José Ángel Peláez

INTEGRABILITY OF THE DERIVATIVE OF A BLASCHKE PRODUCT

We study the membership of derivatives of Blaschke products in Hardy and Bergman spaces, especially for the the interpolating Blaschke products and for those whose zeros lie in a Stolz domain. We obtain new and very simple proofs of some known results and prove new theorems that complement or extend the earlier works of Ahern, Clark, Cohn, Kim, Newman, Protas, Rudin, Vinogradov and others.

Sharp results on the integrability of the derivative of an interpolating Blaschke product

Abstract The Schwarz-Pick lemma readily implies that the derivative of any Blaschke product belongs to all the Bergman spaces Ap with 0 < p < 1. It is also well known that this result is sharp: there exist a Blaschke product whose derivative does not belong to A 1. However, the question of whether there exists an interpolating Blaschke product B with B′ ∉ A 1 remained open. In this paper we give an explicit construction of such an interpolating Blaschke product B. A result of W. S. Cohn asserts that if and B is an interpolating Blaschke product with sequence of zeros of , then B′ ∈ Hp if and only if (1 – |ak |)1–p < ∞. We prove that Cohns result is no longer true for . Indeed, we construct: (a) an interpolating Blaschke product B whose sequence of zeros of satisfies (1 – |ak |)1/2 < ∞ but B′ ∉ H 1/2, and (b) an interpolating Blaschke products B whose sequence of zeros of satisfies (1 – |ak |)1–p < ∞, for all p ∈ (0, 1/2), whose derivative B′ does not belong to the Nevanlinna class.

GENERALIZED HILBERT OPERATORS

If g is an analytic function in the unit disc D, we consider the generalized Hilbert operator Hg defined by

On the zeros of functions in Dirichlet-type spaces

We study the sequences of zeros for functions in the Dirichlet spaces D s . Using Carleson-Newman sequences we prove that there are great similarities for this problem in the case 0 < s < 1 with that for the classical Dirichlet space.

Uniformly discrete sequences in regions with tangential approach to the unit circle

A known result of Newman and Tse asserts that every uniformly discrete sequence contained in a Stolz angle is uniformly separated (see Newman, D.J., 1959, Interpolation in . Transactions of the American Mathematical Society, 92(3), 501–507; Tse, K.-F., 1971, Nontangential interpolating sequences and interpolation by normal functions. Proceedings of the American Mathematical Society, 29, 351–354). We prove that this statement no longer holds if the sequence is located in a tangential region of certain kind. It is well known that a uniformly discrete sequence need not be a Blaschke sequence. We show, however, that every uniformly discrete sequence inside a disc tangential to the unit circle must be a Blaschke sequence.

BOUNDARY BEHAVIOUR OF ANALYTIC FUNCTIONS IN SPACES OF DIRICHLET TYPE

For and, we let be the space of all analytic functions in such that belongs to the weighted Bergman space. We obtain a number of sharp results concerning the existence of tangential limits for functions in the spaces. We also study the size of the exceptional set, where denotes the radial variation of along the radius, for functions.

On the boundedness of Bergman projection

The main purpose of this survey is to gather results on the boundedness of the Bergman projection. First, we shall go over some equivalent norms on weighted Bergman spaces

REMARKS ON THE AREA THEOREM IN THE THEORY OF UNIVALENT FUNCTIONS

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Multiplication and division by inner functions in the space of bloch functions

which are useful in the study of this question. In particular, we shall focus on a decomposition norm theorem for radial weights~

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

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