On global fractional Calderón–Zygmund regularity for the fractional Dirichlet problem on Lipschitz domains

Let n≥2 , s∈(0,1) , and Ωⁿ be a bounded Lipschitz domain. In this paper, we investigate the global (higher-order) Sobolev regularity of weak solutions to the fractional Dirichlet problem {equation*} {cases} (-Δ)^su=f \ \ & in\ \ Ω,\\ u=0 \ \ & in\ \ R^nΩ. {cases} {equation*} Precisely, we prove that there exists a positive constant ε∈(0,s] depending on n , s , and the Lipschitz constant of Ω such that, for any t∈[ε,min\1+ε,2s\) , when f∈ Lq(Ω) with some q∈(n/2s-t,∞] , the weak solution u satisfies {equation*}\|u\|_{Wt,p(R^n)}≤ C\|f\|L^q(Ω){equation*} for all p∈[1,1/t-ε) . In particular, when Ω is a bounded C 1 domain or a bounded Lipschitz domain satisfying the uniform exterior ball condition, the aforementioned global regularity estimates hold with ε=s and they are sharp in this case. Moreover, if Ω is a bounded C1,κ domain with κ∈(0,s) or a bounded Lipschitz domain satisfying the uniform exterior ball condition, we further show the global BMO-Sobolev regularity estimate {equation*}\|(-Δ)s/2u\|BMO(R^n) +\|∇ˢu\|BMO(R^n)≤ C\|f\|L^q(Ω) {equation*} for some q∈(n/s,∞] , which is sharp in the sense that the BMO norm can not be improved to the L^∞ norm.