Riesz Potential Estimates Under Co-canceling Constraints

Inequalities for Riesz potentials are well-known to be equivalent to Sobolev inequalities of the same order for domain norms “far” from L¹ L 1 , but to be weaker otherwise. Recent contributions by Van Schaftingen, by Hernandez, Raiţă and Spector, and by Stolyarov proved that this gap can be filled in Riesz potential inequalities for vector-valued functions in L¹ L 1 fulfilling a co-canceling differential condition. The present work demonstrates that such a property is not just peculiar to the space L¹ L 1 , but is shared with any other rearrangement-invariant space. As a consequence, Riesz potential inequalities under the co-canceling constraint are offered for families of spaces of this kind, such as the Orlicz spaces and the Lorentz-Zygmund spaces. Especially relevant instances of inequalities for domain spaces neighboring L¹ L 1 are singled out.