Constrained differential operators, Sobolev inequalities, and Riesz potentials (2501.07874v3)

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. This work demonstrates that such a property is not just peculiar to the space $L^1$. Indeed, under the same differential constraint, a Riesz potential inequality is shown to hold for any domain and target rearrangement-invariant norms that render a Sobolev inequality of the same order true. This is based on a new interpolation inequality, which, via a kind of duality argument, yields a parallel property of Sobolev inequalities for any linear homogeneous elliptic canceling differential operator. Specifically, Sobolev inequalities involving the full gradient of a certain order share the same rearrangement-invariant domain and target spaces as their analogs for any other homogeneous elliptic canceling differential operator of equal order. As a consequence, Riesz potential inequalities under the co-canceling constraint and Sobolev inequalities for homogeneous elliptic canceling differential operators 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.