Atomic Decomposition of Weighted Besov Spaces

Abstract

We find the atomic decomposition of functions in the weighted Besov spaces under certain factorization conditions on the weight. Introduction. After achieving the atomic decomposition of Hardy spaces (see [8,22, 33]), many of the function saces have been shown to admit similar decompositions. Let us mention the decomposition of B.M.O. (see [32, 25]), Bergman spaces (see [9, 23]), the predual of Bloch space (see [ 11]), Besov spaces (see [15, 4, 10]), Lipschitz spaces (see [18]), Triebel-Lizorkin spaces (see [16, 31]),... They are obtained by quite different methods, but there is a unified and beautiful approach to get the decomposition for most of the spaces. This is the use of a formula due to A.P. Calderon (see [6, 7]). The reader is referred to the book by M. Frazier, B. Jawerth and G. Weiss [18] for a collection of spaces where the Calderon’s formula produces the atomic decomposition and applications of it. Atomic decompositions of weighted versions of different spaces have been also considered in the literature (see [27] for weighted Hardy spaces, [5] for Lipschitz spaces,...). In this paper we shall be concerned with weighted Besov spaces B φ,w. We shall find some conditions on the weights to have atomic decomposition on the spaces. We refer the reader to [19, 29, 18] for general notions and applications of atomic decomposition and to [1, 24, 30] for different formulations and properties of Besov classes. The classes of weights where the results are achieved consist of those which factorize through powers of Dini and b1 weights. Our arguments for the cases 1 < p, q < ∞ will be based upon two main points: Calderon’s formula and a quite simple Schur Lemma. To obtain the other extreme cases p, q ∈ {1,∞} we need some new results on the classes of weights which enable us to apply the same procedure as in the previous cases. The reader should be aware that the case 1 < q < ∞ could have been shown by interpolation with the extreme cases, but a direct proof is presented in the paper. 1991 Mathematics Subject Classification. 42A45, 42B25, 42C15.