A Leray-Trudinger Inequality

TL;DR

This paper extends a Leray-type inequality to multiple dimensions for the case p=n≥2, providing an optimal Sobolev-type enhancement similar to known improvements for other p-values, advancing the understanding of Hardy-type inequalities.

Contribution

It introduces an optimal Sobolev-type improvement of a multidimensional Leray inequality for p=n≥2, filling a gap in the existing Hardy inequality literature.

Findings

Established an optimal Sobolev-type improvement for the Leray inequality in multiple dimensions.

Extended the class of Hardy-type inequalities applicable for p=n≥2.

Provided a theoretical framework analogous to known improvements for other p-values.

Abstract

We consider a multidimensional version of an inequality due to Leray as a substitute for Hardy's inequality in the case $p=n\geq2.$ In this paper we provide an optimal Sobolev-type improvement of this substitute, analogous to the corresponding improvements obtained for $p=2<n$ in S. Filippas, A. Tertikas, Optimizing improved Hardy inequalities, J. Funct. Anal. 192 (1) (2002) 186--233, and for $p>n\geq1$ in G. Psaradakis, An optimal Hardy-Morrey inequality, Calc. Var. Partial Differential Equations 45 (3-4) (2012) 421--441.