Remarks on the Bourgain-Brezis-Mironescu Approach to Sobolev Spaces

摘要

Summary. For a function f ∈ L_(loc)~p(R~n) the notion of p-mean variation of order 1, V_1~p(f, R~n) is defined. It generalizes the concept of F. Riesz variation of functions on the real line R~1 to R~n, n > 1. The characterisation of the Sobolev space W~(1,p)(R~n) in terms of V_1~p(f,R~n) is directly related to the characterisation of W_(1,p)(R~n) by Lipschitz type pointwise inequalities of Bojarski, Hajlasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.