Riesz Transform Characterization and Fefferman-Stein Decomposition of Triebel-Lizorkin Spaces

TL;DR

This paper characterizes Triebel-Lizorkin spaces using Riesz transforms and establishes their Fefferman-Stein decomposition in higher dimensions, introducing new techniques beyond the one-dimensional case.

Contribution

It extends the Riesz transform characterization and Fefferman-Stein decomposition of Triebel-Lizorkin spaces to dimensions greater than one, using novel methods.

Findings

Established Riesz transform characterization of Triebel-Lizorkin spaces in rac{D}{D} dimensions.

Obtained Fefferman-Stein decomposition for these spaces.

Showed strict inclusions between Triebel-Lizorkin and wavelet-based function spaces.

Abstract

Let $D\in\mathbb{N}$, $q\in[2,\infty)$ and $(\mathbb{R}^D,|\cdot|,dx)$ be the Euclidean space equipped with the $D$-dimensional Lebesgue measure. In this article, via an auxiliary function space $\mathrm{WE}^{1,\,q}(\mathbb R^D)$ defined via wavelet expansions, the authors establish the Riesz transform characterization of Triebel-Lizorkin spaces $\dot{F}^0_{1,\,q}(\mathbb{R}^D)$. As a consequence, the authors obtain the Fefferman-Stein decomposition of Triebel-Lizorkin spaces $\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D)$. Finally, the authors give an explicit example to show that $\dot{F}^0_{1,\,q}(\mathbb{R}^D)$ is strictly contained in $\mathrm{WE}^{1,\,q}(\mathbb{R}^D)$ and, by duality, $\mathrm{WE}^{\infty,\,q'}(\mathbb{R}^D)$ is strictly contained in $\dot{F}^0_{\infty,\,q'}(\mathbb{R}^D)$. Although all results when $D=1$ were obtained by C.-C. Lin et al. [Michigan Math. J. 62 (2013), 691-703], as was pointed out by C.-C. Lin et al., the approach used in the case $D=1$ can not be applied to the case $D\ge2$, which needs some new skills.