Sharp Sobolev inequalities via projection averages

TL;DR

This paper introduces a family of sharp Sobolev inequalities based on projection averages, unifying and extending classical inequalities, and characterizes extremal functions especially for the case p=1.

Contribution

It establishes a new family of sharp Sobolev inequalities via projection averages and shows their relation to classical and affine Sobolev inequalities, including extremal functions for p=1.

Findings

New family of sharp Sobolev inequalities via projection averages

Each new inequality implies classical Sobolev inequalities

Complete classification of extremal functions for p=1

Abstract

A family of sharp $L^p$ Sobolev inequalities is established by averaging the length of $i$-dimensional projections of the gradient of a function. Moreover, it is shown that each of these new inequalities directly implies the classical $L^p$ Sobolev inequality of Aubin and Talenti and that the strongest member of this family is the only affine invariant one among them -- the affine $L^p$ Sobolev inequality of Lutwak, Yang, and Zhang. When $p = 1$, the entire family of new Sobolev inequalities is extended to functions of bounded variation to also allow for a complete classification of all extremal functions in this case.