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Sobolev inequalities for canceling operators

Dominic Breit, Andrea Cianchi, Daniel Spector

TL;DR

The paper develops a unified framework for Sobolev-type embeddings with rearrangement-invariant norms for elliptic canceling operators, showing that their domain and target spaces coincide with those for the gradient of the same order. It reduces these embeddings to one-dimensional Hardy-type inequalities and identifies optimal rearrangement-invariant target spaces, including Orlicz and Lorentz-Zygmund scales, with explicit targets. Key technical advances include a sharp K-functional analysis for spaces of k-th order divergence-free vector fields and a pivotal rearrangement inequality for Riesz potentials composed with singular integral operators. The results cover endpoint $L^1$ cases, finite-measure domains, and provide sharp, domain-appropriate target spaces, thereby unifying Sobolev-type embeddings across the class of elliptic canceling operators.

Abstract

Sobolev type inequalities involving homogeneous elliptic canceling differential operators and rearrangement-invariant norms on the Euclidean space are considered. They are characterized via considerably simpler one-dimensional Hardy type inequalities. As a consequence, they are shown to hold exactly for the same norms as their counterparts depending on the standard gradient operator of the same order. The results offered provide a unified framework for the theory of Sobolev embeddings for the elliptic canceling operators. They build upon and incorporate earlier fundamental contributions dealing with the endpoint case of $L^1$-norms. They also include previously available results for the symmetric gradient, a prominent instance of an elliptic canceling operator. In particular, the optimal rearrangement-invariant target norm associated with any given domain norm in a Sobolev inequality for any elliptic canceling operator is exhibited. Its explicit form is detected for specific families of rearrangement-invariant spaces, such as the Orlicz spaces and the Lorentz-Zygmund spaces. Especially relevant instances of inequalities for domain spaces neighboring $L^1$ are singled out.

Sobolev inequalities for canceling operators

TL;DR

The paper develops a unified framework for Sobolev-type embeddings with rearrangement-invariant norms for elliptic canceling operators, showing that their domain and target spaces coincide with those for the gradient of the same order. It reduces these embeddings to one-dimensional Hardy-type inequalities and identifies optimal rearrangement-invariant target spaces, including Orlicz and Lorentz-Zygmund scales, with explicit targets. Key technical advances include a sharp K-functional analysis for spaces of k-th order divergence-free vector fields and a pivotal rearrangement inequality for Riesz potentials composed with singular integral operators. The results cover endpoint cases, finite-measure domains, and provide sharp, domain-appropriate target spaces, thereby unifying Sobolev-type embeddings across the class of elliptic canceling operators.

Abstract

Sobolev type inequalities involving homogeneous elliptic canceling differential operators and rearrangement-invariant norms on the Euclidean space are considered. They are characterized via considerably simpler one-dimensional Hardy type inequalities. As a consequence, they are shown to hold exactly for the same norms as their counterparts depending on the standard gradient operator of the same order. The results offered provide a unified framework for the theory of Sobolev embeddings for the elliptic canceling operators. They build upon and incorporate earlier fundamental contributions dealing with the endpoint case of -norms. They also include previously available results for the symmetric gradient, a prominent instance of an elliptic canceling operator. In particular, the optimal rearrangement-invariant target norm associated with any given domain norm in a Sobolev inequality for any elliptic canceling operator is exhibited. Its explicit form is detected for specific families of rearrangement-invariant spaces, such as the Orlicz spaces and the Lorentz-Zygmund spaces. Especially relevant instances of inequalities for domain spaces neighboring are singled out.
Paper Structure (7 sections, 10 theorems, 189 equations)

This paper contains 7 sections, 10 theorems, 189 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Sobolev type spaces
  4. Sobolev inequalities for elliptic canceling operators
  5. K-functionals for spaces of $k$-th order divergence-free vector fields
  6. A pivotal rearrangement inequality
  7. Proofs of the main results

Key Result

Theorem 3.1

Let $n, m, \ell, k \in {\mathbb N}$, with $n \geq 2$ and $1 \leq k <n$. Assume that $\mathcal{A}_k(D)$ is a linear homogeneous $k$-th order elliptic canceling operator. Let $\|\cdot\|_{X(0,\infty)}$ and $\|\cdot\|_{Y(0,\infty)}$ be rearrangement-invariant function norms. The following facts are equi for every $u \in V^{\mathcal{A}_k}_{\circ} X(\mathbb{R}^n, \mathbb{R}^\ell)$. (ii) The embedding $V

Theorems & Definitions (31)

  • Definition 1: Elliptic operator
  • Definition 2: Canceling operator
  • Theorem 3.1: Reduction principle for Sobolev inequalities for canceling operators in $\mathbb{R}^n$
  • Theorem 3.2: Optimal target space in Sobolev inequalities in $\mathbb{R}^n$
  • Corollary 3.3: Reduction principle for Sobolev inequalities in domains
  • Corollary 3.4: Optimal target space in Sobolev inequalities in domains
  • Theorem 3.5: Optimal Orlicz-Sobolev inequalities
  • Remark 3.6
  • Example 3.7
  • Example 3.8
  • ...and 21 more