Sobolev embeddings, rearrangement-invariant spaces and Frostman measures

Abstract

Sobolev embeddings, of arbitrary order, are considered into function spaces on domains of ${\mathbb{R}}^n$ endowed with measures whose decay on balls is dominated by a power $d$ of their radius. Norms in arbitrary rearrangement-invariant spaces are contemplated. A comprehensive approach is proposed based on the reduction of the relevant $n$-dimensional embeddings to one-dimensional Hardy-type inequalities. Interestingly, the latter inequalities depend on the involved measure only through the power $d$. Our results allow for the detection of the optimal target space in Sobolev embeddings, for broad families of norms, in situations where customary techniques do not apply. In particular, new embeddings, with augmented target spaces, are deduced even for standard Sobolev spaces.

arXiv commentto appear in Ann. Inst. H. Poincaré Anal. Non Linéaire

References in corpus (1)

• Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces

Cited by in corpus (9)

• On the properties of quasi-Banach function spaces
• Weighted inequalities for a superposition of the Copson operator and the Hardy operator
• Basic functional properties of certain scale of rearrangement-invariant spaces
• Discretization and antidiscretization of Lorentz norms with no restrictions on weights
• Optimal behavior of weighted Hardy operators on rearrangement-invariant spaces
• Non-improvability of sharp endpoint estimates
• Potential trace inequalities via a Calderón-type theorem
• Optimal function spaces in weighted Sobolev embeddings with $\alpha$-homogeneous weights
• Non-strict singularity of optimal Sobolev embeddings