On the interpolation space $(L^p(\Omega ),W^{1,p}(\Omega ){)}_{s,p}$ in non-smooth domains

Abstract

We show that, for certain non-smooth bounded domains $\Omega \subset {\mathbb{R}}^n$, the real interpolation space $(L^p(\Omega ),W^{1,p}(\Omega ){)}_{s,p}$ is the subspace ${\tilde{W}}^{s,p}(\Omega )\subset L^p(\Omega )$ induced by the restricted fractional seminorm $$|f{|}_{{\tilde{W}}^{s,p}(\Omega )}=({\int }_{\Omega }{\int }_{|x-y|<\frac{d(x)}{2}}\frac{|f(x)-f(y){|}^p}{|x-y{|}^{n+sp}}dydx{)}^{\frac{1}{p}}.$$ In particular, the above result includes simply connected uniform domains in the plane, for which a characterization of the interpolation space was previously unknown.