Optimal Global Sobolev Estimates for Fractional Dirichlet Problems in Bounded Lipschitz Domains

Let $n\ge 2$, $s\in (0,1)$, and let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be a bounded Lipschitz domain. In this paper, we establish optimal global Sobolev estimates for the fractional Dirichlet problem \begin{equation*} \left{\begin{aligned} (-Δ)^su&=f & & $\text{in}$\ \ Ω, u&=0 & & $\text{in}$\ \ $\mathbb{R}$^n\setminusΩ. \end{aligned}\right. \end{equation*} More precisely, for any given $t\in [s,\min \{2s,s+1\mathrm{/}2\})$, we prove that there exists a small positive constant $\eta =\eta (n,s,t,\mathrm{\Omega })$ such that the following holds. If $n=2$ and $s\in (1\mathrm{/}2,1)$, then, for any $q\in (1,\frac{4}{1+2s}+\eta )$ and $p\in (1,\frac{nq}{n-(2s-t)q}]$, \begin{equation*} |u|{W^{t,p}($\mathbb{R}$^n)}\le C|f|{L^q(Ω)}, \end{equation*} and, if either $n=2$ and $s\in (0,1\mathrm{/}2]$ or $n\ge 3$ and $s\in (0,1)$, then, for any $q\in (1,\frac{n(2s+1)}{nt+(2s-t)(2s+1)}+\eta )$ and $p\in (1,\frac{nq}{n-(2s-t)q}]$, \begin{equation*} |u|{W^{t,p}($\mathbb{R}$^n)}\le C|f|{L^q(Ω)}. \end{equation*} Here the positive constant $C$ depends only on $n$, $s$, $t$, $p$, $q$, and $\mathrm{\Omega }$. The stated universal baselines are optimal in Dahlberg’s sense: for any larger target exponent $p$ there is a bounded Lipschitz domain whose solution with right-hand side identically $1$ is not in $W^{t,p}({\mathbb{R}}^n)$. The counterexamples are constructed using regular Cantor sets on the boundary and, in the planar exceptional case, homogeneous solutions in sectors.