# A direct proof of Sobolev embeddings for quasi-homogeneous Lizorkin--Triebel spaces with mixed norms

## TL;DR

This paper provides a simplified proof of Sobolev embeddings for Lizorkin--Triebel spaces, extending classical results to spaces with mixed norms and quasi-homogeneous types, using a novel inequality for sequences of functions.

## Contribution

It introduces a new, streamlined proof technique for Sobolev embeddings in Lizorkin--Triebel spaces with mixed norms, including quasi-homogeneous cases.

## Key findings

- Extended Sobolev embedding results to mixed norm Lizorkin--Triebel spaces.
- Proved a Nikol'skij--Plancherel--Polya inequality for geometric rectangle sequences.
- Results apply to quasi-homogeneous Lizorkin--Triebel spaces.

## Abstract

The article deals with a simplified proof of the Sobolev embedding theorem for Lizorkin--Triebel spaces (that contain the $L_p$-Sobolev spaces $H^s_p$ as special cases). The method extends to a proof of the corresponding fact for general Lizorkin--Triebel spaces based on mixed $L_p$-norms. In this context a Nikol'skij--Plancherel--Polya inequality for sequences of functions satisfying a geometric rectangle condition is proved. The results extend also to spaces of the quasi-homogeneous type.