Daniel Girela

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.

GENERALIZED HILBERT OPERATORS

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

Growth of the derivative of bounded analytic functions

A well known result of Littlewood and Paley states that if 2 ≦p ≦ ∞ and f is a function analytic in the unit disc Δ which belongs to the Hardy space HP then This result does not remain true for 0  p  2. In this paper we prove that if 0  p  2 then there exists a function f analytic in δ and continuous on δ such that

Inner Functions in Lipschitz, Besov, and Sobolev Spaces

We study the membership of inner functions in Besov, Lipschitz, and Hardy-Sobolev spaces, finding conditions that enable an inner function to be in one of these spaces. Several results in this direction are given that complement or extend previous works on the subject from different authors. In particular, we prove that the only inner functions in either any of the Hardy-Sobolev spaces 𝐻𝑝𝛼 with 1/𝑝≤𝛼<∞ or any of the Besov spaces 𝐵𝛼𝑝,𝑞 with 0<𝑝,𝑞≤∞ and 𝛼≥1/𝑝, except when 𝑝=∞, 𝛼=0, and 2<𝑞≤∞ or when 0<𝑝<∞, 𝑞=∞, and 𝛼=1/𝑝 are finite Blaschke products. Our assertion for the spaces 𝐵0∞,𝑞, 0<𝑞≤2, follows from the fact that they are included in the space VMOA. We prove also that for 2<𝑞<∞, VMOA is not contained in 𝐵0∞,𝑞 and that this space contains infinite Blaschke products. Furthermore, we obtain distinct results for other values of 𝛼 relating the membership of an inner function 𝐼 in the spaces under consideration with the distribution of the sequences of preimages {𝐼−1(𝑎)}, |𝑎|<1. In addition, we include a section devoted to Blaschke products with zeros in a Stolz angle.

On a theorem of Privalov and normal functions

A well known result of Privalov asserts that if f is a function which is analytic in the unit disc ∆ = {z ∈ C : |z| < 1}, then f has a continuous extension to the closed unit disc and its boundary function f(eiθ) is absolutely continuous if and only if f ′ belongs to the Hardy space H1. In this paper we prove that this result is sharp in a very strong sense. Indeed, if, as usual, M1(r, f ′) = 1 2π ∫ π −π ∣∣f ′(reiθ)∣∣ dθ, we prove that for any positive continuous function φ defined in (0, 1) with φ(r) → ∞, as r → 1, there exists a function f analytic in ∆ which is not a normal function and with the property that M1(r, f ′) ≤ φ(r), for all r sufficiently close to 1.

Mean growth of the derivative of infinite blaschke products

If f is an analytic function in the unit disc and 0<r<1, we set and we let n(r,f) denote the number of zeros of f; in the disc . We prove that if B is an interpolating Blaschke product with positive zeros then the quantities M 1(r,B 1) and n(r,B) are comparable and we use this result to prove that for any positive continuous function φ defined in [0,1) with φ(r)∞, as r←1, there exists an infinite Blaschke product B such that as r←1. This latter result was previously known but our proof is simpler than the original one. We also obtain simplified proofs of some other related results.

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.

A generalized Hilbert operator acting on conformally invariant spaces

If

Embedding derivatives of weighted Hardy spaces into Lebesgue spaces

\mu