arXiv subjects

José Ángel Peláez

Publications and source records attributed to José Ángel Peláez.

40 records · Page 3Linked to original sources

arXiv2013

Jointly maximal products in weighted growth spaces

It is shown that for any non-decreasing, continuous and unbounded doubling function $\om$ on $[0,1)$, there exist two analytic infinite products $f_0$ and $f_1$ such that the asymptotic relation $|f_0(z)| + |f_1(z)| \asymp \om(|z|)$ is satisfied for all $z$ in the unit disc. It is also shown that both functions $f_j$ for $j=0,1$ satisfy $T(r,f_j)\asymp\logω(r)$, as $r\to1^-$, and hence give examples of analytic functions for which the Nevanlinna characteristic admits the regular slow growth induced by $ω$.

math.CV↗

arXiv2012

Multipliers of Dirichlet subspaces of the Bloch space

For $0<p<\infty $ we let $\mathcal D^p_{p-1}$ denote the space of those functions $f$ which are analytic in the unit disc $\mathbb D $ and satisfy $\int_\mathbb D (1-| z|)\sp {p-1}| f'(z)| \sp p\,dA(z)<\infty $. It is known that, whenever $p\neq q$, the only multiplier from $\mathcal D^p_{p-1} $ to $\mathcal D^q_{q-1} $ is the trivial one. However, if $X$ is a subspace of the Bloch space and $0<p\le q<\infty$, then $X \cap \mathcal D^p_{p-1}\subset X\cap \mathcal D^q_{q-1} $, a fact which implies that the space of multipliers $\M(\mathcal D^p_{p-1}\cap X, \mathcal D^q_{q-1} \cap X)$ is non-trivial. In this paper we study the spaces of multipliers $\M(\mathcal D^p_{p-1}\cap X, v\cap X)$ ($0<p,q<\infty $) for distinct classical subspaces $X$ of the Bloch space. Specifically, we shall take $X$ to be $H^\infty $, $BMOA$ and the Bloch space $\mathcal B $.

math.CV↗

arXiv2012

Weighted Bergman spaces induced by rapidly incresing weights

This monograph is devoted to the study of the weighted Bergman space $A^p_\om$ of the unit disc $\D$ that is induced by a radial continuous weight $\om$ satisfying {equation}\label{absteq} \lim_{r\to 1^-}\frac{\int_r^1\om(s)\,ds}{\om(r)(1-r)}=\infty.\tag† {equation} Every such $A^p_\om$ lies between the Hardy space $H^p$ and every classical weighted Bergman space $A^p_\a$. Even if it is well known that $H^p$ is the limit of $A^p_\a$, as $\a\to-1$, in many respects, it is shown that $A^p_\om$ lies "closer" to $H^p$ than any $A^p_\a$, and that several finer function-theoretic properties of $A^p_\a$ do not carry over to $A^p_\om$. As to concrete objects to be studied, positive Borel measures $μ$ on $\D$ such that $A^p_\om\subset L^q(μ)$, $0<p\le q<\infty $, are characterized in terms of a neat geometric condition involving Carleson squares. It is also proved that each $f\in A^p_\om$ can be represented in the form $f=f_1\cdot f_2$, where $f_1\in A^{p_1}_\om$, $f_2\in A^{p_2}_\om$ and $\frac{1}{p_1}+ \frac{1}{p_2}=\frac{1}{p}$. Because of the tricky nature of $A^p_\om$ several new concepts are introduced. It gives raise to a some what new approach to the study of the integral operator $$ T_g(f)(z)=\int_{0}^{z}f(ζ)\,g'(ζ)\,dζ. $$ This study reveals the fact that $T_g:A^p_\om\to A^p_\om$ is bounded if and only if $g$ belongs to a certain space of analytic functions that is not conformally invariant. The symbols $g$ for which $T_g$ belongs to the Schatten $p$-class $\SSS_p(A^2_\om)$ are also described. Furthermore, techniques developed are applied to the study of the growth and the oscillation of analytic solutions of (linear) differential equations.

math.CV↗

arXiv2012

Generalized Hilbert Operators

If $g$ is an analytic function in the unit disc $\D $ we consider the generalized Hilbert operator $\hg$ defined by {equation*}\label{H-g} \mathcal{H}_g(f)(z)=\int_0^1f(t)g'(tz)\,dt. {equation*} We study these operators acting on classical spaces of analytic functions in $\D $. More precisely, we address the question of characterizing the functions $g$ for which the operator $\hg $ is bounded (compact) on the Hardy spaces $H^p$, on the weighted Bergman spaces $A^p_α$ or on the spaces of Dirichlet type $\mathcal D^p_α$.

math.CV↗