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On the properties of rearrangement-invariant quasi-Banach function spaces

Anna Musilová, Aleš Nekvinda, Dalimil Peša, Hana Turčinová

TL;DR

The paper extends the Luxemburg representation framework to rearrangement-invariant quasi-Banach function spaces on resonant measure spaces by constructing a representation norm $\|\cdot\|_{\overline{X}}$ with $\|f\|_X = \|f^*\|_{\overline{X}}$, addressing both non-atomic and atomic cases. It develops the fundamental-function/endpoint theory for ri quasi-Banach spaces, introducing weak Marcinkiewicz endpoint spaces $m_{\varphi}$, Lorentz endpoints $\Lambda_{\varphi}$, and their duality with associate spaces, and studies how these interact with Wiener--Luxemburg amalgams. The main technical contribution is Theorem 3 (Representation) and its corollaries, plus a detailed treatment of endpoints (Section 4) and the finite-measure modification (Section 4.2), which together yield a robust framework for embeddings, duality, and optimal target/ domains in this broader setting. By combining measure-preserving representations with amalgam tools, the work opens pathways for applications to Sobolev-type embeddings, operator bounds, and interpolation in the quasi-Banach regime, enabling precise control of local versus global behavior via endpoint and fundamental-function data.

Abstract

This paper explores some important aspects of the theory of rearrangement-invariant quasi-Banach function spaces. We focus on two main topics. Firstly, we prove an analogue of the Luxemburg representation theorem for rearrangement-invariant quasi-Banach function spaces over resonant measure spaces. Secondly, we develop the theory of fundamental functions and endpoint spaces.

On the properties of rearrangement-invariant quasi-Banach function spaces

TL;DR

The paper extends the Luxemburg representation framework to rearrangement-invariant quasi-Banach function spaces on resonant measure spaces by constructing a representation norm with , addressing both non-atomic and atomic cases. It develops the fundamental-function/endpoint theory for ri quasi-Banach spaces, introducing weak Marcinkiewicz endpoint spaces , Lorentz endpoints , and their duality with associate spaces, and studies how these interact with Wiener--Luxemburg amalgams. The main technical contribution is Theorem 3 (Representation) and its corollaries, plus a detailed treatment of endpoints (Section 4) and the finite-measure modification (Section 4.2), which together yield a robust framework for embeddings, duality, and optimal target/ domains in this broader setting. By combining measure-preserving representations with amalgam tools, the work opens pathways for applications to Sobolev-type embeddings, operator bounds, and interpolation in the quasi-Banach regime, enabling precise control of local versus global behavior via endpoint and fundamental-function data.

Abstract

This paper explores some important aspects of the theory of rearrangement-invariant quasi-Banach function spaces. We focus on two main topics. Firstly, we prove an analogue of the Luxemburg representation theorem for rearrangement-invariant quasi-Banach function spaces over resonant measure spaces. Secondly, we develop the theory of fundamental functions and endpoint spaces.
Paper Structure (12 sections, 50 theorems, 172 equations)

This paper contains 12 sections, 50 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Non-increasing rearrangement
  4. Banach function norms and quasinorms
  5. Luxemburg representation theorem
  6. Associate spaces
  7. Fundamental function and endpoint spaces
  8. Wiener--Luxemburg amalgam spaces
  9. Representation theorem
  10. Fundamental function and endpoint spaces
  11. Fundamental function of the associate space
  12. The case of finite measure

Key Result

Proposition 2.2

Let $\varphi: [0, \infty) \to [0, \infty)$ be a non-decreasing function that satisfies the $\Delta_2$-condition with constant $C_{\varphi}$. Then it is also subadditive up to this constant, i.e. it holds for every $s,t \in [0, \infty)$ that

Theorems & Definitions (105)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Definition 2.7
  • Theorem 2.8
  • Definition 2.9
  • ...and 95 more