A coherent theory of tent spaces and homogeneous Triebel-Lizorkin spaces

TL;DR

This paper develops a comprehensive theory of tent spaces that characterize homogeneous Triebel-Lizorkin spaces, extending classical results and providing new endpoint space characterizations.

Contribution

It introduces a systematic framework linking tent spaces with homogeneous Triebel-Lizorkin spaces, including duality, embeddings, and endpoint characterizations.

Findings

Tent spaces characterize homogeneous Triebel-Lizorkin spaces.

Equivalence with weighted tent spaces with Whitney averages.

New characterization of endpoint spaces ^eta_{\u221e,q}.

Abstract

We introduce and systematically investigate a scale of tent spaces that characterizes homogeneous Triebel-Lizorkin spaces $\mathrm{\dot F}^{\beta}_{p,q}$. These spaces generalize the classical spaces of Coifman, Meyer, and Stein, and are shown to be equivalent to the weighted tent spaces with Whitney averages developed by Huang. We show that these tent spaces follow a functional analytic theory that mirrors that of Triebel-Lizorkin spaces, including duality, embeddings, discrete characterizations, John-Nirenberg-type properties, as well as real and complex interpolation. Furthermore, we provide a novel characterization of the endpoint spaces $\mathrm{\dot F}^\beta_{\infty,q}$, completing earlier work by Auscher, Bechtel, and the author.