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Characterization for Campanato norm via quasi-Banach function spaces not assuming the Fatou property

Naoya Hatano

TL;DR

This work extends Campanato-norm characterizations to quasi-Banach function spaces without the Fatou property by developing a direct, duality-free framework. Using sparse domination and carefully crafted assumptions on the space $X$ and the operator $M$, it achieves two-sided characterizations of ${\mathcal{L}}_{1,\phi}$ in terms of $X$-averages over cubes, encompassing a wide class of spaces beyond Banach function spaces. The results recover and generalize known cases for weighted Lebesgue, Lorentz, Orlicz, variable Lebesgue, and Morrey spaces, and they include an alternative characterization via dilated $X$-norms. The approach broadens applicability to settings where Fatou’s property fails or is difficult to exploit, with potential impact on harmonic analysis and related areas that leverage Campanato-BMO-type norms. Overall, the paper provides a flexible, robust toolkit for norm characterizations of Campanato spaces in a broad quasi-Banach context.

Abstract

It is well known that the BMO and Campanato norms can be characterized using the $L^p$-average. These characterizations were later generalized to averages taken over various types of function spaces. In particular, generalizations using Banach function spaces were provided by Ho, Izuki, Noi, and Sawano. In this paper, as a further generalization, we provide similar characterizations using quasi-Banach function spaces that do not assume the Fatou property.

Characterization for Campanato norm via quasi-Banach function spaces not assuming the Fatou property

TL;DR

This work extends Campanato-norm characterizations to quasi-Banach function spaces without the Fatou property by developing a direct, duality-free framework. Using sparse domination and carefully crafted assumptions on the space and the operator , it achieves two-sided characterizations of in terms of -averages over cubes, encompassing a wide class of spaces beyond Banach function spaces. The results recover and generalize known cases for weighted Lebesgue, Lorentz, Orlicz, variable Lebesgue, and Morrey spaces, and they include an alternative characterization via dilated -norms. The approach broadens applicability to settings where Fatou’s property fails or is difficult to exploit, with potential impact on harmonic analysis and related areas that leverage Campanato-BMO-type norms. Overall, the paper provides a flexible, robust toolkit for norm characterizations of Campanato spaces in a broad quasi-Banach context.

Abstract

It is well known that the BMO and Campanato norms can be characterized using the -average. These characterizations were later generalized to averages taken over various types of function spaces. In particular, generalizations using Banach function spaces were provided by Ho, Izuki, Noi, and Sawano. In this paper, as a further generalization, we provide similar characterizations using quasi-Banach function spaces that do not assume the Fatou property.

Paper Structure

This paper contains 11 sections, 20 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['thm:main']}
  4. Remarks
  5. Examples and comparison with the known results
  6. Weighted Lebesgue spaces
  7. Lorentz and weighted Lorentz spaces
  8. Orlicz spaces
  9. Variable Lebesgue spaces
  10. Morrey spaces and their generalizations
  11. Another type of characterizations

Key Result

Theorem 1.3

Let $\theta\in(0,1)$, let $X({\mathbb R}^n)\subset L^0({\mathbb R}^n)$ be a Banach function space, and set $\phi_\theta:{\mathbb R}^n\times(0,\infty)\to(0,\infty)$ by $\phi_\theta(Q)=\ell(Q)^{-\theta}$. If $M$ is bounded on $X'({\mathbb R}^n)$, then,

Theorems & Definitions (29)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Lemma 1.6
  • Theorem 1.7
  • Lemma 2.1
  • Definition 4.1: LoNi24
  • Proposition 4.2
  • ...and 19 more