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Composition followed by differentiation between Bergman and Hardy spaces. (English) Zbl 1079.47031
Let $\phi$ be a non-constant analytic self map on the unit disc $D$ of the complex plane. The purpose of this note is to characterize maps $\phi$ for which the operator $(D C_{\phi})(f)=(f \circ \phi)'$ is bounded or compact between weighted Bergman spaces $A^p_{\alpha}$ and $A^q_{\beta}$, $1 \leq p \leq q$, $\alpha, \beta > -1$. This operator is bounded if and only if a related measure satisfies a Carleson type condition. A result of Luecking which uses Khinchine’s inequality plays an important role in the proofs. The methods developed in the paper are also utilized to study the operator $C_{\phi} \circ D$.

##### MSC:
 47B33 Composition operators 47B38 Operators on function spaces (general) 30H05 Bounded analytic functions
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##### References:
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