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

AbstractWe completely describe those positive Borel measures μ in the unit disc D such that the Bergman space Ap(w)⊂Lq(μ), 0<p,q<∞, where w belongs to a large class W of rapidly decreasing weights which includes the exponential weights wα(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 Tg:Ap(w)→Aq(w), 0<p,q<∞, w∈W, where Tg is the integration operator(Tgf)(z)=∫0zf(ζ)g′(ζ)dζ. The particular choice of the weight wα(r) answers an open question posed by A. Aleman and A. Siskakis. We also describe those analytic functions in D for which Tg belongs to the Schatten p-class of A2(w), 0<p<∞