Interpolation of weighted Sobolev spaces

TL;DR

This paper develops a new approach to interpolate weighted Sobolev spaces using the complex method, extending classical results and providing new cases where the interpolation formula holds, with potential applications in evolution equations.

Contribution

It introduces novel interpolation results for weighted Sobolev spaces, including cases not covered by previous methods, and establishes conditions under which the interpolation formula is valid.

Findings

Interpolation formula holds for certain continuous weights with Lipschitz quotients.

Extension of Stein-Weiss theorem to weighted Sobolev spaces.

Potential applications in convergence analysis of evolution equations.

Abstract

In this work we present a newly developed study of the interpolation of weighted Sobolev spaces by the complex method. We show that in some cases, one can obtain an analogue of the famous Stein-Weiss theorem for weighted $L^{p}$ spaces. We consider an example which gives some indication that this may not be possible in all cases. Our results apply in cases which cannot be treated by methods in earlier papers about interpolation of weighted Sobolev spaces. They include, for example, a proof that $\left[W^{1,p}(\mathbb{R}^{d},\omega_{0}),W^{1,p}(\mathbb{R}^{d},\omega_{1})\right]_{\theta}=W^{1,p}(\mathbb{R}^{d},\omega_{0}^{1-\theta}\omega_{1}^{\theta})$ whenever $\omega_{0}$ and $\omega_{1}$ are continuous and their quotient is the exponential of a Lipschitz function. We also mention some possible applications of such interpolation in the study of convergence in evolution equations.