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Nonhomogeneous div-curl type estimates for system of complex vector fields on local Hardy spaces

Catarina Machado, Tiago Picon

TL;DR

This work extends the classical div-curl framework to a nonhomogeneous setting on local Hardy spaces for elliptic systems of complex vector fields with constant coefficients. It proves a global bound $\|V\cdot W\|_{h^1} \le C(\|V\|_{L^p}\|W\|_{L^{p'}} + \|{\rm div}_{\mathcal{L}^*} V\|_{L^p}\|W\|_{L^{p'}} + \|V\|_{L^p}\|{\rm curl}_{\mathcal{L}} W\|_{L^{p'}})$, and derives a local $bmo$-decomposition via the same div-curl structure. The authors reduce the proof to curl-zero and div-zero canonical cases using a Hodge-type decomposition and employ a priori estimates and maximal-function techniques to control the local Hardy norm. They then connect these estimates to a dual characterization of $bmo$ and provide a corollary giving a DC-based decomposition of $h^1$, along with a corollary that yields a precise div-curl representation for the $bmo$-duality. Overall, the results unify global nonhomogeneous div-curl phenomena with local Hardy space and $bmo$ duality for systems of constant-coefficient vector fields.

Abstract

In this work, we present a nonhomogeneous version of the classical div-curl type estimates in the setup of elliptic system of complex vector fields with constant coefficients on local Hardy space $h^1$. As an application, we obtain a decomposition of the local $bmo$ space via a family of vector fields depending on div-curl terms.

Nonhomogeneous div-curl type estimates for system of complex vector fields on local Hardy spaces

TL;DR

This work extends the classical div-curl framework to a nonhomogeneous setting on local Hardy spaces for elliptic systems of complex vector fields with constant coefficients. It proves a global bound , and derives a local -decomposition via the same div-curl structure. The authors reduce the proof to curl-zero and div-zero canonical cases using a Hodge-type decomposition and employ a priori estimates and maximal-function techniques to control the local Hardy norm. They then connect these estimates to a dual characterization of and provide a corollary giving a DC-based decomposition of , along with a corollary that yields a precise div-curl representation for the -duality. Overall, the results unify global nonhomogeneous div-curl phenomena with local Hardy space and duality for systems of constant-coefficient vector fields.

Abstract

In this work, we present a nonhomogeneous version of the classical div-curl type estimates in the setup of elliptic system of complex vector fields with constant coefficients on local Hardy space . As an application, we obtain a decomposition of the local space via a family of vector fields depending on div-curl terms.
Paper Structure (7 sections, 9 theorems, 100 equations)

This paper contains 7 sections, 9 theorems, 100 equations.

Table of Contents

  1. Introduction
  2. Elliptic system of complex vector fields
  3. Proof of Theorem A
  4. Proof of Theorem \ref{['prop_curl=0const']}
  5. Proof of Theorem \ref{['prop_div=0const']}
  6. Proof of the Theorem B
  7. Proof of Corolary \ref{['coro1']}

Key Result

Theorem 1.1

Suppose $V$ and $W$ are vector fields on $\mathbb R^N$ satisfying If there exists a function $f \in L^p(\mathbb R^N)$ and a matrix-valued function $A$ with components in $L^{p'}(\mathbb R^N)$ such that, in the sense of distributions, then $V \cdot W$ belongs to the local Hardy space $h^1(\mathbb R^N)$, with

Theorems & Definitions (9)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 3.1
  • Lemma 4.1