Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Luca Haardt

Abstract

We introduce and systematically investigate a scale of tent spaces that characterizes homogeneous Triebel-Lizorkin spaces $\mathrm{\dot F}^β_{p,q}$. 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 $\mathrm{\dot F}^β_{\infty,q}$, completing earlier work by Auscher, Bechtel, and the author.

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

Abstract

We introduce and systematically investigate a scale of tent spaces that characterizes homogeneous Triebel-Lizorkin spaces . 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 , completing earlier work by Auscher, Bechtel, and the author.
Paper Structure (16 sections, 28 theorems, 267 equations)

This paper contains 16 sections, 28 theorems, 267 equations.

Table of Contents

  1. Introduction
  2. Tent spaces and Triebel--Lizorkin spaces
  3. Interpolation
  4. Discrete characterizations
  5. Duality
  6. Embeddings
  7. Roadmap and methods
  8. Acknowledgments
  9. Notation
  10. Characterization of endpoint Triebel--Lizorkin spaces
  11. Functional analytic properties of tent spaces
  12. Definitions, relations to known spaces and basic properties
  13. Discrete descriptions
  14. Embedding theory
  15. Duality: non-Banach range
  16. ...and 1 more sections

Key Result

Lemma 2.4

Let $0<q,\alpha<\infty$ and $\delta \in(\frac{d}{\alpha},\infty)$. Suppose that $(g_l)_{l\in\mathbb{Z}}$ is a family of measurable functions in $\mathbb{R}^d$. Then one has

Theorems & Definitions (56)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Remark 2.6
  • proof : Proof of Theorem \ref{['thm: char of lifted Triebel data']}
  • Proposition 2.7: Gauss--Weierstrass characterization of $\dot \mathrm{F}^\beta_{\infty,q}$
  • Definition 3.1: Tent spaces
  • Definition 3.2: $\mathrm{Z}$-spaces
  • ...and 46 more