Embedding theorems and integration operators on Bergman spaces with rapidly decreasing weights

Abstract

We completely describe those positive Borel measures μ in the unit disc D such that the Bergman space A p ( w ) ⊂ L q ( μ ) , 0 < p , q < ∞ , where w belongs to a large class W of rapidly decreasing weights which includes the exponential weights w α ( r ) = exp ( − 1 ( 1 − r ) α ) , α > 0 , and some double exponential type weights. As an application of that result, we characterize the boundedness and the compactness of T g : A p ( w ) → A q ( w ) , 0 < p , q < ∞ , w ∈ W , where T g is the integration operator ( T g f ) ( z ) = ∫ 0 z f ( ζ ) g ′ ( ζ ) d ζ . The particular choice of the weight w α ( r ) answers an open question posed by A. Aleman and A. Siskakis. We also describe those analytic functions in D for which T g belongs to the Schatten p-class of A 2 ( w ) , 0 < p < ∞ .