A new characterization of the Sobolev space
The purpose of this paper is to provide a new characterization of the Sobolev space W 1,1(Rn). We also show a new proof of the characterization of the Sobolev space W 1,p(Rn), 1 ≤ p <∞, in terms of Poincaré inequalities. The Sobolev space W 1,p(Rn), 1 ≤ p < ∞, consists of functions u ∈ Lp(Rn) such that |∇u| ∈ Lp(Rn). It is a Banach space with respect to the norm ‖u‖1,p = ‖u‖p + ‖∇u‖p. Let us point out that from the point of view of Banach spaces the structure of various Sobolev type spaces, with the particular emphasis on W 1,1, has been investigated by A. Pe lczyński, M. Wojciechowski and others; see e.g. [4], [40]–[43] and references therein. The purpose of this paper is to provide a new characterization of the Sobolev space W 1,1(Rn). Here, however, we emphasize future applications in geometric analysis and analysis on metric spaces rather than the theory of Banach spaces. Actually one of the reasons for finding new characterizations of the Sobolev space is the development of analysis on metric spaces; see e.g. [1], [2], [5], [6], [12], [13], [15]–[19], [21], [23], [24], [28]–[31], [38], [47]–[49], [51]. More references will be given later. In order to define a Sobolev type space on a metric-measure space we need a characterization of the space W 1,p(Rn) that does not involve derivatives. One such characterization is given in the following result. Theorem 1 ([19]). u ∈W 1,p(Rn), 1 < p <∞, if and only if u ∈ Lp(Rn) and there exists 0 ≤ g ∈ Lp(Rn) such that |u(x)− u(y)| ≤ |x− y|(g(x) + g(y)) a.e. (1) 2000 Mathematics Subject Classification: Primary 46E35. This work was supported by the KBN grant 2 PO3A 028 22. The research was completed during the stay of the author in Department of Mathematics at University of Michigan. The author wishes to thank the University for the support and hospitality.