We investigate function spaces of generalised smoothness of Besov and TriebelLizorkin type. Equivalent quasi-norms in terms of maximal functions and local means are given. An atomic decomposition theorem for this type of spaces is proved.
Mathematics Subject Classification (2000). 46E35
We investigate the asymptotic behaviour of the entropy numbers of the compact embed-denotes a weighted Besov space. We present a general approach which allows us to work with a large class of weights.
The paper is concerned with function spaces of Sobolev type and of Besov type of variable order of differentiation. The definition and the study of these spaces is closely connected with an appropriate class of pseudodifferential operators. 1980 Mathematics Subject Classification (1985 Revision): 46E35, 47G05; 35S99. The paper deals with function spaces of variable order of differentiation on the Euclidean w-space IR n . The definition of these spaces is closely connected with an appropriate class of pseudodifferential operators. In 1965-75 several papers had been published which relate function spaces and classes of pseudodifferential operators, see for example Unterberger and Bokobza [20], [21], ViSik and Eskin [22], [23]. Volevi and Kagan [24], Unterberger [19], Kumano-go and Tsutsumi [8], Beals [1]. Later on, in 1981, Beals [2] considered weighted distribution spaces based on the very general Weyl calculus of pseudodifferential operators. But all the function spaces defined and studied in these papers are spaces of Sobolev type or Besselpotential type. In contrast to this, we defined in [10] and [11] function spaces of Besov type of variable order of differentiation. Their definition is based on decompositions of R" x IR% which are induced by Symbols a(x, ξ) of appropriate pseudodifferential operators belonging to a subclass S(m,m';d) of the hypoelliptic Symbols of slowly varying strength. In this paper we define now function spaces of Sobolev type of variable order of differentiation for the same class S (m, m'\ δ) of pseudodifferential operators, and we consider their connection with the Besov spaces of variable order of differentiation defined in [l 1]. The paper is organized s follows:In the first section we recall some facts about pseudodifferential operators, the definition of the class S (m, m'; δ) of appropriate Symbols, some examples and those properties which will be needed in the sequel. In section 2 we introduce the function spaces W J p ' a (R") of Sobolev type of variable order of differentiation and describe properties of these spaces. Section 3 contains the definition and some properties of the Brought to you by |
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