Integral isoperimetric transference and dimensionless Sobolev inequalities

TL;DR

This paper introduces a new concept called Gaussian integral isoperimetric transference, leading to sharp, dimension-independent Sobolev-Poincaré inequalities, extending recent results in $L^q$ spaces on the unit cube.

Contribution

It presents a novel approach to derive sharp Sobolev inequalities with dimension-independent constants using Gaussian isoperimetric transference.

Findings

Established a new class of sharp Sobolev-Poincaré inequalities

Constants are independent of the dimension

Extended recent inequalities in $L^q$ spaces on the cube

Abstract

We introduce the concept of Gaussian integral isoperimetric transference and show how it can be applied to obtain a new class of sharp Sobolev-Poincar\'{e} inequalities with constants independent of the dimension. In the special case of $L^{q}$ spaces on the unit $n-$dimensional cube our results extend the recent inequalities that were obtained in \cite{FKS} using extrapolation.