Global endpoint regularity estimates for the fractional Dirichlet problem

Let \(n\ge 2\) $n\ge 2$ , \(s\in (0,1)\) $s\in (0,1)$ , and \(\Omega \subset {\mathbb {R}}^n\) $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a bounded \(C^2\) $C^2$ domain. The aim of this paper is to study the global endpoint regularity estimates for the fractional Dirichlet problem \(\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^su=f \ \ {} &{} \text {in}\ \ \Omega ,\\ u=0 \ \ {} &{} \text {in}\ \ {\mathbb {R}}^n\setminus \Omega . \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{(-\mathrm{\Delta })}^{s}u=f\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=0\hfill & \text{in}{\mathbb{R}}^n\backslash \mathrm{\Omega }.\hfill \end{array}\right)\end{array}$$ More precisely, the authors prove the optimal global BMO regularity estimates \(\begin{aligned} \left\| (-\Delta )^\frac{s}{2} u\right\| _{\textrm{BMO}({\mathbb {R}}^n)}\le C\Vert f\Vert _{L^q(\Omega )}\ \ \text {and}\ \ \left\| \nabla ^s u\right\| _{\textrm{BMO}({\mathbb {R}}^n)}\le C\Vert f\Vert _{L^q(\Omega )} \end{aligned}\) $$\begin{array}{c}\hfill {∥{(-\mathrm{\Delta })}^{\frac{s}{2}},u∥}_{\text{BMO}({\mathbb{R}}^n)}\le {C\Vert f\Vert }_{L^q(\mathrm{\Omega })}\text{and}{∥{\mathrm{\nabla }}^s,u∥}_{\text{BMO}({\mathbb{R}}^n)}\le C{\Vert f\Vert }_{L^q(\mathrm{\Omega })}\end{array}$$ for some \(q\in (\frac{n}{s},\infty )\) $q\in (\frac{n}{s},\infty )$ , and the global regularity estimates \(\begin{aligned} \left\| (-\Delta )^\frac{s}{2} u\right\| _{L^p({\mathbb {R}}^n)}\le C\Vert f\Vert _{H^q(\Omega )}\ \ \text {and}\ \ \left\| \nabla ^s u\right\| _{L^p({\mathbb {R}}^n)}\le C\Vert f\Vert _{H^q(\Omega )} \end{aligned}\) $$\begin{array}{c}\hfill {∥{(-\mathrm{\Delta })}^{\frac{s}{2}},u∥}_{L^p({\mathbb{R}}^n)}\le {C\Vert f\Vert }_{H^q(\mathrm{\Omega })}\text{and}{∥{\mathrm{\nabla }}^s,u∥}_{L^p({\mathbb{R}}^n)}\le C{\Vert f\Vert }_{H^q(\mathrm{\Omega })}\end{array}$$ for any given \(q\in (\frac{n}{n+s},1]\) $q\in (\frac{n}{n+s},1]$ and \(p\in [1,\frac{qn}{n-qs})\) $p\in [1,\frac{\mathit{qn}}{n-qs})$ . Here, \(\nabla ^s\) ${\mathrm{\nabla }}^s$ denotes the Riesz gradient of order s and \(H^q(\Omega )\) $H^q(\mathrm{\Omega })$ denotes the Hardy space on \(\Omega \) $\mathrm{\Omega }$ . Moreover, the authors also obtain the global regularity estimates \(\begin{aligned} \left\| (-\Delta )^\frac{t}{2}u\right\| _{L^p({\mathbb {R}}^n)}\le C\Vert f\Vert _{H^q(\Omega )} \ \ \text {and}\ \ \left\| \nabla ^t u\right\| _{L^p({\mathbb {R}}^n)}\le C\Vert f\Vert _{H^q(\Omega )} \end{aligned}\) $$\begin{array}{c}\hfill {∥{(-\mathrm{\Delta })}^{\frac{t}{2}},u∥}_{L^p({\mathbb{R}}^n)}\le {C\Vert f\Vert }_{H^q(\mathrm{\Omega })}\text{and}{∥{\mathrm{\nabla }}^t,u∥}_{L^p({\mathbb{R}}^n)}\le C{\Vert f\Vert }_{H^q(\mathrm{\Omega })}\end{array}$$ for any given \(t\in (s,\min \{1,s+\frac{s}{n}\})\) $t\in (s,min\{1,s+\frac{s}{n}\})$ , \(q\in (\frac{n}{n+s-n(t-s)}, 1]\) $q\in (\frac{n}{n+s-n(t-s)},1]$ , and \(p\in [1,\frac{qn}{n-qs+qn(t-s)})\) $p\in [1,\frac{\mathit{qn}}{n-qs+qn(t-s)})$ . The global regularity estimates given in this paper are further devolvement for the corresponding results in the scale of Lebesgue spaces, established by B. Abdellaoui, A. J. Fernández, T. Leonori, and A. Younes [arXiv: 2107.06535], in the endpoint case.