Homogeneity Property of Besov and Triebel-Lizorkin Spaces

Cornelia Schneider,Jan Vybíral

doi:10.1155/2012/281085

Abstract

We consider the classical Besov and Triebel-Lizorkin spaces defined via differences and prove a homogeneity property for functions with bounded support in the frame of these spaces. As the proof is based on compact embeddings between the studied function spaces, we present also some results on the entropy numbers of these embeddings. Moreover, we derive some applications in terms of pointwise multipliers.

Highlights

IntroductionThe present note deals with classical Besov spaces Bsp,q Rn and Triebel-Lizorkin spaces Fsp,q Rn defined via differences, briefly denoted as B- and F-spaces in the sequel
As the proof is based on compact embeddings between the studied function spaces, we present some results on the entropy numbers of these embeddings
The present note deals with classical Besov spaces Bsp,q Rn and Triebel-Lizorkin spaces Fsp,q Rn defined via differences, briefly denoted as B- and F-spaces in the sequel

Summary

Introduction

The present note deals with classical Besov spaces Bsp,q Rn and Triebel-Lizorkin spaces Fsp,q Rn defined via differences, briefly denoted as B- and F-spaces in the sequel. We remark that the homogeneity property is closely related with questions concerning refined localization, nonsmooth atoms, local polynomial approximation, and scaling properties. This is out of our scope for the time being. Our proof of 1.2 is based on compactness of embeddings between the function spaces under investigation We use this opportunity to present some closely related results on entropy numbers of such embeddings. The last section states some applications in terms of pointwise multipliers

Preliminaries

Function Spaces on Domains

Entropy Numbers

Homogeneity

A straightforward calculation shows again that fj

Pointwise Multipliers