Let $A^p_\omega$ denote the Bergman space in the unit disc induced by a
radial weight~$\omega$ with the doubling property $\int_{r}^1\omega(s)\,ds\le
C\int_{\frac{1+r}{2}}^1\omega(s)\,ds$. The positive Borel measures such that
the differentiation operator of order $n\in\mathbb{N}\cup\{0\}$ is bounded from
$A^p_\omega$ into $L^q(\mu)$ are characterized in terms of geometric conditions
when $0
The motivation of this paper comes from the two weight inequalityfor the Bergman projection Pω in the unit disc. We show that the boundedness of Pω on L p v is characterized in terms of self-improving Muckenhoupt and Bekollé-Bonami type conditions when the radial weights v and ω admit certain smoothness. En route to the proof we describe the asymptotic behavior of the L p -means and the L p v -integrability of the reproducing kernels of the weighted Bergman space A 2 ω .Date: December 16, 2014.
For n ∈ N, the n-order of an analytic function f in the unit disc D is defined bywhere log + x = max{log x, 0}, log + 1 x = log + x, log + n+1 x = log + log + n x, and M (r, f ) is the maximum modulus of f on the circle of radius r centered at the origin. It is shown, for example, that the solutions f of the complex linear differential equationwhere the coefficients are analytic in D, satisfy σM,n+1(f ) ≤ α if and only if σM,n(aj) ≤ α for all j = 0, . . . , k − 1. Moreover, if q ∈ {0, . . . , k − 1} is the largest index for which σM,n(aq) = max 0≤j≤k−1 σM,n(aj) , then there are at least k − q linearly independent solutions f of ( †) such that σM,n+1(f ) = σM,n(aq).Some refinements of these results in terms of the n-type of an analytic function in D are also given.
Mathematics Subject Classification (2000). Primary 34M10; Secondary 30D35.
Abstract. We address the problem of studying the boundedness, compactness and weak compactness of the integral operators Tg(f )(z) = z 0 f (ζ)g ′ (ζ) dζ acting from a Banach space X into H ∞ . We obtain a collection of general results which are appropriately applied and mixed with specific techniques in order to solve the posed questions to particular choices of X.
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