Inequalities In Homogeneous Triebel-Lizorkin And Besov-Lipschitz Spaces

TL;DR

This paper characterizes homogeneous Triebel-Lizorkin and Besov-Lipschitz spaces using maximal functions of iterated differences, providing new inequalities and equivalences based on Fourier analysis and maximal function techniques.

Contribution

It offers new equivalence characterizations and inequalities for these function spaces in terms of iterated differences and maximal functions, advancing the theoretical understanding.

Findings

Equivalence characterizations of spaces using maximal functions

Inequalities involving iterated differences along axes

Fourier analytic techniques applied to function space analysis

Abstract

This paper provides equivalence characterizations of homogeneous Triebel-Lizorkin and Besov-Lipschitz spaces, denoted by $\dot{F}^s_{p,q}(\mathbb{R}^n)$ and $\dot{B}^s_{p,q}(\mathbb{R}^n)$ respectively, in terms of maximal functions of the mean values of iterated difference. It also furnishes the reader with inequalities in $\dot{F}^s_{p,q}(\mathbb{R}^n)$ in terms of iterated difference and in terms of iterated difference along coordinate axes. The corresponding inequalities in $\dot{B}^s_{p,q}(\mathbb{R}^n)$ in terms of iterated difference and in terms of iterated difference along coordinate axes are also considered. The techniques used in this paper are of Fourier analytic nature and the Hardy-Littlewood and Peetre-Fefferman-Stein maximal functions.