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 operatorThe 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 < ∞.
ABSTRACT. We completely describe the boundedness of the Volterra type operator J g between Hardy spaces in the unit ball of C n . The proof of the one dimensional case used tools, such as the strong factorization for Hardy spaces, that are not available in higher dimensions, and therefore other techniques must be used. In particular, a generalized version of the description of Hardy spaces in terms of the area function is needed.
The space Q s , 0 ≤ s < ∞, consists of those f which are analytic in the unit disc D and such thatgive a complete description of M(Q s ), the algebra of pointwise multipliers of Q s by proving Xiao's conjecture which says that f ∈ M(Q s ) if and only if f ∈ H ∞ and sup a∈D log 2 1 − |a| 2 D | f (z)| 2 (1 − |ϕ a (z)| 2 ) s d A(z) < ∞.
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