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Composition followed by differentiation between Bergman and Hardy spaces. (English) Zbl 1079.47031
Let φ 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 φ for which the operator (DC φ )(f)=(fφ) ' is bounded or compact between weighted Bergman spaces A α p and A β q , 1pq, α,β>-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 φ D.
MSC:
47B33Composition operators
47B38Operators on function spaces (general)
30H05Bounded analytic functions