Cristóbal González

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.

Asymptotic behavior of solutions of difference equations in Banach spaces

In this paper we consider the first order difference equation = Zan f (Xn+i), i=O and give necessary and sufficient conditions so that there exist solutions which are asymptotically constant. These results generalize those given earlier by Popenda and Schmeidel. As an application we give necessary and sufficient conditions for the second order difference equation A (qnAxn) + pnf (Xn) = 0 to have asymptotically constant solutions.

EXISTENCE OF MONOTONIC ASYMPTOTICALLY CONSTANT SOLUTIONS FOR SECOND ORDER DIFFERENTIAL EQUATIONS

Starting from results of Dube and Mingarelli, Wahlen, and Ehrstrom, who give conditions that ensure the existence and uniqueness of nonnegative nondecreasing solutions asymptotically constant of the equation we have been able to reduce their hypotheses in order to obtain the same existence results, at the expense of losing the uniqueness part. The main tool they used is the Banach Fixed Point Theorem, while ours has been the Schauder Fixed Point Theorem together with one version of the Arzela-Ascoli Theorem.

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

A subspace X of the Hardy space H 1 is said to have the f-property if h/I ∈ X whenever h ∈ X and I is an inner function with h/I e H 1 . We let B denote the space of Bloch functions and β 0 the little Bloch space. Anderson proved in 1979 that the space β 0 ∩H ∞ does not have the f-property. However, the question of whether or not B ∩ H p (1 < p < ∞) has the f-property was open. We prove that for every p ∈ [1, oo) the space β ∩ H p does not have the f-property. We also prove that if B is any infinite Blaschke product with positive zeros and G is a Bloch function with |G(z)| → ∞, as z → 1, then the product BG is not a Bloch function.

Asymptotic Behavior of Solutions to a Vector Integral Equation with Deviating Arguments

In this paper, we propose the study of an integral equation, with deviating arguments, of the type in the context of Banach spaces, with the intention of giving sufficient conditions that ensure the existence of solutions with the same asymptotic behavior at as . A similar equation, but requiring a little less restrictive hypotheses, is In the case of , its solutions with asymptotic behavior given by yield solutions of the second order nonlinear abstract differential equation with the same asymptotic behavior at as .

Oscillatory Solutions for Second-Order Difference Equations in Hilbert Spaces

We consider the difference equation , , in the context of a Hilbert space. In this setting, we propose a concept of oscillation with respect to a direction and give sufficient conditions so that all its solutions be directionally oscillatory, as well as conditions which guarantee the existence of directionally positive monotone increasing solutions.