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Composition operators between generally weighted Bloch spaces and Q log q space. (English) Zbl 1163.47019

Let H(𝔻) be the set of all holomorphic functions f in the open unit disk 𝔻. For p,q>0, the weighted Bloch space B log p is the set of all functions in H(𝔻) for which

f B log p =|f(0)|+sup z𝔻 |f ' (z)|(1-|z| 2 ) p log2 1-|z| 2 <,

and Q log q is the space of all fH(𝔻) for which

f * =sup I𝔻 log2 |I| 2 |I| q S(I) |f ' (z)| 2 log1 |z| q dm(z)<,

where dm is planar Lebesgue measure. Here, as usual, S(I) is the Carleson square {z𝔻:1-|I||z|<1,z |z|I} associated with the arc I𝔻.

For analytic selfmaps φ of 𝔻, the author characterizes those composition operators C φ :B log p Q log q , ffϕ, that are bounded, respectively compact. Also, necessary and sufficient conditions on the Taylor coefficients of a lacunary Taylor series f are given that imply that fB log p .

MSC:
47B33Composition operators
47B38Operators on function spaces (general)
30H05Bounded analytic functions