Sobolev inequalities for 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.