Factoring Sobolev inequalities through classes of functions

TL;DR

This paper establishes a precise link between two recent approaches to improving Sobolev inequalities, one from Real Analysis and the other from Convex Geometry, highlighting their underlying connection.

Contribution

It proves a sharp connection between the Real Analysis and Convex Geometry approaches to Sobolev inequalities, unifying these perspectives.

Findings

Established a sharp connection between two approaches to Sobolev inequalities.

Unified the Real Analysis and Convex Geometry methods.

Enhanced understanding of the structure of Sobolev inequalities.

Abstract

We recall two approaches to recent improvements of the classical Sobolev inequality. The first one follows the point of view of Real Analysis, while the second one relies on tools from Convex Geometry. In this paper we prove a (sharp) connection between them.