Riesz potential estimates under co-canceling constraints

TL;DR

This paper extends Riesz potential inequalities under co-canceling constraints from the space L^1 to a broader class of rearrangement-invariant spaces, including Orlicz and Lorentz-Zygmund spaces, broadening their applicability.

Contribution

It demonstrates that co-canceling properties for Riesz potential inequalities are valid beyond L^1, applying to various rearrangement-invariant spaces.

Findings

Riesz potential inequalities hold under co-canceling constraints in Orlicz spaces.

Extension of inequalities to Lorentz-Zygmund spaces.

Identification of key spaces near L^1 where inequalities are especially relevant.

Abstract

Inequalities for Riesz potentials are well-known to be equivalent to Sobolev inequalities of the same order for domain norms ``far" from $L^1$, but to be weaker otherwise. Recent contributions by Van Schaftingen, by Hernandez, Rai\c{t}\u{a} and Spector, and by Stolyarov proved that this gap can be filled in Riesz potential inequalities for vector-valued functions in $L^1$ fulfilling a co-canceling differential condition. The present work demonstrates that such a property is not just peculiar to the space $L^1$. As a consequence, Riesz potential inequalities under the co-canceling constraint are offered for general 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.