Upcoming Terminology in the World of Function Spaces

TL;DR

This paper compares various concepts of function spaces, focusing on Banach and solid BF-spaces, and discusses recent developments like ball quasi-Banach spaces, providing a comprehensive overview in harmonic analysis.

Contribution

It offers a comparative analysis of different function space concepts, clarifying their relationships and recent advancements in the context of harmonic analysis.

Findings

Comparison of Banach function spaces and solid BF-spaces

Discussion of recent developments in ball quasi-Banach spaces

Clarification of concepts used in Time-Frequency Analysis

Abstract

It is the purpose of this article to compare various concepts of ``function spaces''. In particular we compare notions of the concept of Banach Function Spaces (in the spirit of Luxemburg-Zaanen) to the setting of solid BF-spaces as it is widely used in a number of papers in Time-Frequency Analysis or Coorbit Theory. More recently ball quasi-Banach function spaces come into the picture. In a way this note is a companion to the authors' article entitled ``Choosing Function Spaces in Harmonic Analysis'' (from 2015, February Fourier Talks at the Norbert Wiener Center, Vol.4) and the recent article by E.~Lorist and Z.Nieraeth, entitled ``Banach Function Spaces done right'' (Indag. Math., 2023). We restrict our attention to the Banach space case, avoiding the complications arising in the quasi-Banach case.