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New characterization of Hardy-Fofana spaces and temperature equation

Martial Agbly Dakoury, Justin Feuto

TL;DR

The paper addresses characterizing Hardy-Fofana spaces $\\mathcal{H}^{(p,q,\alpha)}(\\mathbb{R}^d)$ through Riesz transforms and by viewing their elements as boundary values of bounded solutions to generalized Cauchy-Riemann (temperature) equations. It develops a Riesz-transform framework and a temperature-CR PDE approach, establishing equivalences between maximal-function criteria, harmonic extensions, and boundary data in $$(L^p,\\ell^q)^{\\alpha}$$-type spaces, including a topological isomorphism with a temperature Hardy-type space. The main contributions are (i) a complete CR/Riesz characterization for Hardy-Fofana spaces and (ii) a robust dilation-invariant temperature-CR description via an explicit heat-kernel mapping that links $\\mathcal{H}^{(p,q,\alpha)}(\\mathbb{R}^d)$ with $\\mathbb{H}^{(p,q,\alpha)}(\\mathbb{R}^{d+1}_+)$. This framework clarifies boundary behavior, operator mappings, and provides a PDE-analytic perspective that extends known results from Hardy-Amalgam spaces to the Hardy-Fofana setting, potentially aiding applications requiring precise function-space control under dilations and transforms.

Abstract

The aim of this paper is to give a characterization of Hardy-Fofana spaces via Riesz trasforms. This characterization allow us to describ the distributions belonging to these spaces as a bounded solutions of Cauchy-Riemann's general temperature equations.

New characterization of Hardy-Fofana spaces and temperature equation

TL;DR

The paper addresses characterizing Hardy-Fofana spaces through Riesz transforms and by viewing their elements as boundary values of bounded solutions to generalized Cauchy-Riemann (temperature) equations. It develops a Riesz-transform framework and a temperature-CR PDE approach, establishing equivalences between maximal-function criteria, harmonic extensions, and boundary data in -type spaces, including a topological isomorphism with a temperature Hardy-type space. The main contributions are (i) a complete CR/Riesz characterization for Hardy-Fofana spaces and (ii) a robust dilation-invariant temperature-CR description via an explicit heat-kernel mapping that links with . This framework clarifies boundary behavior, operator mappings, and provides a PDE-analytic perspective that extends known results from Hardy-Amalgam spaces to the Hardy-Fofana setting, potentially aiding applications requiring precise function-space control under dilations and transforms.

Abstract

The aim of this paper is to give a characterization of Hardy-Fofana spaces via Riesz trasforms. This characterization allow us to describ the distributions belonging to these spaces as a bounded solutions of Cauchy-Riemann's general temperature equations.
Paper Structure (4 sections, 9 theorems, 65 equations)

This paper contains 4 sections, 9 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Prerequisites for Hardy-Fofana spaces
  3. Cauchy-Riemann equations, Riesz transforms and Hardy-Fofana spaces
  4. Temperature Cauchy-Riemann equations and Hardy-Fofana spaces

Key Result

Proposition 2.1

Let $1<p\leq\alpha\leq q<+\infty$ and $1<u\leq+\infty$. For all sequences $\left\{f_n\right\}_{n\geq0}$ of measurable functions, we have with the equivalence constants not depending on the sequence $\left\{f_n\right\}_{n\geq0}$.

Theorems & Definitions (14)

  • Proposition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 3.1
  • proof
  • Proposition 3.2: AjCf, Proposition 2.3
  • Proposition 3.3
  • proof
  • Definition 3.4
  • ...and 4 more