Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Coorbit spaces associated to quasi-Banach function spaces and their molecular decomposition

Jordy Timo van Velthoven, Felix Voigtlaender

Abstract

This paper provides a self-contained exposition of coorbit spaces associated to integrable group representations and quasi-Banach function spaces, and at the same time extends and simplifies previous work. The main results provide an extension of the theory in [Studia Math., 180(3):237-253, 2007] from groups admitting a compact, conjugation-invariant unit neighborhood to arbitrary (possibly nonunimodular) locally compact groups. In addition, the present paper establishes the existence of molecular dual frames and Riesz sequences as in [J. Funct. Anal., 280(10):56, 2021] for the full scale of quasi-Banach function spaces. The theory is developed for possibly projective and reducible unitary representations in order to be easily applicable to well-studied function spaces not satisfying the classical assumptions of coorbit theory. Compared to the existing literature on quasi-Banach coorbit spaces, all our results apply under significantly weaker integrability conditions on the analyzing vectors, which allows for obtaining sharp results in concrete settings

Coorbit spaces associated to quasi-Banach function spaces and their molecular decomposition

Abstract

This paper provides a self-contained exposition of coorbit spaces associated to integrable group representations and quasi-Banach function spaces, and at the same time extends and simplifies previous work. The main results provide an extension of the theory in [Studia Math., 180(3):237-253, 2007] from groups admitting a compact, conjugation-invariant unit neighborhood to arbitrary (possibly nonunimodular) locally compact groups. In addition, the present paper establishes the existence of molecular dual frames and Riesz sequences as in [J. Funct. Anal., 280(10):56, 2021] for the full scale of quasi-Banach function spaces. The theory is developed for possibly projective and reducible unitary representations in order to be easily applicable to well-studied function spaces not satisfying the classical assumptions of coorbit theory. Compared to the existing literature on quasi-Banach coorbit spaces, all our results apply under significantly weaker integrability conditions on the analyzing vectors, which allows for obtaining sharp results in concrete settings
Paper Structure (35 sections, 45 theorems, 297 equations)

This paper contains 35 sections, 45 theorems, 297 equations.

Table of Contents

  1. Introduction
  2. Function spaces and local maximal functions
  3. General notation
  4. Function spaces
  5. Discrete sets
  6. Sequence spaces
  7. Cocycles and twisted convolution
  8. Local maximal functions
  9. Convolution relations for Wiener amalgam spaces
  10. Embeddings into Lebesgue spaces
  11. Convolution relations
  12. Counterexamples
  13. Coorbit spaces associated with integrable group representations
  14. Admissible vectors and reproducing formulae
  15. Integrable vectors
  16. ...and 20 more sections

Key Result

Lemma 3.1

Let $\Lambda = (\lambda_i)_{i \in I} \subseteq G$ be relatively separated. If $F_1, F_2 : G \to [0,\infty)$ are continuous functions, then for all $x,y \in G$. In addition, if $F_1 \in L^1(G)$ is continuous and satisfies $F_1^{\vee} \in W^L(L^1)$, then the mapping is well-defined and bounded, with absolute convergence of the defining series a.e. on $G$. Its operator norm satisfies $\|D_{F_1, \La

Theorems & Definitions (65)

  • Lemma 3.1
  • Definition 4.1
  • Remark 4.2
  • Lemma 4.3
  • Remark 4.4
  • Definition 4.5
  • Remark 4.6
  • Theorem 4.7
  • Remark 4.8
  • Corollary 4.9
  • ...and 55 more