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

TL;DR

This paper characterizes the real interpolation space between L^p and W^{1,p} in certain non-smooth domains, showing it equals a subspace defined by a restricted fractional seminorm, including previously uncharacterized simply connected uniform domains.

Contribution

It provides a new characterization of the interpolation space in non-smooth domains, extending known results to simply connected uniform domains in the plane.

Findings

Interpolation space equals a subspace defined by a restricted fractional seminorm

Includes simply connected uniform domains in the plane

Extends previous characterizations to 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.