Interpolation, embeddings and traces of anisotropic fractional Sobolev spaces with temporal weights

TL;DR

This paper studies weighted anisotropic fractional Sobolev spaces with temporal weights, focusing on their properties and applications to maximal regularity in parabolic PDEs using advanced operator theory.

Contribution

It introduces new insights into the structure and interpolation of weighted anisotropic Sobolev spaces relevant for parabolic boundary value problems.

Findings

Characterization of weighted vector-valued Sobolev spaces

Development of interpolation results for these spaces

Application to maximal regularity in parabolic PDEs

Abstract

We investigate the properties of a class of weighted vector-valued $L_p$-spaces and the corresponding (an)isotropic Sobolev-Slobodetskii spaces. These spaces arise naturally in the context of maximal $L_p$-regularity for parabolic initial-boundary value problems. Our main tools are operators with a bounded $\calH^\infty$-calculus, interpolation theory, and operator sums.