arXiv subjects

Noel Merchán

Publications and source records attributed to Noel Merchán.

9 recordsLinked to original sources

arXiv2024

Generalized Cesàro operator acting on Hilbert spaces of analytic functions

Let $\mathbb{D}$ denote the unit disc in $\mathbb{C}$. We define the generalized Cesàro operator as follows $$ C_ω(f)(z)=\int_0^1 f(tz)\left(\frac{1}{z}\int_0^z B^ω_t(u)\,du\right)\,ω(t)dt,$$ where $\{B^ω_ζ\}_{ζ\in\mathbb{D}}$ are the reproducing kernels of the Bergman space $A^2_ω$ induced by a radial weight $ω$ in the unit disc $\mathbb{D}$. We study the action of the operator $C_ω$ on weighted Hardy spaces of analytic functions $\mathcal{H}_γ$, $γ>0$ and on general weighted Bergman spaces $A^2_μ$.

math.CV↗

arXiv2022

Hilbert-type operator induced by radial weight on Hardy spaces

We consider the Hilbert-type operator defined by $$ H_ω(f)(z)=\int_0^1 f(t)\left(\frac{1}{z}\int_0^z B^ω_t(u)\,du\right)\,ω(t)dt,$$ where $\{B^ω_ζ\}_{ζ\in\mathbb{D}}$ are the reproducing kernels of the Bergman space $A^2_ω$ induced by a radial weight $ω$ in the unit disc $\mathbb{D}$. We prove that $H_ω$ is bounded on the Hardy space $H^p$, $1<p<\infty$, if and only if \begin{equation} \label{abs1} \sup_{0\le r<1} \frac{\widehatω(r)}{\widehatω\left( \frac{1+r}{2}\right)}<\infty, \tag† \end{equation} and \begin{equation*} \sup\limits_{0<r<1}\left(\int_0^r \frac{1}{\widehatω(t)^p} dt\right)^{\frac{1}{p}} \left(\int_r^1 \left(\frac{\widehatω(t)}{1-t}\right)^{p'}\,dt\right)^{\frac{1}{p'}} <\infty, \end{equation*} where $\widehatω(r)=\int_r^1 ω(s)\,ds$. We also prove that $H_ω: H^1\to H^1$ is bounded if and only if \eqref{abs1} holds and $$ \sup\limits_{r \in [0,1)} \frac{\widehatω(r)}{1-r} \left(\int_0^r \frac{ds}{\widehatω(s)}\right)<\infty.$$ As for the case $p=\infty$, $H_ω$ is bounded from $H^\infty$ to $BMOA$, or to the Bloch space, if and only if \eqref{abs1} holds. In addition, we prove that there does not exist radial weights $ω$ such that $H_ω: H^p \to H^p $, $1\le p<\infty$, is compact and we consider the action of $H_ω$ on some spaces of analytic functions closely related to Hardy spaces.

math.CV↗

arXiv2020

Multipliers and integration operators between conformally invariant spaces

In this paper we are concerned with two classes of conformally invariant spaces of analytic functions in the unit disc $\D$, the Besov spaces $B^p$ $(1\le p<\infty )$ and the $Q_s$ spaces $(0<s<\infty )$. Our main objective is to characterize for a given pair $(X, Y)$ of spaces in these classes, the space of pointwise multipliers $M(X, Y)$, as well as to study the related questions of obtaining characterizations of those $g$ analytic in $\D $ such that the Volterra operator $T_g$ or the companion operator $I_g$ with symbol $g$ is a bounded operator from $X$ into $Y$.

math.CV↗

arXiv2019

Semigroups of composition operators in analytic Morrey spaces

Analytic Morrey spaces belong to the class of function spaces which, like BMOA, are defined in terms of the degree of oscillation on the boundary of functions analytic in the unit disc. We consider semigroups of composition operators on these spaces and focus on the question of strong continuity. It is shown that these semigroups behave like on BMOA.

math.CV↗

arXiv2018

Hankel matrices acting on the Hardy space $H^1$ and on Dirichlet spaces

If $\,μ\,$ is a finite positive Borel measure on the interval $\,[0,1)$, we let $\,\mathcal H_μ\,$ be the Hankel matrix $\,(μ_{n, k})_{n,k\ge 0}\,$ with entries $\,μ_{n, k}=μ_{n+k}$, where, for $\,n\,=\,0, 1, 2, \dots $, $μ_n\,$ denotes the moment of order $\,n\,$ of $\,μ$. This matrix induces formally the operator $\,\mathcal{H}_μ(f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} μ_{n,k}{a_k}\right)z^n\,$ on the space of all analytic functions $\,f(z)=\sum_{k=0}^\infty a_kz^k\,$, in the unit disc $\,\mathbb D $. When $\,μ\,$ is the Lebesgue measure on $\,[0,1)\,$ the operator $\,\mathcal H_μ\,$ is the classical Hilbert operator $\,\mathcal H\,$ which is bounded on $\,H^p\,$ if $\,1<p<\infty $, but not on $\,H^1$. J. Cima has recently proved that $\,\mathcal H\,$ is an injective bounded operator from $\,H^1\,$ into the space $\,\mathscr C\,$ of Cauchy transforms of measures on the unit circle. \par The operator $\,\mathcal H_μ\,$ is known to be well defined on $\,H^1\,$ if and only if $\,μ\,$ is a Carleson measure and in such a case we have that $\mathcal H_μ(H^1)\subset \,\mathscr C$. Furthermore, it is bounded from $\,H^1\,$ into itself if and only if $\,μ\,$ is a $1$-logarithmic $1$-Carleson measure. \par In this paper we prove that when $\,μ\,$ is a $1$-logarithmic $1$-Carleson measure then $\,\mathcal H_μ\,$ actually maps $\,H^1\,$ into the space of Dirichlet type $\,\mathcal D^1_0\,$. We discuss also the range of $\,\mathcal H_μ\,$ on $\,H^1\,$ when $\,μ\,$ is an $α$-logarithmic $1$-Carleson measure ($0<α<1$). We study also the action of the operators $\,\mathcal H_μ\,$ on Bergman spaces and on Dirichlet spaces.

math.CV↗

arXiv2018

A family of Dirichlet-Morrey spaces

To each weighted Dirichlet space $\mathcal{D}_p$, $0<p<1$, we associate a family of Morrey-type spaces ${\mathcal{D}}_p^λ$, $0< λ< 1$, constructed by imposing growth conditions on the norm of hyperbolic translates of functions. We indicate some of the properties of these spaces, mention the characterization in terms of boundary values, and study integration and multiplication operators on them.

math.CV↗

arXiv2017

Mean Lipschitz spaces and a generalized Hilbert operator

If $μ$ is a positive Borel measure on the interval $[0, 1)$ we let $\mathcal H_μ$ be the Hankel matrix $\mathcal H_μ=(μ_{n, k})_{n,k\ge 0}$ with entries $μ_{n, k}=μ_{n+k}$, where, for $n\,=\,0, 1, 2, \dots $, $μ_n$ denotes the moment of order $n$ of $μ$. This matrix induces formally the operator $$\mathcal{H}_μ(f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} μ_{n,k}{a_k}\right)z^n$$ on the space of all analytic functions $f(z)=\sum_{k=0}^\infty a_kz^k$, in the unit disc $\mathbb{D} $. This is a natural generalization of the classical Hilbert operator. In this paper we study the action of the operators $\mathcal H_μ$ on mean Lipschitz spaces of analytic functions.

math.CV↗

arXiv2017

A Hankel matrix acting on spaces of analytic functions

If $μ$ is a positive Borel measure on the interval $[0, 1)$ we let $\mathcal H_μ$ be the Hankel matrix $\mathcal H_μ=(μ_{n, k})_{n,k\ge 0}$ with entries $μ_{n, k}=μ_{n+k}$, where, for $n\,=\,0, 1, 2, \dots $, $μ_n$ denotes the moment of order $n$ of $μ$. This matrix induces formally the operator $$\mathcal{H}_μ(f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} μ_{n,k}{a_k}\right)z^n$$ on the space of all analytic functions $f(z)=\sum_{k=0}^\infty a_kz^k$, in the unit disc $\mathbb D $. This is a natural generalization of the classical Hilbert operator. In this paper we improve the results obtained in some recent papers concerning the action of the operators $H_μ$ on Hardy spaces and on Möbius invariant spaces.

math.CV↗

arXiv2017

A generalized Hilbert operator acting on conformally invariant spaces

If $μ$ is a positive Borel measure on the interval $[0, 1)$ we let $\mathcal H_μ$ be the Hankel matrix $\mathcal H_μ=(μ_{n, k})_{n,k\ge 0}$ with entries $μ_{n, k}=μ_{n+k}$, where, for $n\,=\,0, 1, 2, \dots $, $μ_n$ denotes the moment of orden $n$ of $μ$. This matrix induces formally the operator $$\mathcal{H}_μ(f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} μ_{n,k}{a_k}\right)z^n$$ on the space of all analytic functions $f(z)=\sum_{k=0}^\infty a_kz^k$, in the unit disc $\D $. This is a natural generalization of the classical Hilbert operator. The action of the operators $H_{μ}$ on Hardy spaces has been recently studied. This paper is devoted to study the operators $H_μ$ acting on certain conformally invariant spaces of analytic functions on the disc such as the Bloch space, $BMOA$, the analytic Besov spaces, and the $Q_s$ spaces.

math.CV↗