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Composition operators on small spaces. (English) Zbl 1114.47028
Let A α,s p denote the space of holomorphic functions f on the unit disk such that 𝔻 | s f(z)| p (1-|z| 2 ) α dA(z)<, that is to say that its s-fractional derivative s f(z)= n=0 (n+1) α a n z n belongs to A α p . The authors analyze the boundedness and compactness of C φ on the spaces A α,s p in some particular situations. Motivated by the notion of “suitably small” Banach space, they denote by the set of parameters (s,p,α) such that either sp>α+2 or sp=α+2,0<p1. Their main results establish that if (s-1,p,α) (where s is assumed to be a positive integer for α=-1) and φA α,s p , then the composition operator C φ is bounded on A α,s-k p for all k + ,ks. In particular, they obtain in such a case that C φ is bounded on A α,s p if and only if φA α,s p . As a byproduct, they get that A α,s p is an algebra under multiplication for (s,p,α). Among other interesting results in the paper, the authors also give a complete description of symbols for the boundedness of C φ on the Zygmund class Λ 1 of functions such that |f '' (z)|=O1 1-|z|. Finally, some examples concerning the boundedness of C φ on A α,s p if α+1spα+2+p are provided.
MSC:
47B33Composition operators
46E15Banach spaces of continuous, differentiable or analytic functions
30D55H (sup p)-classes (MSC2000)