Sobolev spaces and Lagrange interpolation

TL;DR

This paper provides a geometric proof of a new inequality related to Sobolev spaces, extending pointwise characterization and discussing Sobolev spaces in domains, with comments on prior work and additional definitions.

Contribution

It introduces a direct geometric proof of a novel inequality for Sobolev spaces and discusses pointwise characterization and domain-specific Sobolev spaces.

Findings

New geometric proof of inequality involving m-th difference

Extension of pointwise characterization of Sobolev functions

Definition of Sobolev spaces in domains G

Abstract

In this short paper the discussion of the pointwise characterization of functions $f$ in the Sobolev space $W^{m,p}(\R^n)$ given in the recent paper (Bojarski) is supplemented in \SS1 by a direct, essentially geometric, proof of the novel inequality (for $m>1$), appearing in Bojarski apparently for the first time, and involving the use of the $m$-th difference of the function $f$. Moreover in \SS2 some additional comments to the text in Bojarski are given and a natural class of Sobolev spaces in domains $G$ in $\R^n$ is defined. \SS3 contains some final remarks.