Sobolev Inequalities, Riesz Transforms and the Ricci Flow

TL;DR

This paper explores methods to derive advanced Sobolev inequalities from existing ones and applies these results to extend inequalities along the Ricci flow for various p-values.

Contribution

It introduces new techniques using Bessel potentials and Riesz transforms to extend Sobolev inequalities in the context of Ricci flow.

Findings

Extended Sobolev inequalities to W^{1,p} and W^{2,p} along Ricci flow

Developed methods for deriving Sobolev inequalities using Riesz transforms

Applied results to improve understanding of geometric analysis under Ricci flow

Abstract

In this paper we study the problem of deriving further Sobolev inequalities from a given Sobolev inequality. We use several different methods, including Bessel potentials and Riesz transforms. We apply the results to the Ricci flow to extend the author's results on the $W^{1,2}$ Sobolev inequality along the Ricci flow to $W^{1,p}$ and $W^{2,p}$ Sobolev inequalities for general p.