,t)}\int_{{{{\mathbb R}}^n}} \frac{(x-y)[u(x)-u(y)]}{|x-y|^{n+t+1}}\,dy, \end{equation*}

where $\mu_{(n,t)}:=\frac{2^t{\Gamma}(\frac{n+t+1}{2})}{\pi^{\frac n2}{\Gamma}(\frac{1-t}{2})}$ (see, for instance, [Reference Abdellaoui, Fernández, Leonori and Younes2]).

When some additional assumptions are exerted on the domain Ω, such as, Ω is a bounded Lipschitz domain with small Lipschitz constant, a bounded C ¹ domain, a bounded $C^{1,\kappa}$ domain with some $\kappa\in(0,1)$, or a bounded Lipschitz domain satisfying the uniform exterior ball condition (see Definition 2.1), then we have the following improved regularity for the solution to the problem (1.1).

We prove (i) and (ii) of Theorem 1.7 by applying an approach used in the proof of Theorem 1.5. More precisely, by first showing that $u\in C^{\delta}({{{\mathbb R}}^n})$ with any given $\delta\in(0,s)$ when the Lipschitz constant of Ω is small enough, and then repeating the proof of Theorem 1.5, we prove (i) of Theorem 1.7. The proof of (ii) is similar to that of (i). Moreover, we establish the endpoint regularity estimate as in (iii) of Theorem 1.7 by applying the method used in the proof of [Reference Ma and Yang31, Theorem 1.3] and proving some delicate estimate for the integral involving the distance function in bounded Lipschitz domains (see Lemma 4.3). Meanwhile, to show (iii) of Theorem 1.7, we need to establish the Hölder type continuity estimates of the fractional Laplacian $(-\Delta)_x^{\frac{s}{2}}G_s(x,y)$ with respect to x (see Lemmas 4.4, 4.5, and 4.6), based on the Hölder continuity of the Green function G_s. We also remark that when Ω is a bounded C ² domain, the estimate (1.6) was proved in [Reference Ma and Yang31, Theorems 1.3 and 1.7].

The remainder of this paper is organized as follows. In Section 2, we present some properties of Lipschitz domains and recall some known pointwise estimates related to the Green function G_s. In Section 3, we first give the global Hölder estimate for the solution of the problem (1.1), and then by using the global Hölder estimate for the solution of the problem (1.1) and the interior estimates for the solution of the problem (1.1) and its gradient, we prove Theorem 1.5. In Section 4, we prove Theorem 1.7 via establishing the Hölder type continuity of $(-\Delta)_x^{\frac{s}{2}}G_s(x,y)$.

Finally, we make some conventions on symbols. Throughout the whole paper, we always denote by C a positive constant which is independent of the main parameters, but it may vary from line to line. The symbol $f\lesssim g$ means that $f\le Cg$. If $f\lesssim g$ and $g\lesssim f$, then we write $f\sim g$. If $f\le Cg$ and g = h or $g\le h$, we then write $f\lesssim g=h$ or $f\lesssim g\le h$. Denote by ${\mathbb N}$ the set of all positive integers. For any $x\in{{{\mathbb R}}^n}$ and $r\in(0,\infty)$, denote by $B_r(x)$ the ball with the centre x and the radius r. For any given domain $\Omega\subset{{{\mathbb R}}^n}$ and any $x\in{{{\mathbb R}}^n}$, let $\Omega^{c}:={{{\mathbb R}}^n}\setminus\Omega$ and $\delta(x):={\mathrm{\,dist\,}}(x,\Omega^{c})$. For any given $q\in[1,\infty]$, we denote by q' its conjugate exponent, that is, $1/q + 1/q'= 1$.

2. Preliminaries

In this section, we first recall the definition and some properties of Lipschitz domains in ${{{\mathbb R}}^n}$; and then present some estimates on Green functions associated with the problem (1.1).

Next, we recall a basic geometric property of Lipschitz domains (see, for instance, [Reference Bogdan8, Reference Jerison and Kenig25]).

The next lemma provides useful estimates for the decay rate of s-harmonic functions near the boundary $\partial\Omega$, which was established in [Reference Bogdan9].

Lemma 2.4. Let $n\ge2$, $s\in(0,1)$, $\Omega\subset{{{\mathbb R}}^n}$ be a bounded Lipschitz domain, and let α and β be as in Remark 2.2(i). Then there exists a positive constant $\varepsilon\in(0,s)$ depending only on the Lipschitz constant of Ω, n, and s such that, for any $x_0\in\partial\Omega$, $r\in(0,\min\{\alpha/2,\beta/2\})$, and any function u satisfying that $(-\Delta)^s u=0$ in $\Omega\cap B(x_0,2r)$ and $u\equiv 0$ in $\Omega^{c}\cap B(x_0,2r)$, there exists a positive constant C independent of u, x ₀, and r such that, for any $x\in\Omega\cap B(x_0,r)$,

\begin{equation*} C^{-1}u\left(A_{x_0,r}\right)\left[\frac{\delta(x)}{r}\right]^{2s-\varepsilon}\le u(x)\le Cu\left(A_{x_0,r}\right)\left[\frac{\delta(x)}{r}\right]^{\varepsilon}, \end{equation*}

where the point $A_{x_0,r}\in\Omega\cap B(x_0,r)$ is the same as in Proposition 2.3 and $\delta(x):={\mathrm{\,dist\,}}(x,\Omega^{c})$.

Now, we present some known pointwise estimates of the Green function G_s and its gradient as follows (see, for instance, [Reference Bogdan9, Reference Bogdan, Kulczycki and Nowak10]).

Lemma 2.5. Let $n\ge2$, $s\in(0,1)$, $\Omega\subset{{{\mathbb R}}^n}$ be a bounded Lipschitz domain, and let G_s be the Green function as in (1.2). Then there exists a positive constant C depending only on n, s, and Ω such that, for any $x,y\in\Omega$ with x ≠ y,

(2.2)\begin{equation} 0\le G_s(x,y)\le \frac{C}{|x-y|^{n-2s}}. \end{equation}

Moreover, for any $x,y\in\Omega$ with x ≠ y,

(2.3)\begin{equation} |\nabla_{x}G_{s}(x,y)|\le nG_{s}(x,y)\max\left\{\frac{1}{|x-y|},\frac{1}{\delta(x)}\right\}, \end{equation}

where $\nabla_{x}$ denotes the gradient with respect to x.

Let $s\in[0,\infty)$ and $p\in[1,\infty)$. Recall that the Bessel potential space $L^{s,p}({{{\mathbb R}}^n})$ is defined as the completion of $C^{\infty}_{\rm c}({{{\mathbb R}}^n})$ with with respect to the norm

(2.4)\begin{equation} \|u\|_{L^{s,p}({{{\mathbb R}}^n})}=\left\|(1-\Delta)^{\frac{s}{2}}u\right\|_{L^p({{{\mathbb R}}^n})}. \end{equation}

We point out that, for any $p\in(1,\infty)$ and $s\in[0,2)$, the norm $\|\cdot\|_{L^{s,p}({{{\mathbb R}}^n})}$ defined as in (2.4) is equivalent to

\begin{equation*} \||\cdot\||_{L^{s,p}({{{\mathbb R}}^n})}:=\|\cdot\|_{L^{p}({{{\mathbb R}}^n})}+\left\|(-\Delta)^{\frac{s}{2}}\cdot\right\|_{L^{p}({{{\mathbb R}}^n})} \end{equation*}

(see, for instance, [Reference Adams5, Reference Maz’ya32]). Moreover, it is known that, for any $s\in(0,2)$, $p\in(1,\infty)$, and $\varepsilon\in(0,s)$,

(2.5)\begin{equation} L^{s+\varepsilon,p}({{{\mathbb R}}^n})\subset W^{s,p}({{{\mathbb R}}^n})\subset L^{s-\varepsilon,p}({{{\mathbb R}}^n}) \end{equation}

(see, for instance, [Reference Adams5, Theorem 7.63(g)]).

3. Proof of Theorem 1.5

In this section, we show Theorem 1.5. We begin with presenting the global Hölder estimate for the weak solution of the problem (1.1).

Proposition 3.1. Let $n\ge2$, $s\in(0,1)$, and $\Omega\subset{{{\mathbb R}}^n}$ be a bounded Lipschitz domain. There exists a constant $\varepsilon\in(0,s)$, depending only on the Lipschitz constant of Ω, n, and s, such that the weak solution u of the problem (1.1), with $f\in L^q(\Omega)$ for some $q\in(\frac{n}{2s-\varepsilon},\infty]$, satisfies $u\in C^{\varepsilon}({{{\mathbb R}}^n})$. Moreover, there exists a positive constant C, independent of both u and f, such that

(3.1)\begin{equation} \|u\|_{C^{\varepsilon}({{{\mathbb R}}^n})}\le C\|f\|_{L^q(\Omega)}. \end{equation}

The conclusion presented in Proposition 3.1 is known to the expert. For instance, when f = 0 in Ω and replacing the condition u = 0 in $\mathbb{R}^n\backslash\Omega$ by u = g in $\mathbb{R}^n\backslash\Omega$, with that the function g is Hölder continuous in $\mathbb{R}^n\backslash\Omega$, in (1.1), the conclusion of Proposition 3.1 was obtained in [Reference Korvenpää, Kuusi and Palatucci27, Theorem 8]; when Ω is a bounded $C^{1,\kappa}$ domain with some $\kappa\in(0,1]$ and $f\in L^\infty(\Omega)$, the conclusion of Proposition 3.1 was established in [Reference Fernández-Real and Ros-Oton23, Proposition 2.6.9]. However, for the sake of completeness, we give the proof of Proposition 3.1 in this section.

Lemma 3.2. Let $n\ge2$, $s\in(0,1)$, $\Omega\subset{{{\mathbb R}}^n}$ be a bounded Lipschitz domain, G_s be the Green function as in (1.2), and let $\varepsilon\in(0,s)$ be the same as in Lemma 2.4. Then there exists a positive constant C depending only on n, s, and Ω such that, for any $x,y\in\Omega$ with x ≠ y,

(3.2)\begin{equation} 0\le G_{s}(x,y)\le C\frac{[\delta(x)]^{\varepsilon}}{|x-y|^{n-2s+\varepsilon}}, \end{equation}

where, for any $x\in\Omega$, $\delta(x):={\mathrm{\,dist\,}}(x,\Omega^{c})$.

Proof. Fix $x,y\in\Omega$ with x ≠ y. We prove (3.2) by considering the following two cases on $|x-y|$ and $\delta(x)$.

Case 1) $|x-y|\le4\delta(x)$. In this case, by (2.2), we can directly conclude that

\begin{equation*} G_{s}(x,y)\lesssim\frac{|x-y|^{\varepsilon}}{|x-y|^{n-2s+\varepsilon}}\lesssim\frac{[\delta(x)]^{\varepsilon}}{|x-y|^{n-2s+\varepsilon}}. \end{equation*}

Case 2) $|x-y| \gt 4\delta(x)$. Take $x_0\in\partial\Omega$ satisfy $|x-x_0|=\delta(x)$. Applying Lemma 2.4 to the function $G_{s}(\cdot,y)$, we deduce that, for any $z\in \Omega\cap B(x_0,\frac{1}{4}|x-y|)$,

(3.3)\begin{equation} G_{s}(z,y)\lesssim G_{s}\left(A_{x_0, \frac14|x-y|},y\right)\left[\frac{\delta(z)}{|x-y|}\right]^{\varepsilon}, \end{equation}

where the point $A_{x_0, \frac14|x-y|}\in\Omega\cap B(x_0,\frac{1}{4}|x-y|)$ is as in Proposition 2.3. By Proposition 2.3, we find that

\begin{align*} \left|A_{x_0, \frac14|x-y|}-y\right| &\ge|y-x_0|-\left|A_{x_0, \frac14|x-y|}-x_0\right|\ge|x-y|-|x-x_0|-\\ &\left|A_{x_0, \frac14|x-y|}-x_0\right| \ge|x-y|-\frac{1}{4}|x-y|-\frac{1}{4}|x-y|\ge\frac{1}{2}|x-y|, \end{align*}

which, together with (2.2) and (3.3), further implies that

\begin{equation*} G_{s}(x,y)\lesssim \frac{1}{|A_{x_0, \frac14|x-y|}-y|^{n-2s}}\left[\frac{\delta(x)}{|x-y|}\right]^{\varepsilon}\lesssim \frac{[\delta(x)]^{\varepsilon}}{|x-y|^{n-2s+\varepsilon}}. \end{equation*}

This finishes the proof of Lemma 3.2.

Next, we recall the interior Hölder estimate for the problem (1.1) (see, for instance, [Reference Ros-Oton and Serra34, Reference Ros-Oton and Serra35, Reference Silvestre37]). We first recall the definition of the Hölder space. Let $O\subset{{{\mathbb R}}^n}$ be a domain and ${\gamma}\in(0,2)$. When ${\gamma}\in(0,1]$, the Hölder space $C^{{\gamma}}(O)$ is defined by setting

\begin{equation*} C^{{\gamma}}(O):=\left\{u\in L^{\infty}(O):\ [u]_{C^{{\gamma}}(O)}:=\sup_{\overset{x,\,y\in O}{x\neq y}}\frac{|u(x)-u(y)|}{|x-y|^{{\gamma}}} \lt \infty\right\} \end{equation*}

with the norm

\begin{equation*} \|u\|_{C^{{\gamma}}(O)}:=\|u\|_{L^{\infty}(O)}+[u]_{C^{{\gamma}}(O)}. \end{equation*}

When ${\gamma}\in(1,2)$, the Hölder space $C^{{\gamma}}(O)$ is defined by setting

\begin{equation*} C^{{\gamma}}(O):=\left\{u\in C^1(O):\ [\nabla u]_{C^{{\gamma}-1}(O)}:=\sup_{\overset{x,\,y\in O}{x\neq y}}\frac{|\nabla u(x)-\nabla u(y)|}{|x-y|^{{\gamma}-1}} \lt \infty\right\} \end{equation*}

with the norm

\begin{equation*} \|u\|_{C^{{\gamma}}(O)}:=\|\nabla u\|_{L^\infty(O)}+[\nabla u]_{C^{{\gamma}-1}(O)}. \end{equation*}

Lemma 3.3. Let $n\ge2$, $s\in(0,1)$, and $\Omega\subset{{{\mathbb R}}^n}$ be a bounded Lipschitz domain. Assume that $u\in L^{\infty}(B_1(0))$ satisfies $(1+|\cdot|)^{-n-2s}u(\cdot)\in L^1({{{\mathbb R}}^n})$ and

\begin{equation*} (-\Delta)^{s}u=f \ \ \text{in} \ \ B_1(0), \end{equation*}

for $f\in L^q(B_1(0))$ with $q\in(\frac{n}{2s},\infty)$. Then $u\in C^{{\gamma}}(B_{1/2}(0))$ for any ${\gamma}\in(0,2s-\frac{n}{q}]$, and

\begin{equation*} \|u\|_{C^{{\gamma}}(B_{1/2}(0))}\le C\left[\|f\|_{L^q(B_1(0))}+\left\|(1+|\cdot|)^{-n-2s}u (\cdot)\right\|_{L^1({{{\mathbb R}}^n})}+\|u\|_{L^{\infty}(B_1(0))}\right], \end{equation*}

where C is a positive constant depending only on n, s, q, and γ.

Lemma 3.4. Let $n\ge2$, $s\in(0,1)$, $\Omega\subset{{{\mathbb R}}^n}$ be a bounded Lipschitz domain, $\varepsilon\in(0,s)$ be the same as in Lemma 2.4, and let u be the weak solution to the problem (1.1) with $f\in L^q(\Omega)$ for some $q\in(\frac{n}{2s-\varepsilon},\infty]$. Then $u\in C^{{\gamma}}(\Omega)$ with any given ${\gamma}\in(0,2s-\frac{n}{q}]$ and, for any $x_{0}\in\Omega$, there exists a positive constant C depending only on Ω, s, n, q, and γ such that

(3.4)\begin{equation} [u]_{C^{{\gamma}}(B_{\delta(x_{0})/2}(x_{0}))}\le C[\delta(x_{0})]^{\varepsilon-{\gamma}}\|f\|_{L^q(\Omega)}. \end{equation}

Moreover, if $s\in(\frac{1}{2},1)$ and $f\in L^q(\Omega)$ with $q\in(\frac{n}{2s-1},\infty]$, then, for any ${\gamma}\in[1,2s-\frac{n}{q}]$, there exists a positive constant C depending only on Ω, s, n, q, and γ such that

(3.5)\begin{equation} [\nabla u]_{C^{{\gamma}-1}(B_{\delta(x_{0})/2}(x_{0}))}\le C[\delta(x_{0})]^{\varepsilon-{\gamma}}\|f\|_{L^q(\Omega)}. \end{equation}

It is worth pointing out that the conclusion of Lemma 3.4 in some special cases is known. For instance, when Ω is a bounded Lipschitz domain satisfying the uniform exterior ball condition and $f\in L^\infty(\Omega)$, the conclusion of Lemma 3.4 with $\varepsilon:=s$ was established in [Reference Ros-Oton and Serra34, Lemma 2.9].

Proof of Lemma 3.4

We first show (3.4). Let $x_0\in\Omega$, $R:=\delta(x_0)$, and $\widetilde{u}(\cdot):=u(x_{0}+R\cdot)$. From (1.1) and a scaling argument, it follows that

(3.6)\begin{equation} (-\Delta)^{s}\widetilde{u}(\cdot)=R^{2s}f(x_{0}+R\cdot)\ \ \text{in}\ \ B_1(0). \end{equation}

By (1.3) and (3.2), we conclude that, for any $x\in\Omega$,

(3.7)\begin{align} |u(x)|&\le\left|\int_{\Omega}G_{s}(x,y)f(y)\,dy\right|\nonumber\\ &\lesssim[\delta(x)]^{\varepsilon}\int_{\Omega}\frac{1}{|x-y|^{n-2s+\varepsilon}}|f(y)|\,dy \lesssim\|f\|_{L^q(\Omega)}[\delta(x)]^{\varepsilon}. \end{align}

From this, we deduce that, for any $y\in B_1(0)$,

\begin{align*} \left|\widetilde{u}(y)\right|&=\left|u(x_0+Ry)\right|\lesssim \|f\|_{L^q(\Omega)}[\delta(x_0+Ry)]^{\varepsilon}\\ &\lesssim\|f\|_{L^q(\Omega)}[\delta(x_0)+Ry]^\varepsilon \lesssim\|f\|_{L^q(\Omega)}R^\varepsilon [1+|y|^\varepsilon], \end{align*}

which, combined with Lemma 3.3, (3.6), (3.7), the assumption that $q\in(\frac{n}{2s-\varepsilon},\infty]$, and the fact that, for any $x\in B_{\delta(x_0)/2}(x_0)$, $\delta(x)\le2\delta(x_0)$, further implies that

\begin{align*} \left\|\widetilde{u}\right\|_{C^{\gamma}(B_{1/2}(0))} &\lesssim R^{2s}\|f(x_0+R\cdot)\|_{L^q(B_1(0))} +\left\|(1+|\cdot|)^{-n-2s}\widetilde{u}(\cdot) \right\|_{L^1({{{\mathbb R}}^n})}\\ \nonumber &\quad+ \left\|\widetilde{u}\right\|_{L^\infty(B_1(0))}\\ \nonumber &\lesssim R^{2s-\frac nq}\|f\|_{L^q(B_R(x_0))}+ R^\varepsilon\|f\|_{L^q(\Omega)}\lesssim R^\varepsilon\|f\|_{L^q(\Omega)}. \end{align*}

Finally, by this and an observation that

\begin{equation*} [u]_{C^{{\gamma}}(B_{R/2}(x_{0}))}=R^{-{\gamma}}[\widetilde{u}]_{C^{{\gamma}}(B_{1/2}(0))}\ \ \text{and}\ \ [\nabla u]_{C^{{\gamma}-1}(B_{R/2}(x_{0}))} =R^{-{\gamma}}[\nabla\widetilde{u}]_{C^{{\gamma}-1}(B_{1/2}(0))}, \end{equation*}

we finish the proofs of (3.4) and (3.5) for the $C^{{\gamma}}$ seminorm of u and the $C^{{\gamma}-1}$ seminorm of $\nabla u$ in $B_{R/2}(x_{0})$. This finishes the proof of Lemma 3.4.

Now, we show Proposition 3.1 by using Lemma 3.4.

Proof of Proposition 3.1

Let $x,y\in{{{\mathbb R}}^n}$ with x ≠ y. We prove (3.1) by considering the following three cases on x and y.

Case 1) $x\in\Omega^c$ or $y\in\Omega^c$. If $x,y\in\Omega^c$, using the fact that $u\equiv0$ in $\Omega^c$, we find that $|u(x)-u(y)|=0\le|x-y|^\varepsilon\|f\|_{L^q(\Omega)}$. If $x\in\Omega$ and $y\in\Omega^c$, by (3.7), we conclude that

\begin{equation*} |u(x)-u(y)|=|u(x)|\lesssim[\delta(x)]^\varepsilon\|f\|_{L^q(\Omega)}\le|x-y|^\varepsilon\|f\|_{L^q(\Omega)}. \end{equation*}

If $x\in\Omega^c$ and $y\in\Omega$, this estimate also holds. Therefore, in this case, we have

(3.8)\begin{equation} \frac{|u(x)-u(y)|}{|x-y|^{\varepsilon}}\lesssim\|f\|_{L^q(\Omega)}. \end{equation}

Case 2) $x,y\in\Omega$ and $|x-y|\ge\frac{1}{2}\max\{\delta(x),\delta(y)\}$. In this case, from (3.7), it follows that

\begin{equation*} \frac{|u(x)-u(y)|}{|x-y|^{\varepsilon}}\le\frac{|u(x)|+|u(y)|}{|x-y|^{\varepsilon}} \lesssim\frac{|u(x)|}{[\delta(x)]^{\varepsilon}}+\frac{|u(y)|}{[\delta(y)]^{\varepsilon}}\lesssim\|f\|_{L^q(\Omega)}. \end{equation*}

Case 3) $x,y\in\Omega$ and $|x-y| \lt \frac{1}{2}\max\{\delta(x),\delta(y)\}$. In this case, we take ${\gamma}:=\varepsilon$ in Lemma 3.4 and obtain that

\begin{equation*} [u]_{C^{\varepsilon}(B_{\delta(x)/2}(x))}\lesssim\|f\|_{L^q(\Omega)} \ \ \text{and}\ \ [u]_{C^{\varepsilon}(B_{\delta(y)/2}(y))}\lesssim\|f\|_{L^q(\Omega)}, \end{equation*}

which implies that the estimate (3.8) also holds in this case.

From (3.7), we deduce that $\|u\|_{L^{\infty}({{{\mathbb R}}^n})}\lesssim\|f\|_{L^q(\Omega)}.$ By this and the conclusions obtained in Cases 1)–3), we find that $u\in C^\varepsilon({{{\mathbb R}}^n})$ and $\|u\|_{C^\varepsilon({{{\mathbb R}}^n})}\lesssim\|f\|_{L^q(\Omega)}$. This completes the proof of Proposition 3.1.

To show Theorem 1.5, we recall an estimate on the integrability of powers of the distance function (see, for instance, [Reference Carbery, Maz’ya, Mitrea and Rule16, Lemma 2.14]).

Lemma 3.5. Let $n\ge2$ and Ω be a bounded Lipschitz domain of ${{{\mathbb R}}^n}$. For any $x\in\Omega$, let $\delta(x):={\mathrm{\,dist\,}}(x,\Omega^{c})$. Then there exists a positive constant C such that, for any given $\gamma\in(0,1)$,

(3.9)\begin{equation} \int_{\Omega}[\delta(x)]^{-\gamma}\,dx\le\frac{C}{1-\gamma}. \end{equation}

Next, we prove Theorem 1.5 by using Proposition 3.1 and Lemmas 3.2, 3.4, and 3.5.

Proof of Theorem 1.5

Let $\varepsilon\in(0,s)$ be the same as in Lemma 2.4, $t\in[\varepsilon,\min\{1+\varepsilon,2s\})$, $q\in(\frac{n}{2s-t},\infty]$, and $p\in[1,\frac{1}{t-\varepsilon})$. We only need to prove $u\in W^{t,p}({{{\mathbb R}}^n})$ and the estimate (1.4) for $t\in(\varepsilon,\min\{1+\varepsilon,2s\})$, because the conclusion in the case of $t=\varepsilon$ can be obtained by the embedding theorem for the fractional Sobolev space (see, for instance, [Reference Leoni29, Theorems 7.32 and 7.34]) and the estimate (1.4) in the case of $t\in(\varepsilon,\min\{1+\varepsilon,2s\})$.

Now, we assume $t\in(\varepsilon,\min\{1+\varepsilon,2s\})$ and show (1.4) by considering the following three cases on t.

Case 1) $t\in(\varepsilon,\min\{1,2s\})$. In this case, we first estimate the seminorm $[u]_{W^{t,p}({{{\mathbb R}}^n})}$. Let

\begin{equation*} D:=\{(x,y)\in\Omega\times\Omega:\ \delta(x)\le\delta(y)\}. \end{equation*}

By symmetry, we find that

(3.10)\begin{equation} \iint_{\Omega\times\Omega}\frac{|u(x)-u(y)|^p}{|x-y|^{n+tp}}\,dydx=2\iint_{D}\frac{|u(x)-u(y)|^p}{|x-y|^{n+tp}}\,dydx \end{equation}

and

(3.11)\begin{equation} \iint_{[\Omega\times\Omega^c]\cup[\Omega^c\times\Omega]}\frac{|u(x)-u(y)|^p}{|x-y|^{n+tp}}\,dydx =2\iint_{\Omega\times\Omega^c}\frac{|u(x)-u(y)|^p}{|x-y|^{n+tp}}\,dydx. \end{equation}

Furthermore, let

\begin{equation*} A:=\left\{(x,y)\in D:\ |x-y| \lt \frac{\delta(x)}{2}\right\}\ \ \text{and}\ \ B:=[D\setminus A]\cup[\Omega\times\Omega^c]. \end{equation*}

Then, using (3.10) and (3.11), we can rewrite $[u]^p_{W^{t,p}({{{\mathbb R}}^n})}$ as

(3.12)\begin{align} [u]^p_{W^{t,p}({{{\mathbb R}}^n})}&=2\iint_{A}\frac{|u(x)-u(y)|^p}{|x-y|^{n+tp}}\,dydx+2\iint_{B}\frac{|u(x)-u(y)|^p}{|x-y|^{n+tp}}\,dydx\nonumber\\ &=:\mathrm{I}_1+\mathrm{I}_2. \end{align}

For the term $\mathrm{I}_2$, using the Hölder estimate (3.1) and the estimate (3.9), we obtain that

(3.13)\begin{align} \mathrm{I}_2&\le C\|f\|^p_{L^q(\Omega)}\int_{\Omega}\int_{[B(x,\delta(x)/2)]^c}|x-y|^{-n-tp+\varepsilon p}\,dydx \nonumber\\ &\le \frac{C}{p(t-\varepsilon)}\|f\|^p_{L^q(\Omega)}\int_{\Omega}[\delta(x)]^{p(-t+\varepsilon)}\,dx \nonumber\\ &\le \frac{C}{p(t-\varepsilon)(1-p(t-\varepsilon))}\|f\|^p_{L^q(\Omega)}. \end{align}

To deal with $\mathrm{I}_1$, taking ${\gamma}\in(t,2s-\frac{n}{q}]$ and using (3.4) and (3.9), we get

\begin{align*} \mathrm{I}_1&\le C\|f\|^p_{L^q(\Omega)}\int_{\Omega}[\delta(x)]^{p(\varepsilon-{\gamma})}\int_{B(x,\delta(x)/2)}|x-y|^{-n-(t-{\gamma})p}\,dydx \nonumber\\ &\le \frac{C}{p({\gamma}-t)}\|f\|^p_{L^q(\Omega)}\int_{\Omega}[\delta(x)]^{p(-t+\varepsilon)}\,dx \nonumber\\ &\le \frac{C}{p({\gamma}-t)(1-p(t-\varepsilon))}\|f\|^p_{L^q(\Omega)}. \end{align*}

From this, (3.12), and (3.13), we deduce that

(3.14)\begin{equation} [u]^p_{W^{t,p}({{{\mathbb R}}^n})}\le \frac{C({\gamma}-\varepsilon)}{p(t-\varepsilon)({\gamma}-t)(1-p(t-\varepsilon))}\|f\|^p_{L^q(\Omega)}. \end{equation}

Moreover, by (3.7), the fact that $u\equiv0$ in $\Omega^c$, and the assumption that Ω is bounded, we conclude that $u\in L^m({{{\mathbb R}}^n})$ for any $m\in(1,\infty]$ and

(3.15)\begin{equation} \|u\|_{L^m({{{\mathbb R}}^n})}\le C\|f\|_{L^q(\Omega)}. \end{equation}

This, combined with (3.14), further implies that $u\in W^{t,p}({{{\mathbb R}}^n})$ and the estimate (1.4) holds in this case.

Case 2) t = 1. In this case, take $\delta\in(0,(1-s)/2)$. Let $t_\delta:=1-\delta/2$ and ${\gamma}_\delta:=1+\delta/2$. Then, from (3.14), it follows that

\begin{equation*} [u]^p_{W^{t_\delta,p}({{{\mathbb R}}^n})}\le \frac{C({\gamma}_\delta-\varepsilon)}{p\delta(t_\delta-\varepsilon)(1-p(t_\delta-\varepsilon))} \|f\|^p_{L^q(\Omega)}, \end{equation*}

which implies that

\begin{equation*}\liminf_{\delta\to0}\,\delta[u]^p_{W^{t_\delta,p}({{{\mathbb R}}^n})}\le \frac{C(1-\varepsilon)}{p(1-\varepsilon)(1-p(1-\varepsilon))}\|f\|^p_{L^q(\Omega)}. \end{equation*}

Thus, applying the Bourgain–Brezis–Mironescu formula (see, for instance, [Reference Bourgain, Brezis and Mironescu13, Reference Bourgain, Brezis and Mironescu14] and [Reference Leoni29, Theorem 7.39]), we conclude that $|\nabla u|\in L^p({{{\mathbb R}}^n})$ and $\|\nabla u\|_{L^p({{{\mathbb R}}^n})}\lesssim \|f\|_{L^q(\Omega)}$, which, combined with (3.15), further implies that $u\in W^{1,p}({{{\mathbb R}}^n})$ and the estimate (1.4) holds in this case.

Case 3) $t\in(1,\min\{1+\varepsilon,2s\})$. In this case, the assumption $2s \gt 1$ is automatic, which implies $s\in(1/2,1)$. By the proof in Case 2), we find that $u\in W^{1,p}({{{\mathbb R}}^n})$ and $\|u\|_{W^{1,p}({{{\mathbb R}}^n})}\lesssim\|f\|_{L^q(\Omega)}$. Now, we show that $\nabla u\in W^{t-1,p}({{{\mathbb R}}^n})$. Similarly to (3.12), we have

(3.16)\begin{align} [\nabla u]^p_{W^{t-1,p}({{{\mathbb R}}^n})} &=2\iint_{A}\frac{|\nabla u(x)-\nabla u(y)|^p}{|x-y|^{n+(t-1)p}}\,dydx+2\iint_{B}\frac{|\nabla u(x)-\nabla u(y)|^p}{|x-y|^{n+(t-1)p}}\,dydx \nonumber\\ &=:\mathrm{I}_3+\mathrm{I}_4. \end{align}

For $\mathrm{I}_3$, using the Hölder continuity estimate on $\nabla u$ given in (3.5) with some ${\gamma}\in(t,2s-\frac{n}{q}]$ and (3.9), we find that

(3.17)\begin{align} \mathrm{I}_3&\le C\|f\|^p_{L^q(\Omega)}\int_{\Omega}[\delta(x)]^{p(\varepsilon-{\gamma})}\int_{B(x,\delta(x)/2)}|x-y|^{-n-(t-{\gamma})p}\,dydx \nonumber\\ &\le \frac{C}{p({\gamma}-t)}\|f\|^p_{L^q(\Omega)}\int_{\Omega}[\delta(x)]^{p(-t+\varepsilon)}\,dx \nonumber\\ &\le \frac{C}{p({\gamma}-t)(1-p(t-\varepsilon))}\|f\|^p_{L^q(\Omega)}. \end{align}

To estimate $\mathrm{I}_4$, by the formula $u(x)=\int_{\Omega}G_s(x,y)f(y)\,dy$, (2.2), (2.3), and (3.2), we conclude that, for any $x\in\Omega$,

(3.18)\begin{align} |\nabla u(x)| &\lesssim\left|\int_{\Omega}\nabla_{x}G_{s}(x,y)f(y)\,dy\right| \nonumber\\ &\lesssim\int_{\Omega\setminus B_{\delta(x)}(x)}\frac{1}{|x-y|^{n-2s+\varepsilon}}[\delta(x)]^{\varepsilon-1}|f(y)|\,dy \nonumber\\ &\quad+\int_{B_{\delta(x)}(x)}\frac{1}{|x-y|^{n-2s+1}}|f(y)|\,dy =:E+F. \end{align}

From the assumptions $q \gt \frac{n}{2s-t}$ and $t\in(1,\min\{1+\varepsilon,2s\})$, we deduce that $q'\in[1,\frac{n}{n-2s+t})$ and $q'(n-2s+\varepsilon) \lt n$, which, together with Hölder’s inequality, implies that

(3.19)\begin{align} E\le[\delta(x)]^{\varepsilon-1}\|f\|_{L^q(\Omega)}\left[\int_\Omega\frac{1}{|x-y|^{(n-2s+\varepsilon)q'}}\,dy\right]^{\frac1{q'}}\lesssim[\delta(x)]^{\varepsilon-1}\|f\|_{L^q(\Omega)}. \end{align}

Meanwhile, by $q'\in[1,\frac{n}{n-2s+t})$ and t > 1, we find that $n-2s+1 \lt n/q'$, which, combined with Hölder’s inequality, the assumption that Ω is bounded, and $\varepsilon \lt s \lt 1$, implies that

\begin{align*} F&\le\|f\|_{L^q(\Omega)}\left[\int_{B_{\delta(x)}(x)}\frac{1}{|x-y|^{(n-2s+1)q'}}\,dy\right]^{\frac1{q'}} \nonumber \\ &\lesssim[\delta(x)]^{\frac{n}{q'}-(n-2s+1)} \|f\|_{L^q(\Omega)}\lesssim[\delta(x)]^{\varepsilon-1}\|f\|_{L^q(\Omega)}. \end{align*}

From this, (3.18), and (3.19), it follows that, for any $x\in\Omega$,

\begin{equation*}|\nabla u(x)|\lesssim\|f\|_{L^q(\Omega)}[\delta(x)]^{\varepsilon-1},\end{equation*}

which further implies that, for any $(x,y)\in D\setminus A$,

\begin{equation*} |\nabla u(x)- \nabla u(y)|\le|\nabla u(x)|+|\nabla u(y)|\lesssim[\delta(x)]^{\varepsilon-1}\|f\|_{L^q(\Omega)}. \end{equation*}

This, together with the fact that $\nabla u\equiv0$ in ${{{\mathbb R}}^n}\setminus\overline{\Omega}$, implies that

(3.20)\begin{align} \iint_{B}\frac{|\nabla u(x)-\nabla u(y)|^p}{|x-y|^{n+(t-1)p}}\,dydx &\le C\|f\|^p_{L^q(\Omega)}\int_{\Omega}[\delta(x)]^{p(\varepsilon-1)}\int_{B(x,\delta(x)/2)^c}\nonumber\\ &\quad|x-y|^{-n-(t-1)p}\,dydx \nonumber\\ &\le \frac{C}{p(t-1)}\|f\|^p_{L^q(\Omega)}\int_{\Omega}[\delta(x)]^{p(\varepsilon-t)}\,dx \nonumber\\ &\le \frac{C}{p(t-1)(1-p(t-\varepsilon))}\|f\|^p_{L^q(\Omega)}. \end{align}

Therefore, collecting the estimates (3.16), (3.17), and (3.20), we deduce that

\begin{equation*}[\nabla u]_{W^{t-1,p}({{{\mathbb R}}^n})}\lesssim\|f\|_{L^q(\Omega)},\end{equation*}

which, combined with $\|u\|_{W^{1,p}({{{\mathbb R}}^n})}\lesssim\|f\|_{L^q(\Omega)}$, further implies that $u\in W^{t,p}({{{\mathbb R}}^n})$ and $\|u\|_{W^{t,p}({{{\mathbb R}}^n})}\lesssim\|f\|_{L^q(\Omega)}$. This finishes the proof of (1.4) in this case and hence of Theorem 1.5.

4. Proof of Theorem 1.7

In this section, we give the proof of Theorem 1.7. We begin with the following pointwise estimates and Hölder continuity estimates for the Green function G_s.

Lemma 4.1. Let $s\in(0,1)$, $n\ge2$, $\Omega\subset{{{\mathbb R}}^n}$ be a bounded Lipschitz domain, and let G_s be the Green function as in (1.2). Then, for any given $\delta\in(0,s)$, there exists a positive constant M ₀ such that, if the Lipschitz constant M of Ω satisfies $M\le M_0$, then, for any $x,y\in\Omega$ with x ≠ y,

(4.1)\begin{equation} 0\le G_s(x,y)\le C\min\left\{\frac{1}{|x-y|^{n-2s}},\frac{[\delta(x)]^{\delta}}{|x-y|^{n-2s+\delta}},\frac{[\delta(y)]^{\delta}}{|x-y|^{n-2s+\delta}}\right\}, \end{equation}

where C is a positive constant depending only on n, s, δ, and Ω. In particular, when Ω is a bounded C ¹ domain in ${{{\mathbb R}}^n}$, the estimate (4.1) holds for any $\delta\in(0,s)$. Moreover, when $\Omega\subset{{{\mathbb R}}^n}$ is a bounded $C^{1,\kappa}$ domain with some $\kappa\in(0,s)$ or a bounded Lipschitz domain satisfying the uniform exterior ball condition, the estimate (4.1) holds for $\delta=s$.

When the Lipschitz constant M of Ω is small enough, the estimate (4.1) with any given $\delta\in(0,s)$ was established in [Reference Kim and Weidner26, Lemma 9.3 and Remark 9.4]. When $\Omega\subset{{{\mathbb R}}^n}$ is a bounded $C^{1,\kappa}$ domain with some $\kappa\in(0,s)$, the estimate (4.1) with $\delta=s$ was obtained in [Reference Kim and Weidner26, Theorem 1.1 or Remark 9.4]. When $\Omega\subset{{{\mathbb R}}^n}$ is a bounded Lipschitz domain satisfying the uniform exterior ball condition, the estimate (4.1) with $\delta=s$ was given in [Reference Liu and Zhuo30, Proposition 4.4 and Remark 8].

Using the pointwise estimate for G_s in Lemma 4.1 and repeating the proof of [Reference Ma and Yang31, Lemma 3.1], we have the following Hölder estimate for the Green function G_s.

Lemma 4.2. Let $s\in(0,1)$, $n\ge2$, $\Omega\subset{{{\mathbb R}}^n}$ be a bounded $C^{1,\kappa}$ domain with some $\kappa\in(0,s)$ or a bounded Lipschitz domain satisfying the uniform exterior ball condition, and let G_s be the Green function as in (1.2). Then there exists a positive constant C, depending only on n, s, and Ω, such that, for any $x,y_1,y_2\in{{{\mathbb R}}^n}$,

\begin{equation*} |G_s(x,y_1)-G_s(x,y_2)|\le C|y_1-y_2|^{s}\left[\frac{1}{|x-y_1|^{n-s}}+\frac{1}{|x-y_2|^{n-s}}\right]. \end{equation*}

To show Theorem 1.7, we need the following estimate for the distance function in bounded Lipschitz domains. In particular, when $\Omega\subset{{{\mathbb R}}^n}$ is a bounded C ² domain, the conclusion of the following Lemma 4.3 was obtained in [Reference Abdellaoui, Fernández, Leonori and Younes2, Lemma 2.4].

Lemma 4.3. Let $n\ge2$, $\lambda,a\in(0,1)$, and $\Omega\subset{{{\mathbb R}}^n}$ be a bounded Lipschitz domain. Then there exists a positive constant C, depending only on n, Ω, λ, and a, such that, for any $x\in\Omega$,

\begin{equation*} \int_\Omega\frac{1}{|x-y|^{n-\lambda}[\delta(y)]^a}\,dy \le C\left[1+\omega_{a,\lambda}(x)\right], \end{equation*}

where

\begin{align*} \omega_{a,\lambda}(x):= \begin{cases} 1\ \ \ \ &\text{if}\ \ \lambda \gt a,\\ \left|\ln\delta(x)\right| &\text{if}\ \ \lambda=a,\\ [\delta(x)]^{\lambda-a}\ \ \ \ &\text{if}\ \ \lambda \lt a. \end{cases} \end{align*}

Proof. Fix $x\in\Omega$. Then we have

(4.2)\begin{align} \int_\Omega\frac{1}{|x-y|^{n-\lambda}[\delta(y)]^a}\,dy &=\int_{B_{{\delta(x)}/{2}}(x)} \frac{1}{|x-y|^{n-\lambda}[\delta(y)]^a}\,dy \nonumber\\ &\quad+\int_{\Omega\setminus B_{{\delta(x)}/{2}}(x)} \frac{1}{|x-y|^{n-\lambda}[\delta(y)]^a}\,dy =:\mathrm{I}_1+\mathrm{I}_2. \end{align}

From the fact that, for any $y\in B_{{\delta(x)}/{2}}(x)$, $\delta(y)\ge{\delta(x)}/{2}$, it follows that

(4.3)\begin{align} \mathrm{I}_1\le\frac{2^a}{[\delta(x)]^a} \int_{B_{{\delta(x)}/{2}}(x)}\frac{dy}{|x-y|^{n-\lambda}} \lesssim[\delta(x)]^{\lambda-a}\lesssim\omega_{a,\lambda}(x). \end{align}

Moreover, take a point $x_0\in\partial\Omega$ such that

(4.4)\begin{equation} \delta(x)=|x-x_0|. \end{equation}

By Definition 2.1 and Remark 2.2(i), we conclude that there exist a Lipschitz function ψ, positive constants α and β, and a cylinder $S_{\alpha,\,\beta}(x_0)$ such that (2.1) holds, where the positive constants α and β are independent of x ₀.

For $\mathrm{I}_2$, we first assume that $x\notin S_{\alpha/2,\,\beta/2}(x_0)$. In this case, from (4.4), it follows that $\delta(x)\ge\mu:=\min\{\alpha/2,\beta/2\}$. By this, we find that, for any $y\in\Omega\setminus B_{\delta(x)/2}(x)$, $|y-x|\ge\delta(x)/2\ge\mu/2$, which implies that

(4.5)\begin{equation} \mathrm{I}_2\le\frac{2^{n-\lambda}}{\mu^{n-\lambda}} \int_\Omega\frac{dy}{[\delta(y)]^{a}}\lesssim1. \end{equation}

Next, we assume that $x\in S_{\alpha/2,\,\beta/2}(x_0)$. In this case, we have

(4.6)\begin{align} \mathrm{I}_2 &\le\int_{\Omega\setminus S_{\alpha,\,\beta}(x_0)}\frac{1}{|x-y|^{n-\lambda}[\delta(y)]^a}\,dy +\int_{S_{\alpha,\,\beta}(x_0)}\frac{1}{|x-y|^{n-\lambda}[\delta(y)]^a}\,dy\nonumber\\ &=:\mathrm{I}_{2,1}+\mathrm{I}_{2,2}. \end{align}

For $\mathrm{I}_{2,1}$, from the fact that, for any $y\in\Omega\setminus S_{\alpha,\,\beta}(x_0)$, $|y-x|\ge\mu$, we deduce that

(4.7)\begin{equation} \mathrm{I}_{2,1}\le\frac{1}{\mu^{n-\lambda}}\int_\Omega\frac{1}{[\delta(y)]^a}\,dy\lesssim1. \end{equation}

Let M be as in Remark 2.2(i). For $\mathrm{I}_{2,2}$, by Definition 2.1, we find that there exists a constant $c\in[1,c_0]$ with $c_0\in(1,\infty)$ depending only on M such that

\begin{align*} \mathrm{I}_{2,2} &\lesssim\int_0^\beta\int_{B'_\alpha(x_0)}\frac{1}{|x-(z',\psi(z')+\rho)|^{n-\lambda}[\delta(\psi(z')+\rho)]^a}\,dz'd\rho\\ &\lesssim\int_0^\beta\int_{B'_\alpha(x_0)}\frac{1}{|(x',\psi(x')+c\delta(x))-(z',\psi(z')+\rho)|^{n-\lambda}\rho^a}\,dz'd\rho\\ &\lesssim\int_0^\beta\int_{B'_\alpha(x_0)}\frac{1}{[(M+1)|x'-z'|+|\psi(x')-\psi(z')+c\delta(x)-\rho|]^{n-\lambda}\rho^a}\,dz'd\rho, \end{align*}

where x' is as in Definition 2.1 and $B'_\alpha(x'_0):=\{z'\in{\mathbb R}^{n-1}:\ |z'-x'_0| \lt \alpha\}$, which, together with the fact that, for any $y,z\in B'_\alpha(x'_0)$,

\begin{equation*} |y'-z'|\ge\frac{1}{M}|\psi(y')-\psi(z')| \end{equation*}

and

\begin{equation*} |\psi(y')-\psi(z')+c\delta(x)-\rho|\ge|\rho-c\delta(x)|-|\psi(y')-\psi(z')|, \end{equation*}

further implies that

\begin{equation*} \mathrm{I}_{2,2} \lesssim\int_0^\beta\int_{B'_\alpha(x'_0)}\frac{1}{[|\rho-c\delta(x)|+|x'-z'|]^{n-\lambda}\rho^a}\,dz'd\rho. \end{equation*}

From this and the fact that, for any $z'\in B'_\alpha(x'_0)$, $z'\in B'_{2\alpha}(x')$, it follows that

(4.8)\begin{align} \mathrm{I}_{2,2} &\lesssim\int_0^\beta\int_{B'_{2\alpha}(x')}\frac{1}{[|\rho-c\delta(x)|+|x'-z'|]^{n-\lambda}\rho^a}\,dz'd\rho\nonumber\\ &\lesssim\int_0^\beta\int_0^{2\alpha}\int_{\partial B'_r(x')}\frac{1}{[|\rho-c\delta(x)|+r]^{n-\lambda}\rho^a}\,d\sigma drd\rho\nonumber\\ &\lesssim\int_0^\beta\int_0^{2\alpha}\frac{r^{n-2}}{[|\rho-c\delta(x)|+r]^{n-\lambda}\rho^a}\,drd\rho\nonumber\\ &\lesssim\int_0^\beta\frac{1}{|\rho-c\delta(x)|^{1-\lambda}\rho^a}\,d\rho\int_0^\infty\frac{r^{n-2}}{(1+r)^{n-\lambda}}\,dr\nonumber\\ &\lesssim\int_0^{\beta/c}\frac{1}{|\rho-\delta(x)|^{1-\lambda}\rho^a}\,d\rho\nonumber\\ &\lesssim\int_0^{\delta(x)}\frac{1}{[\delta(x)-\rho]^{1-\lambda}\rho^a}\,d\rho+\int_{\delta(x)}^{\beta/c}\frac{1}{|\rho-\delta(x)|^{1-\lambda}\rho^a}\,d\rho\nonumber\\ &=:\mathrm{I}_{2,2,1}+\mathrm{I}_{2,2,2}, \end{align}

where $d\sigma$ denotes the $(n-2)$-dimensional surface measure. For $\mathrm{I}_{2,2,1}$, we have

(4.9)\begin{equation} \mathrm{I}_{2,2,1} \lesssim[\delta(x)]^{\lambda-a}\int_0^1\frac{1}{(1-\rho)^{1-\lambda}\rho^a}\,d\rho\lesssim1+\omega_{a,\lambda}(x). \end{equation}

For $\mathrm{I}_{2,2,2}$, by the fact that, for any $\rho \gt 2\delta(x)$, $\rho-\delta(x) \gt \rho/2$, we conclude that

\begin{align*} \mathrm{I}_{2,2,2} &\lesssim\int_{\delta(x)}^{2\delta(x)}\frac{1}{(\rho-\delta(x))^{1-\lambda}\rho^a}\,d\rho+\int_{2\delta(x)}^{\beta/c}\frac{1}{\rho^{1-\lambda+a}}\,d\rho\\ &\lesssim[\delta(x)]^{\lambda-a}\int_1^2\frac{1}{(\rho-1)^{1-\lambda}\rho^a}\,d\rho+1+\omega_{a,\lambda}(x)\\ &\lesssim1+\omega_{a,\lambda}(x). \end{align*}

This, together with (4.2), (4.3), (4.5), (4.6), (4.7), (4.8), and (4.9), further implies that the conclusion of Lemma 4.3 holds. This completes the proof of Lemma 4.3.

Furthermore, to prove Theorem 1.7(iii), we need the following Hölder type continuity estimates for the fractional Laplacian $(-\Delta)^{\frac{s}{2}}_xG_s(\cdot,y)$.

Lemma 4.4. Let $s\in(0,1)$, $n\ge2$, $\Omega\subset{{{\mathbb R}}^n}$ be a bounded $C^{1,\kappa}$ domain with some $\kappa\in(0,s)$ or a bounded Lipschitz domain satisfying the uniform exterior ball condition, and let G_s be the Green function as in (1.2). Assume that $r\in(0,\frac{1}{16}diam(\Omega))$, $x_0\in\Omega$, and $B_r(x_0)$ is a ball satisfying that $B_r(x_0)\subset\Omega$. Then there exists a positive constant C, independent of x ₀ and r, such that, for any $x\in B_r(x_0)$ and $y\in\Omega\setminus B_{4r}(x_0)$,

\begin{equation*} \left|(-\Delta)^{\frac{s}{2}}_xG_s(x,y)-(-\Delta)^{\frac{s}{2}}_xG_s(x_0,y)\right|\le C\frac{1+|\ln d(x)-\ln r|}{|y-x_0|^{n-s}}, \end{equation*}

where, for any $x\in B_r(x_0)$, $d(x):={\mathrm{\,dist\,}}(x,\partial B_r(x_0))$.

Lemma 4.5. Let $s\in(0,1)$, $n\ge2$, and let $\Omega\subset{{{\mathbb R}}^n}$ and G_s be as in Lemma 4.4. Assume further that $r\in(0,\infty)$, $x_0\in{{{\mathbb R}}^n}\setminus\Omega$, and $B_r(x_0)$ is a ball satisfying that $B_r(x_0)\subset{{{\mathbb R}}^n}\setminus\Omega$ and $\Omega\setminus B_{4r}(x_0)\neq\emptyset$. Then there exists a positive constant C, independent of x ₀ and r, such that, for any $x\in B_r(x_0)$ and $y\in\Omega\setminus B_{4r}(x_0)$,

\begin{equation*} \left|(-\Delta)^{\frac{s}{2}}_xG_s(x,y)-(-\Delta)^{\frac{s}{2}}_xG_s(x_0,y)\right|\le C\frac{1+|\ln d(x)-\ln r|}{|y-x_0|^{n-s}}, \end{equation*}

where, for any $x\in B_r(x_0)$, $d(x):={\mathrm{\,dist\,}}(x,\partial B_r(x_0))$.

Lemma 4.6. Let $s\in(0,1)$, $n\ge2$, and let $\Omega\subset{{{\mathbb R}}^n}$ and G_s be as in Lemma 4.4. Assume that $r\in(0, \min\{\alpha/4,\beta/4, r_0\})$, $x_0\in\partial\Omega$, and $B_r(x_0)$ is a ball satisfying that $\Omega\setminus B_{4r}(x_0)\neq\emptyset$, where α and β are as in Remark 2.2(i). Then there exists a positive constant C, independent of x ₀ and r, such that, for any $x\in B_r(x_0)$ and $y\in\Omega\setminus B_{4r}(x_0)$,

\begin{equation*} \left|(-\Delta)^{\frac{s}{2}}_xG_s(x,y)-(-\Delta)^{\frac{s}{2}}_xG_s(z_0,y)\right|\le C\frac{1+|\ln\delta(x)-\ln r|}{|y-x_0|^{n-s}}, \end{equation*}

where $z_0\in B_r(x_0)\setminus\Omega$ is a corkscrew point with respect to x ₀ as in Remark 2.2(ii).

Using Lemmas 4.1, 4.2, and 4.3, and repeating the proofs of [Reference Ma and Yang31, Lemmas 3.5 and 3.8], we can prove Lemmas 4.4 and 4.5; we omit the details here. In order to prove Lemma 4.6, we need the following calculus in Sobolev spaces (see, for instance, [Reference Abdellaoui, Fernández, Leonori and Younes2, Lemma 2.5]).

Lemma 4.7. Let $r\in(0,\infty)$, $x\in{{{\mathbb R}}^n}$, and $f\in W^{1,1}(B_r(x))$. Then, for almost every $y,z\in B_r(x)$,

\begin{equation*} f(y)-f(z)=\int^1_0(y-z)\cdot\nabla f(\tau y+(1-\tau)z)\,d\tau. \end{equation*}

Proof of Lemma 4.6

By the fact that, for any $z\in{{{\mathbb R}}^n}\setminus\Omega$ and $y\in{{{\mathbb R}}^n}$, $G_s(z,y)=0$, we find that

(4.10)\begin{align} (-\Delta)^{\frac{s}{2}}_xG_s(x,y)-(-\Delta)^{\frac{s}{2}}_xG_s(z_0,y) &=\int_{B_{\sigma r}(0)}\frac{G_s(x,y)-G_s(x+z,y)}{|z|^{n+s}}\,dz\nonumber\\ &\quad+\int_{{{{\mathbb R}}^n}\setminus B_{\sigma r}(0)}\frac{G_s(x,y)-G_s(z_0,y)}{|z|^{n+s}}\,dz\nonumber\\ &\quad+\int_{{{{\mathbb R}}^n}\setminus B_{\sigma r}(0)}\frac{G_s(z_0+z,y)-G_s(x+z,y)}{|z|^{n+s}}\,dz\nonumber\\ &=:\mathrm{I}_1+\mathrm{I}_2+\mathrm{I}_3. \end{align}

For $\mathrm{I}_1$, if $x\in B_r(x_0)\setminus\Omega$, from Lemma 4.2 and the fact that, for any $z\in B_{\sigma r}(0)$ and $y\in\Omega\setminus B_{4r}(x_0)$,

\begin{equation*} |y-x|\sim|y-x-z|\sim|y-x_0|, \end{equation*}

it follows that

(4.11)\begin{align} |\mathrm{I}_1| &=\left|\int_{\{z\in B_{\sigma r(0)}:\ |z|\ge\delta(x)\}}\frac{G_s(x,y)-G_s(x+z,y)}{|z|^{n+s}}\,dz\right|\nonumber\\ &\lesssim\int_{\{z\in B_{\sigma r}(0):\ \delta(x)\le|z|\le r\}}\frac{1}{|z|^n|y-x_0|^{n-s}}\,dz\lesssim\frac{|\ln\delta(x)-\ln r|}{|y-x_0|^{n-s}}. \end{align}

From (4.1), (2.3), and the fact that, for any $z\in B_{2r}(x_0)$ and $y\in\Omega\setminus B_{4r}(x_0)$, $|y-z|\ge|y-x_0|-|z-x_0|\ge2r$, we deduce that

\begin{equation*} \nabla_xG_s(\cdot,y)\in L^1(B_{2r}(x_0)). \end{equation*}

By this and Lemma 4.7, we conclude that, for any $x\in B_r(x_0)\cap\Omega$, $z\in B_r(0)$, and $y\in\Omega\setminus B_{4r}(x_0)$,

(4.12)\begin{equation} G_s(x,y)-G_s(x+z,y)=\int^1_0z\cdot\nabla_x G_s(x+\tau z,y)\,d\tau. \end{equation}

Let $\theta_{\tau}:=x+\tau z$, $I_{\Omega}:=\{\tau\in[0,1]:\theta_{\tau}\in\Omega\}$, and $\Omega_{\tau,x}:=\{z\in{{{\mathbb R}}^n}:\theta_{\tau}\in\Omega\}$. Then, by (4.12), (4.1), (2.3), and the fact that, for any $x+\tau z\in\Omega$,

\begin{equation*} |y-x-\tau z|\ge2r\ge\delta(x+\tau z), \end{equation*}

we conclude that

(4.13)\begin{align} |\mathrm{I}_1|=&\left|\int_{B_r(0)}\int^1_0z\cdot\nabla_xG_s(x+\tau z,y)\frac{1}{|z|^{n+s}}\,d\tau dz\right|\nonumber \\ \lesssim&\int_{B_r(0)}\int_{I_{\Omega}}\frac{1}{|z|^{n+s-1}[\delta(x+\tau z)]^{1-s}|y-x-\tau z|^{n-s}}\,d\tau dz \nonumber\\ \lesssim&\int^1_0\int_{\Omega_{\tau,x}\cap B_r(0)}\frac{1}{|z|^{n+s-1}[\delta(x+\tau z)]^{1-s}|y-x-\tau z|^{n-s}}\,dzd\tau. \end{align}

Furthermore, for any $\tau\in[0,1]$, we have that $x+\tau z\in B_{2r}(x_0)$ and $|y-x-\tau z|\sim|y-x_0|$. From this, a change of variable, and (4.13), we deduce that

(4.14)\begin{align} |\mathrm{I}_1| &\lesssim\frac{1}{|y-x_0|^{n-s}}\int^1_0\int_{B_{2r}(x_0)\cap\Omega}\frac{\tau^{s-1}}{|\omega-x|^{n+s-1}[\delta(\omega)]^{1-s}}\,d\omega d\tau \nonumber\\ &\lesssim\frac{1}{|y-x_0|^{n-s}}\int_{B_{2r}(x_0)\cap\Omega}\frac{1}{|\omega-x|^{n+s-1}[\delta(\omega)]^{1-s}}\,d\omega\nonumber\\ &\lesssim\frac{1}{|y-x_0|^{n-s}}\int^{2r}_0\int_{B'_{2r}(x'_0)}\frac{1}{|(z',\psi(z')+\rho)-x|^{n+s-1}[\delta(\psi(z')+\rho)]^{1-s}}\,dz'd\rho, \end{align}

where $B'_{\alpha}(x'_0):=\{z'\in{\mathbb R}^{n-1}:\ |z'-x'_0| \lt \alpha\}$.

Moreover, by Definition 2.1 and (4.14), we find that there exists a constant c depending only on M such that

\begin{align*} |\mathrm{I}_1| &\lesssim\frac{1}{|y-x_0|^{n-s}}\int^{2r}_0\int_{B'_{2r}(x'_0)}\nonumber\\ &\quad\,\frac{1}{|(z',\psi(z')+\rho)-(x',\psi(x')+c\delta(x))|^{n+s-1}[\delta(\psi(z')+\rho)]^{1-s}}\,dz'd\rho\nonumber\\ &\lesssim\frac{1}{|y-x_0|^{n-s}}\int^{2r}_0\int_{B'_{2r}(x'_0)}\nonumber\\ &\quad\,\frac{1}{|(z',\psi(z')+\rho)-(x',\psi(x')+c\delta(x))|^{n+s-1}\rho^{1-s}}\,dz'd\rho\nonumber\\ &\lesssim\frac{1}{|y-x_0|^{n-s}}\int^{2r}_0\int_{B'_{2r}(x'_0)}\nonumber\\ &\quad\,\frac{1}{[(M+1)|z'-x'|+|\psi(z')-\psi(x')+\rho-c\delta(x)|]^{n+s-1}\rho^{1-s}}\,dz'd\rho, \end{align*}

which, together with the fact that

\begin{equation*} |\psi(x')-\psi(z')|\le M|z'-x'| \end{equation*}

and

\begin{equation*} |\psi(z')-\psi(x')+\rho-c\delta(x)|\ge|\rho-c\delta(x)|-|\psi(x')-\psi(z')|, \end{equation*}

further implies that

\begin{align*} |\mathrm{I}_1| \lesssim&\frac{1}{|y-x_0|^{n-s}}\int^{2r}_0\int_{B'_{2r}(x'_0)}\frac{1}{[|\rho-c\delta(x)|+|z'-x'|]^{n+s-1}\rho^{1-s}}\,dz'd\rho. \end{align*}

From this and the fact that, for any $z'\in B'_{2r}(x'_0)$ and $x'\in B'_r(x'_0)$, we have $z'\in B'_{3r}(x')$, it follows that

\begin{align*} |\mathrm{I}_1| &\lesssim\frac{1}{|y-x_0|^{n-s}}\int^{2r}_0\int_{B'_{3r}(x')}\frac{1}{[|\rho-c\delta(x)|+|z'-x'|]^{n+s-1}\rho^{1-s}}\,dz'd\rho\nonumber\\ &\lesssim\frac{1}{|y-x_0|^{n-s}}\int^{2r}_0\int^{3r}_0\int_{\partial B'_{\tau}(x')}\frac{1}{[|\rho-c\delta(x)|+\tau]^{n+s-1}\rho^{1-s}}\,d\sigma d\tau d\rho \nonumber\\ &\lesssim\frac{1}{|y-x_0|^{n-s}}\int^{2r}_0\int^{3r}_0\frac{\tau^{n-2}}{[|\rho-c\delta(x)|+\tau]^{n+s-1}\rho^{1-s}}\,d\tau d\rho\nonumber\\ &\lesssim\frac{1}{|y-x_0|^{n-s}}\int^{2r}_0\frac{1}{|\rho-c\delta(x)|^s\rho^{1-s}}\,d\rho\int^{\infty}_0\frac{\tau^{n-2}}{(1+\tau)^{n+s-1}}\,d\tau\nonumber\\ &\lesssim\frac{1}{|y-x_0|^{n-s}}\int^{2r/c}_0\frac{1}{|\rho-\delta(x)|^s\rho^{1-s}}\,d\rho, \end{align*}

which, combined with the fact that, for any $\rho \gt 2\delta(x)$, $\rho-\delta(x) \gt \rho/2$, we conclude that, if $r/c \gt \delta(x)$,

(4.15)\begin{align} |\mathrm{I}_1| &\lesssim\frac{1}{|y-x_0|^{n-s}}\left\{\int^{\delta(x)}_0\frac{1}{[\delta(x)-\rho]^s\rho^{1-s}}\,d\rho +\int^{2\delta(x)}_{\delta(x)}\frac{1}{[\rho-\delta(x)]^s\rho^{1-s}}\,d\rho\right.\nonumber\\ &\quad\left. +\int^{2r/c}_{2\delta(x)}\frac{1}{[\rho-\delta(x)]^s\rho^{1-s}}\,d\rho\right\}\nonumber\\ &\lesssim\frac{1}{|y-x_0|^{n-s}}\left\{\int^1_0\frac{1}{(1-\rho)^s\rho^{1-s}}\,d\rho +\int^2_1\frac{1}{(\rho-1)^s\rho^{1-s}}\,d\rho +\int^{2r/c}_{2\delta(x)}\frac{1}{\rho}\,d\rho\right\}\nonumber\\ &\lesssim\frac{1+|\ln\delta(x)-\ln\frac{r}{c}|}{|y-x_0|^{n-s}}; \end{align}

otherwise, if $r/c\le\delta(x)$, then

\begin{align*} |\mathrm{I}_1| &\lesssim\frac{1}{|y-x_0|^{n-s}}\int^{2\delta(x)}_0\frac{1}{|\delta(x)-\rho|^s\rho^{1-s}}\,d\rho\nonumber\\ &\lesssim\frac{1}{|y-x_0|^{n-s}}\int^2_0\frac{1}{|1-\rho|^s\rho^{1-s}}\,d\rho \lesssim\frac{1}{|y-x_0|^{n-s}}. \end{align*}

This, together with (4.11) and (4.15), further implies that, for any $x\in B_r(x_0)$,

(4.16)\begin{equation} |\mathrm{I}_1|\lesssim\frac{1+|\ln\delta(x)-\ln r|}{|y-x_0|^{n-s}}. \end{equation}

For $\mathrm{I}_2$, from Lemma 4.2 and the fact that, for any $x\in B_r(x_0)\cap\Omega$ and $y\in\Omega\setminus B_{4r}(x_0)$,

\begin{equation*} |y-z_0|\le2r \end{equation*}

and

\begin{equation*} |y-x|\sim|y-z_0|\sim|y-x_0|, \end{equation*}

we deduce that

(4.17)\begin{equation} |\mathrm{I}_2|\lesssim\frac{|x-z_0|^s}{|y-x_0|^{n-s}}\int_{{{{\mathbb R}}^n}\setminus B_{\sigma r}(0)}\frac{1}{|z|^{n+s}}\,dz\lesssim\frac{1}{|y-x_0|^{n-s}}. \end{equation}

For $\mathrm{I}_3$, by Lemma 4.2, we find that

\begin{equation*} |\mathrm{I}_3|\lesssim|x-z_0|^s\int_{{{{\mathbb R}}^n}\setminus B_{\sigma r}(0)}\frac{1}{|z|^{n+s}}\left[\frac{1}{|y-z_0-z|^{n-s}}+\frac{1}{|y-x-z|^{n-s}}\right]\,dz. \end{equation*}

Furthermore, from the fact that, for any $z\in{{{\mathbb R}}^n}$ satisfying $|y-x-z| \lt \frac{1}{2}|y-x|$,

\begin{equation*} |z|\ge|y-x|-|y-x-z|\ge\frac{1}{2}|y-x|, \end{equation*}

it follows that

(4.18)\begin{align} &\int_{{{{\mathbb R}}^n}\setminus B_{\sigma r}(0)}\frac{1}{|y-x-z|^{n-s}|z|^{n+s}}\,dz \nonumber\\ &=\int_{\{z\in{{{\mathbb R}}^n}\setminus B_{\sigma r}(0):\ |y-x-z|\ge\frac{1}{2}|y-x|\}}\frac{1}{|y-x-z|^{n-s}|z|^{n+s}}\,dz\nonumber\\ &\quad+\int_{\{z\in{{{\mathbb R}}^n}\setminus B_{\sigma r}(0):\ |y-x-z| \lt \frac{1}{2}|y-x|\}}\frac{1}{|y-x-z|^{n-s}|z|^{n+s}}\,dz \nonumber \\ & \lesssim\frac{1}{|y-x_0|^{n-s}}\int_{{{{\mathbb R}}^n}\setminus B_{\sigma r}(0)}\frac{1}{|z|^{n+s}}\,dz\nonumber\\ &\quad+\frac{1}{|y-x_0|^{n+s}}\int_{\{z\in{{{\mathbb R}}^n}\setminus B_{\sigma r}(0):\ |y-x-z| \lt \frac{1}{2}|y-x|\}}\frac{1}{|y-x-z|^{n-s}}\,dz\nonumber\\ &\lesssim\frac{1}{|y-x_0|^{n-s}r^s}+\frac{1}{|y-x_0|^n}. \end{align}

Similar to (4.18), we obtain

\begin{equation*} \int_{{{{\mathbb R}}^n}\setminus B_{\sigma r}(0)}\frac{1}{|y-x-z|^{n-s}|z|^{n+s}}\,dz\lesssim\frac{1}{|y-x_0|^{n-s}r^s}+\frac{1}{|y-x_0|^n}, \end{equation*}

which, combined with (4.18), further implies that

\begin{equation*} |\mathrm{I}_3|\lesssim\frac{1}{|y-x_0|^{n-s}}. \end{equation*}

By this, (4.10), (4.16), and (4.17), we conclude that the conclusion of Lemma 4.6 holds. This finishes the proof of Lemma 4.6.

For the space $\mathrm{BMO}({{{\mathbb R}}^n})$, we have the following basic property (see, for instance, [Reference Grafakos24, Proposition 3.1.2(4)]).

Proposition 4.8. Let $f\in L^1_{{\mathrm{loc}}}({{{\mathbb R}}^n})$. Then

\begin{equation*} \frac{1}{2}\|f\|_{{{\mathrm {BMO}}}({{{\mathbb R}}^n})}\le\sup_{B\subset{{{\mathbb R}}^n}}\inf_{c\in\mathbb{C}}\frac{1}{|B|}\int_{B}|f(x)-c|\,dx\le\|f\|_{{{\mathrm{BMO}}}({{{\mathbb R}}^n})}, \end{equation*}

where the supremum is taken over all balls B in ${{{\mathbb R}}^n}$ and the infimum is taken over all constants in $\mathbb{C}$.

Finally, by using Lemmas 4.1, 4.4, 4.5, and 4.6 and Proposition 4.8, we show Theorem 1.7.

Proof of Theorem 1.7

We first prove (i). Let $t\in[s,2s)$, $p\in[1,\frac{1}{t-s})$, and u be the weak solution of the problem (1.1) with $f\in L^q(\Omega)$ for some $q\in(\frac{n}{2s-t},\infty]$. For any given $\delta\in(0,s)$, by Lemma 4.1, we find that there exists a positive constant M ₀ such that, if the Lipschitz constant of Ω satisfies $M\le M_0$, then, for any $x,y\in\Omega$ with x ≠ y,

(4.19)\begin{equation} 0\le G_s(x,y)\le C\frac{[\delta(x)]^{\delta}}{|x-y|^{n-2s+\delta}}. \end{equation}

Replacing (3.2) by (4.19) and repeating the proof of Proposition 3.1, we conclude that, for any given $\delta\in(0,s)$, $u\in C^{\delta}({{{\mathbb R}}^n})$ and

(4.20)\begin{equation} \|u\|_{C^{\delta}({{{\mathbb R}}^n})}\lesssim\|f\|_{L^q(\Omega)}. \end{equation}

Then, applying (4.20) and similar to the proof of Theorem 1.5, we find that $u\in W^{t,p}({{{\mathbb R}}^n})$ and $\|u\|_{W^{t,p}({{{\mathbb R}}^n})}\lesssim\|f\|_{L^q(\Omega)}$. This finishes the proof of (i).

The proof of (ii) is similar to that of (i), and we omit the details.

Now, we show (iii). Let $q\in(n/s,\infty]$. We first prove

(4.21)\begin{equation} \left\|(-\Delta)^{\frac{s}{2}}u\right\|_{\mathrm{BMO}({{{\mathbb R}}^n})}\lesssim\|f\|_{L^q(\Omega)}. \end{equation}

From the estimate $\|u\|_{W^{t,p}({{{\mathbb R}}^n})}\lesssim\|f\|_{L^q(\Omega)}$ obtained in (ii) and (2.5), we infer that, for any $p\in(1,\infty)$, there exists a positive constant C depending only on n, s, q, and Ω such that

(4.22)\begin{equation} \left\|(-\Delta)^{\frac{s}{2}}u\right\|_{L^p({{{\mathbb R}}^n})}\le C\|f\|_{L^q(\Omega)}. \end{equation}

Using Proposition 4.8, to prove (4.21), it suffices to show that, for any $r\in(0,\infty)$ and $x_0\in{{{\mathbb R}}^n}$, there exists a constant c such that

(4.23)\begin{equation} \frac{1}{|B_r(x_0)|}\int_{B_r(x_0)}\left|(-\Delta)^{\frac{s}{2}}u(x)-c\right|\,dx\le C\|f\|_{L^q(\Omega)}, \end{equation}

where C is a positive constant independent of f, u, r, and x ₀.

Let $\widetilde{R}:=\frac{1}{2}\min\{\frac{{\mathrm{diam\,}}(\Omega)}{16},\frac{\alpha}{4},\frac{\beta}{4}\}$, where α and β are the same as in Definition 2.1. Next, we prove (4.23) by considering the following four cases on r and x ₀.

Case 1) $r\ge\widetilde{R}$. In this case, by (4.22), we find that, for any $x_0\in{{{\mathbb R}}^n}$,

\begin{align*} &\frac{1}{|B_r(x_0)|}\int_{B_r(x_0)}\left|(-\Delta)^{\frac{s}{2}}u(x)\right|\,dx \\ & \nonumber\quad\le|B_r(x_0)|^{-\frac{1}{p}}\left[\int_{B_r(x_0)}\left|(-\Delta)^{\frac{s}{2}}u(x)\right|^p\,dx\right]^{\frac{1}{p}} \le C\|f\|_{L^q(\Omega)}, \end{align*}

where C is a positive constant independent of f, u, r, and x ₀.

Case 2) $r \lt \widetilde{R}$ and $B_r(x_0)\subset\Omega$. In this case, let

\begin{equation*} c:=\int_{\Omega}(-\Delta)^{\frac{s}{2}}_xG_s(x_0,y)f(y)\mathbf{1}_{{{{\mathbb R}}^n}\setminus B_{4r}(x_0)}(y)\,dy. \end{equation*}

Then we derive that

(4.24)\begin{align} &\frac{1}{|B_r(x_0)|}\int_{B_r(x_0)}\left|(-\Delta)^{\frac{s}{2}}u(x)-c\right|\,dx\nonumber\\ &\quad=\frac{1}{|B_r(x_0)|}\int_{B_r(x_0)}\left|\int_{\Omega}(-\Delta)^{\frac{s}{2}}_xG_s(x,y)f(y)\,dy\right.\nonumber\\ &\qquad\left.-\int_{\Omega}(-\Delta)^{\frac{s}{2}}_xG_s(x_0,y)f(y)\mathbf{1}_{{{{\mathbb R}}^n}\setminus B_{4r}(x_0)}(y)\,dy\right|\,dx\nonumber\\ &\quad\le\frac{1}{|B_r(x_0)|}\int_{B_r(x_0)}\int_{\Omega}\left|(-\Delta)^{\frac{s}{2}}_xG_s(x,y)-(-\Delta)^{\frac{s}{2}}_xG_s(x_0,y)\right||f(y)|\nonumber\\ &\qquad\times\mathbf{1}_{{{{\mathbb R}}^n}\setminus B_{4r}(x_0)}(y)\,dydx\nonumber\\ &\qquad+\frac{1}{|B_r(x_0)|}\int_{B_r(x_0)}\left|\int_{\Omega}(-\Delta)^{\frac{s}{2}}_xG_s(x,y)f(y)\mathbf{1}_{B_{4r}(x_0)}(y)\,dy\right|\,dx\nonumber\\ &\quad=:\mathrm{I_1}+\mathrm{I_2}. \end{align}

For $\mathrm{I_1}$, from the assumption that $q\in(\frac{n}{s},\infty)$, Lemma 4.4, and Hölder’s inequality, it follows that

(4.25)\begin{align} \mathrm{I_1} &\lesssim\frac{1}{|B_r(x_0)|}\int_{B_r(x_0)}\int_{\Omega\setminus B_{4r}(x_0)}\frac{1+|\ln d(x)-\ln r|}{|y-x_0|^{n-s}}|f(y)|\,dydx\nonumber\\ &\lesssim\frac{1}{|B_r(x_0)|}\int_{B_r(x_0)}1+|\ln d(x)-\ln r|\,dx\int_{\Omega}\frac{|f(y)|}{|y-x_0|^{n-s}}\,dy\nonumber\\ &\lesssim\left[1+\frac{1}{|B_r(x_0)|}\int_{B_r(x_0)}|\ln d(x)-\ln r|\,dx\right]\left[\int_{B_{{\mathrm{diam\,}}(\Omega)}(x_0)}\frac{1}{|y-x_0|^{\frac{(n-s)q}{q-1}}}\,dy\right]^{1-\frac{1}{q}}\nonumber\\ &\quad\,\|f\|_{L^q(\Omega)}\lesssim\|f\|_{L^q(\Omega)}. \end{align}

Furthermore, we take $m\in(\frac{n}{s},q)$ and $p:=\frac{mq}{q-m}$. From $f\in L^q(\Omega)\subset L^m(\Omega)$, (4.22), and Hölder’s inequality, we deduce that

\begin{align*} \mathrm{I_2} &\lesssim|B_r(x_0)|^{-\frac{1}{p}}\left[\int_{B_r(x_0)}\left|\int_{\Omega}(-\Delta)^{\frac{s}{2}}_xG_s(x,y)f(y) \mathbf{1}_{B_{4r}(x_0)}(y)\,dy\right|^p\right]^{\frac{1}{p}}\nonumber\\ &\lesssim|B_r(x_0)|^{-\frac{1}{p}}\|f\|_{L^m(\Omega\cap B_{4r}(x_0))}\nonumber\\ &\lesssim|B_r(x_0)|^{-\frac{1}{p}}|B_{4r}(x_0)|^{\frac{1}{m}-\frac{1}{q}}\|f\|_{L^q(\Omega)}\lesssim\|f\|_{L^q(\Omega)}, \end{align*}

which, together with (4.24) and (4.25), further implies that (4.23) holds in this case.

Case 3) $r \lt \widetilde{R}$ and $B_r(x_0)\subset{{{\mathbb R}}^n}\setminus\Omega$. In this case, let

\begin{equation*} c:=\int_{\Omega}(-\Delta)^{\frac{s}{2}}_xG_s(x_0,y)f(y)\mathbf{1}_{{{{\mathbb R}}^n}\setminus B_{4r}(x_0)}(y)\,dy. \end{equation*}

Similar to Case 2), by Lemma 4.5, (4.22), and Hölder’s inequality, we conclude that (4.23) holds in this case.

Case 4) $r \lt \widetilde{R}$ and $B_r(x_0)\cap\partial\Omega\neq\emptyset$. In this case, we find that there exists $z\in\partial\Omega$ such that $B_r(x_0)\subset B_{2r}(z)$. Let $z_0\in B_{2r}(z)\setminus\Omega$ be a corkscrew point with respect to z as in Remark 2.2(ii) and

\begin{equation*} c:=\int_{\Omega}(-\Delta)^{\frac{s}{2}}_xG_s(z_0,y)f(y)\mathbf{1}_{{{{\mathbb R}}^n}\setminus B_{8r}(z)}(y)\,dy. \end{equation*}

Then, we have

(4.26)\begin{align} &\frac{1}{|B_r(x_0)|}\int_{B_r(x_0)}\left|(-\Delta)^{\frac{s}{2}}u(x)-c\right|\,dx\nonumber\\ &\quad\lesssim\frac{1}{|B_r(z)|}\int_{B_{2r}(z)}\left|(-\Delta)^{\frac{s}{2}}u(x)-c\right|\,dy\nonumber\\ &\quad\lesssim\frac{1}{|B_r(z)|}\int_{B_{2r}(z)}\int_{\Omega}\left|(-\Delta)^{\frac{s}{2}}_xG_s(x,y) -(-\Delta)^{\frac{s}{2}}_xG_s(z_0,y)\right| \nonumber \\ & \qquad\ |f(y)|\mathbf{1}_{{{{\mathbb R}}^n}\setminus B_{8r}(z)}(y)\,dydx +\frac{1}{|B_r(z)|} \int_{B_{2r}(z)}\nonumber \\ & \qquad\ \left|\int_{\Omega}(-\Delta)^{\frac{s}{2}}_xG_s(x,y)f(y)\mathbf{1}_{B_{8r}(z)}(y)\,dy\right|\,dx\nonumber\\ &\quad=:\mathrm{I_3}+\mathrm{I_4}. \end{align}

For $\mathrm{I_3}$, from Lemma 4.6 and Hölder’s inequality, we deduce that

(4.27)\begin{align} \mathrm{I_3} &\lesssim\frac{1}{|B_r(z)|}\int_{B_{2r}(z)}\int_{\Omega\setminus B_{8r}(z)}\frac{1+|\ln \delta(x)-\ln r|}{|y-z|^{n-s}}|f(y)|\,dydx\nonumber\\ &\lesssim\left[1+\frac{1}{|B_r(z)|}\int_{B_{2r}(z)}|\ln \delta(x)-\ln r|\,dx\right]\|f\|_{L^q(\Omega)}\nonumber\\ &\lesssim\left[1+\frac{1}{|B_r(z)|}\int^{2r}_0\int_{B'_{2r}(z')}|\ln \rho-\ln r|\,dz'd\rho\right]\|f\|_{L^q(\Omega)}\nonumber\\ &\lesssim\left[1+\frac{1}{r}\int^{2r}_0|\ln \rho-\ln r|\,d\rho\right]\|f\|_{L^q(\Omega)}\lesssim\|f\|_{L^q(\Omega)}. \end{align}

For $\mathrm{I_4}$, similar to the estimate of $\mathrm{I_2}$, we derive that $\mathrm{I_4}\lesssim\|f\|_{L^q(\Omega)},$ which, combined with (4.26) and (4.27), further implies that (4.23) holds in this case. This completes the proof of (4.21).

Next, we show

(4.28)\begin{equation} \left\|\nabla^s u\right\|_{\mathrm{BMO}({{{\mathbb R}}^n})}\lesssim\|f\|_{L^q(\Omega)}. \end{equation}

Notice that, for any $x,x_0,y\in{{{\mathbb R}}^n}$, and $r\in(0,\infty)$,

(4.29)\begin{align} \nabla^s_x&G_s(x,y)-\nabla^s_x G_s(x_0,y)\nonumber\\ &=-\int_{{{\mathbb R}}^n}\frac{z[G_s(x,y)-G_s(x+z,y)]}{|z|^{n+s+1}}\,dz +\int_{{{\mathbb R}}^n}\frac{z[G_s(x_0,y)-G_s(x_0+z,y)]}{|z|^{n+s+1}}\,dz\nonumber\\ &=-\int_{B_r}\frac{z[G_s(x,y)-G_s(x+z,y)]}{|z|^{n+s+1}}\,dz +\int_{B_r}\frac{z[G_s(x_0,y)-G_s(x_0+z,y)]}{|z|^{n+s+1}}\,dz\nonumber\\ &\quad-\int_{{{{\mathbb R}}^n}\setminus B_r} \frac{z\left\{\left[G_s(x,y)-G_s(x+z,y)\right]-\left[G_s(x_0,y)-G_s(x_0+z,y)\right]\right\}}{|z|^{n+s+1}}\,dz. \end{align}

Then, by using (4.29) and lightly modifying the proofs of Lemmas 4.4, 4.5, and 4.6, we conclude that the conclusions of Lemmas 4.4, 4.5, and 4.6 also hold with $(-\Delta)^{\frac{s}{2}}$ replaced by $\nabla^s$. By this and similar to the proof of (4.21), we find that the estimate (4.28) holds. We omit the details here. Finally, from (4.21) and (4.28), it follows that (1.6) holds. This finishes the proof of (iii) and hence Theorem 1.7.