Riesz potential estimates under co-canceling constraints

2025/12/06 by Breit, D., Cianchi, A., Spector, D.
#Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2512.06352

Abstract

Inequalities for Riesz potentials are well-known to be equivalent to Sobolev inequalities of the same order for domain norms ``far" from L¹, 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 fulfilling a co-canceling differential condition. The present work demonstrates that such a property is not just peculiar to the space L¹. 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 L¹ are singled out.

Citations

• Sobolev inequalities for canceling operators
• Boundedness of functions in fractional Orlicz-Sobolev spaces
• Sharp trace and Korn inequalities for differential operators
• Hardy--Littlewood--Sobolev inequality for p=1
• Fractional Integration and Optimal Estimates for Elliptic Systems
• Symmetric gradient Sobolev spaces endowed with rearrangement-invariant norms
• Fractional Orlicz-Sobolev embeddings
• Normal and Integral Currents

Related

• Embeddings into Orlicz spaces via the modified Riesz potential
• Potential trace inequalities via a Calderón-type theorem
• Non-improvability of Sharp Endpoint Estimates
• Some remarks concerning potentials on different spaces
• Sobolev inequalities for canceling operators