Sobolev inequalities for canceling operators

TL;DR

This paper develops a unified framework for Sobolev inequalities involving elliptic canceling operators, characterizing them via Hardy inequalities and identifying optimal target norms across various rearrangement-invariant spaces.

Contribution

It introduces a simplified characterization of Sobolev inequalities for elliptic canceling operators using Hardy inequalities and determines optimal norms for these embeddings.

Findings

Sobolev inequalities are characterized via Hardy inequalities.

Optimal rearrangement-invariant target norms are explicitly identified.

Results unify and extend previous work on 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.