Sobolev spaces and averaging I

TL;DR

This paper introduces a new concept of maximal mean difference quotient for functions in Lebesgue spaces, aiming to define a natural class of metric Sobolev spaces applicable to vector-valued functions and general measure metric spaces.

Contribution

It proposes a novel maximal mean difference quotient concept for Lebesgue space functions, extending Sobolev space theory to metric measure spaces and higher-order contexts.

Findings

Defines a new maximal mean difference quotient for Lebesgue functions.

Extends the concept to vector-valued functions on metric spaces.

Discusses higher order Sobolev spaces using generalized Taylor-Whitney jets.

Abstract

An apparently new concept of maximal mean difference quotient is defined for functions in the Lebesgue space $L_{loc}(R^n)$. Our definitions are meaningful for vector valued functions on general measure metric spaces as well and seem to lead to the most natural class of metric Sobolev spaces. The discussion of higher order Sobolev spaces and higher order mean difference quotients on regular subsets of Euclidean spaces is also possible in the context of the generalized Taylor-Whitney jets. This paper is a direct sequel to the papers: B. Bojarski, Taylor expansions and Sobolev spaces, Bull. Georgian Natl. Acad. Sci. (N.S.) 5 (2011), no. 2, 5-10. B. Bojarski, Sobolev spaces and Lagrange interpolation, Proc. A. Razmadze Math. Inst. 158 (2012), 1-12.