Improved Hardy-Sobolev inequalities

TL;DR

This paper presents new Hardy-Sobolev inequalities that connect Sobolev embeddings with heat kernel bounds, using operator semigroup analysis inspired by Ledoux's recent work.

Contribution

It introduces a novel approach applying Ledoux's technique to the operator L, deriving inequalities that unify Hardy, Sobolev, and Gagliardo-Nirenberg types.

Findings

Established a Hardy-type inequality with heat kernel bounds

Connected Sobolev embeddings to heat kernel analysis

Analyzed the semigroup generated by the operator L

Abstract

The main result includes features of a Hardy-type inequality and an inequality of either Sobolev or Gagliardo-Nirenberg type. It is inspired by the method of proof of a recent improved Sobolev inequality derived by M. Ledoux which brings out the connection between Sobolev embeddings and heat kernel bounds. Here Ledoux's technique is applied to the operator $L:= {\bf{x}} \cdot \nabla$ and the analysis requires the determination of the operator semigroup $\{e^{-tL^{*}L}\}_{t>0}$ and its properties.