Characterisation by Local Means of Anisotropic Lizorkin--Triebel Spaces with Mixed Norms

TL;DR

This paper advances the theoretical understanding of anisotropic Lizorkin--Triebel spaces with mixed norms by characterizing them through convolution-based quasi-norms, extending key inequalities to this setting.

Contribution

It introduces a new characterization of Lizorkin--Triebel spaces with mixed norms using local means and extends important inequalities to this context.

Findings

Extended Rychkov's inequalities to mixed norm spaces

Provided a convolution-based characterization of anisotropic Lizorkin--Triebel spaces

Enhanced the theoretical framework for spaces with quasi-homogeneous smoothness

Abstract

This is a contribution to the theory of Lizorkin--Triebel spaces having mixed Lebesgue norms and quasi-homogeneous smoothness. We discuss their characterisation in terms of general quasi-norms based on convolutions. In particular, this covers the case of local means, in Triebel's terminology. The main step is an extension of some crucial inequalities due to Rychkov to the case with mixed norms.