Nuclear Embeddings in Weighted Function Spaces
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Integral Equations and Operator Theory
Nuclear Embeddings in Weighted Function Spaces Dorothee D. Haroske
and Leszek Skrzypczak
Abstract. We study nuclear embeddings for weighted spaces of Besov and Triebel–Lizorkin type where the weight belongs to some Muckenhoupt class and is essentially of polynomial type. Here we can extend our previous results concerning the compactness of corresponding embeddings. The concept of nuclearity was introduced by A. Grothendieck in 1955. Recently there is a refreshed interest to study such questions. This led us to the investigation in the weighted setting. We obtain complete characterisations for the nuclearity of the corresponding embedding. Essential tools are a discretisation in terms of wavelet bases, operator ideal techniques, as well as a very useful result of Tong about the nuclearity of diagonal operators acting in p spaces. In that way we can further contribute to the characterisation of nuclear embeddings of function spaces on domains. Mathematics Subject Classification. Primary 46E35; Secondary 47B10. Keywords. Nuclear embeddings, Weighted Besov spaces, Weighted Triebel–Lizorkin spaces, Radial spaces.
1. Introduction Grothendieck introduced the concept of nuclearity in [14] more than 60 years ago. It paved the way to many famous developments in functional analysis later one, like the theories of nuclear locally convex spaces, operator ideals, eigenvalue distributions, and traces and determinants in Banach spaces. Enflo used nuclearity in his famous solution [10] of the approximation problem, a long-standing problem of Banach from the Scottish Book. We refer to [29,31], and, in particular, to [32] for further historic details. Let X, Y be Banach spaces, T ∈ L(X, Y ) a linear and bounded operator. Then T is called nuclear, denoted by T ∈ N (X, Y ), if there exist Both authors were partially supported by the German Research Foundation (DFG), Grant No. Ha 2794/8-1. The second author was supported by National Science Center, Poland, Grant No. 2013/10/A/ST1/00091. 0123456789().: V,-vol
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elements aj ∈ X , the dual space of X, and yj ∈ Y , j ∈ N, such that ∞ ∞ j=1 aj X yj Y < ∞ and a nuclear representation T x = j=1 aj (x)yj for any x ∈ X. Together with the nuclear norm ν(T ) = inf
∞ j=1
aj X yj Y : T =
∞
aj (·)yj ,
j=1
where the infimum is taken over all nuclear representations of T , the space N (X, Y ) becomes a Banach space. It is obvious that nuclear operators are, in particular, compact. Already in the early years there was a strong interest to study examples of nuclear operators beyond diagonal operators in p sequence spaces, where a complete answer was obtained in [43] (with some partial forerunner in [29]). Concentrating on embedding operators in spaces of Sobolev type, first results can be found, for instance, in [28,33,34]. Though the topic was always studied to a certain extent, we realised an increased interest in the last years. Concentrating on the Sobolev embedding for spaces
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