Sobolev spaces revisited

TL;DR

This paper introduces a new one-parameter family of characterizations for Sobolev and BV functions based on superlevel set sizes of difference quotients, offering an alternative perspective to existing formulas and applications to interpolation inequalities.

Contribution

It presents a novel family of formulas characterizing Sobolev and BV functions, expanding the theoretical framework and connecting to wavelet coefficients and interpolation inequalities.

Findings

New characterization formulas for Sobolev and BV functions

Connections to wavelet coefficient sizes for BV functions

Applications to Gagliardo-Nirenberg interpolation inequalities

Abstract

We describe a recent, one-parameter family of characterizations of Sobolev and BV functions on $\mathbb{R}^n$, using sizes of superlevel sets of suitable difference quotients. This provides an alternative point of view to the BBM formula by Bourgain, Brezis and Mironescu, and complements in the case of BV some results of Cohen, Dahmen, Daubechies and DeVore about the sizes of wavelet coefficients of such functions. An application towards Gagliardo-Nirenberg interpolation inequalities is then given. We also establish a related one-parameter family of formulae for the $L^p$ norm of functions in $L^p(\mathbb{R}^n)$.