Intrinsic characterization and the extension operator in variable exponent function spaces on special Lipschitz domains

TL;DR

This paper provides intrinsic characterizations and a universal extension operator for variable exponent Besov and Triebel-Lizorkin spaces on special Lipschitz domains, advancing the understanding of these function spaces.

Contribution

It introduces two intrinsic characterizations of variable exponent spaces and constructs a universal extension operator, extending previous methods to these complex spaces.

Findings

Two intrinsic characterizations using local means and Peetre maximal operator.

Construction of a linear, bounded, and universal extension operator.

Abstract

We study 2-microlocal Besov and Triebel-Lizorkin spaces with variable exponents on special Lipschitz domains. These spaces are as usual defined by restriction of the corresponding spaces on $\R^n$. In this paper we give two intrinsic characterizations of these spaces using local means and the Peetre maximal operator. Further we construct a linear and bounded extension operator following the approach done by Rychkov, which at the end also turns out to be universal.