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Cancellation conditions and boundedness of Inhomogeneous Calderón-Zygmund operators on local Hardy spaces associate with spaces of homogeneous type

Joel Coacalle, Tiago Picon, Claudio Vasconcelos

TL;DR

This work develops a localized Calderón-Zygmund theory on spaces of homogeneous type by introducing h^p_ #(X), an atomic local Hardy space built from approximate atoms with inhomogeneous moment conditions. The authors establish duality with local Campanato spaces, derive a molecular decomposition, and relate h^p_ #(X) to the existing local Hardy space h^p(X). Central to their results is a sufficient cancellation condition on R^*(1) that guarantees boundedness of inhomogeneous Calderón-Zygmund operators on h^p(X) for 0<p<1, and a stronger condition for p=1, with extensions to L^p(X). The approach harmonizes atomic, molecular, and maximal characterizations in spaces of homogeneous type, enabling boundedness results that generalize classical CZ theory to the local, non-Euclidean setting and highlighting the role of R^*(1) via local Campanato-type estimates. Collectively, the results extend the CZ framework to local Hardy spaces on spaces of homogeneous type, providing tools for analysis in non-smooth geometric contexts and potential applications in harmonic analysis and PDEs on these spaces.

Abstract

In this work, we present sufficient cancellation conditions for the boundedness of inhomogeneous Calderón-Zygmund type operators on local Hardy spaces defined over spaces of homogeneous type in the sense of Coifman & Weiss for $ 0<p\leq 1 $. A new approach to atoms and molecules for local Hardy spaces in this setting are introduced with special moment conditions.

Cancellation conditions and boundedness of Inhomogeneous Calderón-Zygmund operators on local Hardy spaces associate with spaces of homogeneous type

TL;DR

This work develops a localized Calderón-Zygmund theory on spaces of homogeneous type by introducing h^p_ #(X), an atomic local Hardy space built from approximate atoms with inhomogeneous moment conditions. The authors establish duality with local Campanato spaces, derive a molecular decomposition, and relate h^p_ #(X) to the existing local Hardy space h^p(X). Central to their results is a sufficient cancellation condition on R^*(1) that guarantees boundedness of inhomogeneous Calderón-Zygmund operators on h^p(X) for 0<p<1, and a stronger condition for p=1, with extensions to L^p(X). The approach harmonizes atomic, molecular, and maximal characterizations in spaces of homogeneous type, enabling boundedness results that generalize classical CZ theory to the local, non-Euclidean setting and highlighting the role of R^*(1) via local Campanato-type estimates. Collectively, the results extend the CZ framework to local Hardy spaces on spaces of homogeneous type, providing tools for analysis in non-smooth geometric contexts and potential applications in harmonic analysis and PDEs on these spaces.

Abstract

In this work, we present sufficient cancellation conditions for the boundedness of inhomogeneous Calderón-Zygmund type operators on local Hardy spaces defined over spaces of homogeneous type in the sense of Coifman & Weiss for . A new approach to atoms and molecules for local Hardy spaces in this setting are introduced with special moment conditions.

Paper Structure

This paper contains 14 sections, 33 theorems, 255 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spaces of Homogeneous type
  4. Local Lipschitz spaces
  5. Atomic local Hardy spaces $h^{p}_\#(X)$
  6. Approximate atoms
  7. Local atomic Hardy spaces
  8. The local Campanato spaces as the dual of $h^{p}_\#(X)$.
  9. Molecular decomposition
  10. Relation between $h^{p}_{\#}(X)$ and $h^p(X)$
  11. Continuity of inhomogeneous Calderón-Zygmund type operators
  12. On local Hardy spaces $h^{p}(X)$
  13. On Lebesgue spaces $L^p(X)$.
  14. The case $p=1$

Key Result

Theorem 1.1

Let $s\in (0,\eta]$ and $R$ be a Calderón-Zygmund operator of type $s$. Then R extends to a bounded operator on $H^{p}(X)$ for $\frac{\gamma}{\gamma+s}<p\leq 1$ if and only if $R^{\ast}(1) = 0$.

Theorems & Definitions (73)

  • Theorem 1.1: HanHanLi16
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Remark 2.2
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • ...and 63 more