Sharp Adams-type inequality invoking Hardy inequalities

TL;DR

This paper proves a sharp Adams-type inequality that incorporates Hardy inequalities in even dimensions, resulting in a non-compact Sobolev embedding into Orlicz spaces and analyzing the embedding's lack of compactness.

Contribution

It introduces a novel Adams-type inequality involving Hardy inequalities for even dimensions, expanding the understanding of Sobolev embeddings in Orlicz spaces.

Findings

Established a sharp Adams-type inequality with Hardy inequalities in even dimensions

Derived a non-compact Sobolev embedding into Orlicz spaces

Characterized the lack of compactness of the embedding

Abstract

We establish a sharp Adams-type inequality invoking a Hardy inequality for any even dimension. This leads to a non compact Sobolev embedding in some Orlicz space. We also give a description of the lack of compactness of this embedding in the spirit of \cite{Bahouri}.