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Higher order div-curl type estimates for elliptic linear differential operators on localizable Hardy spaces

Catarina Machado, Tiago Picon

TL;DR

This work extends the classical div-curl framework of Coifman–Lions–Meyer–Semmes to higher-order elliptic operators with smooth variable coefficients on localizable Hardy spaces. Central to the approach is a novel smooth atomic decomposition for local Hardy–Sobolev spaces and a Poincaré-type inequality, which together enable sharp local estimates for products of high-order derivatives against divergence-free vector fields. The main result, Theorem A, provides a local Hardy-space bound for A(·,D)φ·v in terms of A(·,D)φ in h^p and v in the Sobolev–Hardy norm, under the condition A^*(·,D)v=0, with a nonhomogeneous version (Theorem B) removing this constraint. The paper further develops the theory for elliptic systems and complexes of vector fields, yielding nonhomogeneous, local div-curl estimates with explicit treatment of vector-field operators and their adjoints, thereby enabling applications to elliptic systems and de Rham-type complexes in local Hardy contexts.

Abstract

In this work, we present higher order div-curl type estimates in the sense of Coifman, Lions, Meyer & Semmes in the local setup of elliptic linear differential operators with smooth coefficients on localizable Hardy spaces. Our version implies and extends results obtained for first order operators associated to elliptic systems and complexes of vector fields. As tools, with own interest, we develop a new smooth atomic decomposition on localizable Hardy-Sobolev spaces and a Poincaré type inequality.

Higher order div-curl type estimates for elliptic linear differential operators on localizable Hardy spaces

TL;DR

This work extends the classical div-curl framework of Coifman–Lions–Meyer–Semmes to higher-order elliptic operators with smooth variable coefficients on localizable Hardy spaces. Central to the approach is a novel smooth atomic decomposition for local Hardy–Sobolev spaces and a Poincaré-type inequality, which together enable sharp local estimates for products of high-order derivatives against divergence-free vector fields. The main result, Theorem A, provides a local Hardy-space bound for A(·,D)φ·v in terms of A(·,D)φ in h^p and v in the Sobolev–Hardy norm, under the condition A^*(·,D)v=0, with a nonhomogeneous version (Theorem B) removing this constraint. The paper further develops the theory for elliptic systems and complexes of vector fields, yielding nonhomogeneous, local div-curl estimates with explicit treatment of vector-field operators and their adjoints, thereby enabling applications to elliptic systems and de Rham-type complexes in local Hardy contexts.

Abstract

In this work, we present higher order div-curl type estimates in the sense of Coifman, Lions, Meyer & Semmes in the local setup of elliptic linear differential operators with smooth coefficients on localizable Hardy spaces. Our version implies and extends results obtained for first order operators associated to elliptic systems and complexes of vector fields. As tools, with own interest, we develop a new smooth atomic decomposition on localizable Hardy-Sobolev spaces and a Poincaré type inequality.
Paper Structure (15 sections, 10 theorems, 160 equations)

This paper contains 15 sections, 10 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Localizable Hardy-Sobolev spaces
  3. Atomic decomposition
  4. Hardy-Sobolev spaces
  5. Poincaré inequality
  6. Generalization of Poincaré Inequality when $\Omega=\mathbb R^{N}$
  7. New smooth atomic decomposition on Hardy-Sobolev spaces
  8. A generalized Calderón-Zygmund decomposition
  9. Proof of Proposition \ref{['czespecial']}
  10. Hardy-Sobolev spaces and elliptic differential operators
  11. The Proof of Theorem A
  12. Proof of control \ref{['atm']}
  13. Non-homogeneous estimates and applications
  14. Elliptic system of vector fields
  15. Elliptic complexes of vector fields

Key Result

Lemma 3.1

Mi Let $m \in \mathbb N$. Then for each cube $Q \subset \mathbb R^N$ there exists a function $\phi_Q$ on $\mathbb R^N\times \mathbb R^N$ which has the following properties:

Theorems & Definitions (18)

  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Definition 3.1
  • Lemma 3.1
  • Proposition 3.1
  • Definition 3.2
  • Remark 4.1
  • Proposition 4.1
  • Proposition 4.2
  • ...and 8 more