Continuous characterizations of Besov-Lizorkin-Triebel spaces and new interpretations as coorbits

TL;DR

This paper provides new continuous characterizations of Besov-Lizorkin-Triebel spaces using local means, Lusin functions, and Peetre maximal functions, enabling atomic decompositions and wavelet bases via coorbit theory.

Contribution

It introduces novel continuous characterizations of these function spaces and connects them with coorbit space theory, leading to new atomic and wavelet decompositions.

Findings

Characterizations in terms of local means, Lusin functions, and Peetre maximal functions.

Atomic decompositions and wavelet bases for homogeneous spaces.

Sufficient conditions for wavelet construction based on moment, decay, and smoothness.

Abstract

We give characterizations for homogeneous and inhomogeneous Besov-Lizorkin-Triebel spaces in terms of continuous local means for the full range of parameters. In particular, we prove characterizations in terms of Lusin functions and spaces involving the Peetre maximal function to apply the classical coorbit space theory due to Feichtinger and Gr\"ochenig. This results in atomic decompositions and wavelet bases for homogeneous spaces. In particular we give sufficient conditions for suitable wavelets in terms of moment, decay and smoothness conditions.