Sobolev spaces and averaging I

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.