The dimension of a closed subset ofR n and related function spaces

Abstract

Let F 6= ∅ be a closed subset of Rn with empty interior. There are several proposals what should be called the dimension of F , globally and locally. Besides the classical Hausdorff dimension there exist nearby but, in general, not identical definitions, better adapted to the needs of measure theory, see [14] and also [9] and [7]: Ch.II,1. One aim of our paper is to contribute to this field of research by introducing two types of a dimension of F , the distributional dimension and the cascade dimension. The first notion is connected with the question whether there exist nontrivial singular distributions with a support on F and belonging to some function spaces of Besov type on Rn. The second notion is related to the eentropy and ecapacity of F and its neighbourhood and is connected with atomic representations of function spaces. Our approach is intimately linked with function spaces on Rn and on F . Hence, the second aim of this paper is to introduce some function spaces of Besov type on F . In that sense this note might be considered as a direct continuation of our paper [13]. On the other hand function spaces on (closed) subsets of Rn have been studied extensively by A.Jonsson and H.Wallin, see [5], [6], [7], [8] and [15]. Our approach is closely related to this work and should also be seen in the context of the theory developed there. The paper has two sections. Section 2 deals with function spaces on Rn and on F and related problems: dimension, extension, duality. In Section 3 we introduce two notions of dimensions. Our main results are the Theorems 2.3 and 3.3. All unimportant positive constants are denoted by c, occasionally with additional subsripts within the same formula or the same step of the proof. Furthermore, (k.l/m) refers to formula (m) in subsection k.l, whereas (j) means formula (j) in the same subsection. Similarly we refer to remarks, theorems etc.