Sharp Sobolev Inequalities via Projection Averages

Philipp Kniefacz,Franz E Schuster

doi:10.1007/s12220-020-00544-6

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.

Highlights

The fruitful interplay between analysis and geometry is probably highlighted most prominently by the rich theory of Sobolev inequalities and, in particular, by its best known representative—the sharp L p Sobolev inequality in Rn
In the case p = 1, it was previously shown by Maz’ya [30] and Federer and Fleming [10] that the sharp L1 Sobolev inequality is equivalent to the isoperimetric inequality
While the explicit knowledge of the optimal constant has proven beneficial in certain areas of mathematical physics, its importance is far outweighed by the classification of the extremal functions in (1.1)

Summary

Introduction

The fruitful interplay between analysis and geometry is probably highlighted most prominently by the rich theory of Sobolev inequalities and, in particular, by its best known representative—the sharp L p Sobolev inequality in Rn. In the case p = 1, inequality (1.5) was extended to BV (Rn) in [17] and it was shown that equality holds precisely for characteristic functions of Euclidean balls (see below, for the value of the optimal constant). In the case i = 1, inequality (1.9) reduces to the affine invariant Zhang–Sobolev inequality from [36,38] which, by (1.10), is the strongest of the Sobolev inequalities provided by Theorem 3 It directly implies the case i = n − 1 of (1.9) obtained in [17], which, in turn, is stronger than the classical Sobolev inequality for functions of bounded variation (1.8)—the case i = n of (1.9)

Background

Auxiliary Results

Proofs of the Main Results