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 $\widetilde W^{s,p}(\Omega) \subset L^p(\Omega)$ induced by the restricted fractional seminorm $$ |f|_{\widetilde W^{s,p}(\Omega)} = \Big( \int_\Omega \int_{|x-y|<\frac{d(x)}2} \frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}} \, dy \,dx \Big)^\frac{1}{p}. $$ In particular, the above result includes simply connected uniform domains in the plane, for which a characterization of the interpolation space was previously unknown.