On Gagliardo-Nirenberg type inequalities

TL;DR

This paper introduces refined Gagliardo-Nirenberg inequalities that relate Lorentz norms to Sobolev and Besov quasinorms, extending classical embeddings and applying heat kernel estimates to cover limiting cases.

Contribution

It presents new Gagliardo-Nirenberg inequalities involving Lorentz, Besov, and Triebel-Lizorkin norms, including limiting cases, using heat kernel methods.

Findings

Established inequalities involving Lorentz and Besov norms.

Extended inequalities to the case p=1 for Sobolev norms.

Provided refinements of Sobolev embeddings.

Abstract

We present a Gagliardo-Nirenberg inequality which bounds Lorentz norms of the function by Sobolev norms and homogeneous Besov quasinorms with negative smoothness. We prove also other versions involving Besov or Triebel-Lizorkin quasinorms. These inequalities can be considered as refinements of Sobolev type embeddings. They can also be applied to obtain Gagliardo-Nirenberg inequalities in some limiting cases. Our methods are based on estimates of rearrangements in terms of heat kernels. These methods enable us to cover also the case of Sobolev norms with p = 1.