Optimal function spaces in weighted Sobolev embeddings with $\alpha$-homogeneous weights

TL;DR

This paper characterizes the best possible rearrangement-invariant function spaces for weighted Sobolev inequalities on convex cones with homogeneous weights, providing a reduction principle and concrete examples including Lorentz--Karamata spaces.

Contribution

It introduces a reduction principle and characterizes optimal function spaces for weighted Sobolev inequalities with $\alpha$-homogeneous weights, including Lorentz--Karamata spaces.

Findings

Established a reduction principle for weighted Sobolev inequalities.

Characterized optimal target and domain function spaces.

Provided concrete examples using Lorentz--Karamata spaces.

Abstract

We study weighted Sobolev inequalities on open convex cones endowed with $\alpha$-homogeneous weights satisfying a certain concavity condition. We establish a so-called reduction principle for these inequalities and characterize optimal rearrangement-invariant function spaces for these weighted Sobolev inequalities. Both optimal target and optimal domain spaces are characterized. Abstract results are accompanied by general yet concrete examples of optimal function spaces. For these examples, the class of so-called Lorentz--Karamata spaces, which contains in particular Lebesgue spaces, Lorentz spaces, and some Orlicz spaces, is used.