Sobolev embeddings, rearrangement-invariant spaces and Frostman measures

TL;DR

This paper develops a unified method to analyze Sobolev embeddings into rearrangement-invariant spaces with measures having polynomial decay, enabling the identification of optimal target spaces and deriving new embeddings.

Contribution

It introduces a reduction of multidimensional Sobolev embeddings to one-dimensional Hardy inequalities depending only on the measure's decay parameter.

Findings

Derived new Sobolev embeddings with larger target spaces.

Identified optimal target spaces for broad classes of norms.

Extended classical results to measures with polynomial decay.

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.