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

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.

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• Boundedness of classical operators on rearrangement-invariant spaces
• Sobolev embeddings, rearrangement-invariant spaces and Frostman measures
• Interpolation theorem for Marcinkiewicz spaces with applications to Lorentz gamma spaces
• Reduction principle for a certain class of kernel-type operators
• Poincaré-Sobolev inequalities with rearrangement-invariant norms on the entire space
• Optimal behavior of weighted Hardy operators on rearrangement-invariant spaces
• On the smoothness of slowly varying functions