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Solvability of elliptic homogeneous linear equations with measure data in weighted Lebesgue spaces

Victor Biliatto, Joel Coacalle, Tiago Picon

TL;DR

This work develops a unifying framework for solving the elliptic, homogeneous, linear equation $A^{*}(D)f=\u03bcmu$ with measure data in weighted Lebesgue spaces. The authors show that solvability in $L^{p}_{w}$ for $1<p<\infty$ is characterized by finite weighted $m$-energy $\|I_m\u03bcmu\|_{L^{p}_{w}}$, and they establish a duality-based existence result when this energy is finite; for the endpoint $p=\infty$ solvability in $L^{\infty}_{1/w}$ is obtained under the operator being canceling and under testing conditions on $\u03bcmu$ and $w$, using a new weighted $L^{1}$ Stein-Weiss inequality for a special class of vector fields. A refined version for $A_1$ weights extends the earlier Moonens–Russ–Tuominen and BP results, providing two testing conditions that imply solvability. The paper also develops two-weighted inequalities, weighted $L^{1}$-type estimates for Riesz transforms, and trace/fractional inequalities, thereby unifying and extending several prior works in this area. Overall, the results offer a robust mechanism to handle measure data in weighted elliptic problems, with potential applications to PDEs with singular sources and anisotropic media.

Abstract

Let $A(D)$ be an elliptic homogeneous linear differential operator with complex constant coefficients, $ μ$ be a vector-valued Borel measure and $w$ be a positive locally integrable function on $\mathbb{R}^N$. In this work, we present sufficient conditions on $μ$ and $w$ for the existence of solutions in the weighted Lebesgue spaces $L^p_w$ for the equation $A^{*}(D)f=μ$, for $ 1\leq p<\infty $. Those conditions are related to a certain control of the Riesz potential of the measure $μ$. We also present sufficient conditions for the solvability when $p=\infty$ adding a canceling condition on the operator. Our method is based on a new weighted $L^1$ Stein-Weiss type inequality on measures for a special class of vector fields.

Solvability of elliptic homogeneous linear equations with measure data in weighted Lebesgue spaces

TL;DR

This work develops a unifying framework for solving the elliptic, homogeneous, linear equation with measure data in weighted Lebesgue spaces. The authors show that solvability in for is characterized by finite weighted -energy , and they establish a duality-based existence result when this energy is finite; for the endpoint solvability in is obtained under the operator being canceling and under testing conditions on and , using a new weighted Stein-Weiss inequality for a special class of vector fields. A refined version for weights extends the earlier Moonens–Russ–Tuominen and BP results, providing two testing conditions that imply solvability. The paper also develops two-weighted inequalities, weighted -type estimates for Riesz transforms, and trace/fractional inequalities, thereby unifying and extending several prior works in this area. Overall, the results offer a robust mechanism to handle measure data in weighted elliptic problems, with potential applications to PDEs with singular sources and anisotropic media.

Abstract

Let be an elliptic homogeneous linear differential operator with complex constant coefficients, be a vector-valued Borel measure and be a positive locally integrable function on . In this work, we present sufficient conditions on and for the existence of solutions in the weighted Lebesgue spaces for the equation , for . Those conditions are related to a certain control of the Riesz potential of the measure . We also present sufficient conditions for the solvability when adding a canceling condition on the operator. Our method is based on a new weighted Stein-Weiss type inequality on measures for a special class of vector fields.

Paper Structure

This paper contains 13 sections, 14 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Muckenhoupt weight classes
  4. Riesz potentials and measures with finite weighted energy
  5. Canceling and cocanceling operators
  6. Proof of Theorem \ref{['theoA']}
  7. Proof of Theorem \ref{['theoB']}
  8. Proof of Theorem \ref{['theoC']}
  9. Applications and general comments
  10. Non-weighted Lebesgue solvability
  11. Two-weighted inequalities
  12. Weighted $L^1$ type estimate for Riesz transform
  13. Trace and fractional inequalities for vector fields

Key Result

Theorem 1.1

Let $A(D)$ be a homogeneous linear differential operator of order $1\leqslant m<N$ on $\mathbb{R}^N$ from $E$ to $F$ and $\mu$ be a Borel measure on $\mathbb{R}^N$ with values in $E^*$. If $A(D)$ is elliptic and canceling, and $\mu$ satisfies and the potential control then there exists $f \in L^\infty(\mathbb{R}^N,F^{*})$ solving eq A(D).

Theorems & Definitions (24)

  • Theorem 1.1: BP
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Lemma 2.4: DNP
  • Lemma 2.5
  • Proposition 3.1
  • proof
  • ...and 14 more