arXiv subjects
Publications and source records attributed to Daniel Girela.
We study the boundedness and the compactness of Rhaly and generalized Hilbert operators when acting between distinct spaces of analytic functions in the unit disc of the complex plane.
In this article we address the question of characterizing the sequences of complex numbers $(\eta )=\{ \eta_n\}_{n=0}^\infty $ whose associated Rhaly operator $\mathcal R_{(\eta )}$ is bounded or compact on the Hardy spaces $H^p$ ($1\le p<\infty $), on the Bergman spaces $A^p_\alpha $, and on the Dirichlet spaces $\mathcal D^p_\alpha $ ($1\le p<\infty $, $\alpha >-1$). We give a number of conditions which are either necessary or sufficient for the boundedness (compactness) of $\mathcal R_{(\eta )}$ on these spaces. These conditions have to do with the membership in certain mean Lipschitz spaces of analytic functions of the function $F_{(\eta )}$ defined by $F_{(\eta )}(z)=\sum_{n=0}^\infty \eta _nz^n$ ($z\in \mathbb D$). \par We prove that if $2\le p<\infty $ and $\eta_n=\og \left (\frac{1}{n}\right )$, then $\mathcal R_{(\eta )}$ is bounded on $H^p$. However, there exists a sequence $(\eta )$ with $\eta_n=\og \left (\frac{1}{n}\right )$ such that the operator $\mathcal R_{(\eta )}$ is not bounded on $H^p$ for $1\le p<2$. \par We deal also with the derivative-Hardy spaces. For $p>0$ the derivative-Hardy space $S^p$ consists of those functions $f$, analytic in the unit disc $\mathbb D$, such that $f^\prime \in H^p$. We prove that if $1\le p<\infty $ and $1<q<\infty $ then $\mathcal R_{(\eta )}$ is a bounded operator from $S^p$ into $S^q$ if and only if it is compact and this happens if and only if $F_{(\eta )}\in S^q$.
If $(\eta )=\{ \eta_n\} _{n=0}^\infty $ is a sequence of complex numbers, the Ces\`aro-type operator $\mathcal C_{(\eta )}$ is formally defined in the space of analytic funtions in the unit disc $\mathbb D$ as follows: If $f$ is an analytic function in $\mathbb D$, $f(z)=\sum_{n=0}^\infty a_nz^n$ ($z\in \mathbb D$), then $\mathcal C_{(\eta )}(f)$ is formally defined by $$\mathcal C_{(\eta )}(f)(z)=\mathcal C_{\{\eta_n\}}(f)(z)=\sum_{n=0}^\infty \eta _n\left (\sum_{k=0}^na_k\right )z^n.$$ The operator $\mathcal C_{(\eta )}$ is a natural generalization of the Ces\`{a}ro operator. For each $\alpha\in \mathbb R$ we let $\mathcal D^2_\alpha$ be the space of functions $f\in\hol(\mathbb D)$ such that $|a_0|^2+\sum_{n=1}^\infty n^{1-\alpha} |a_n|^2<\infty$ where$f(z)=\sum_{n=0}^\infty a_nz^n$. In this paper we give a complete characterization of the sequences of complex numbers $(\eta )$ for which the operator $\mathcal C_{(\eta )}$ is bounded (compact) from $\mathcal D^2_\alpha $ into $\mathcal D^2_\beta $ for any $\alpha , \beta \in \mathbb R$.
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 $Y$.
We prove that for every $p\ge 1$ there exists a bounded function in the analytic Besov space $B^p$ whose derivative is "badly integrable", along every radius. We apply this result to study multipliers and weighted superposition operators acting on the spaces $B^p$.
We give new constructions of pair of functions $(f, g)$, analytic in the unit disc, with $g\in H^\infty $ and $f$ an unbounded Bloch function, such that the product $g\cdot f$ is not a Bloch function.
We use properties of the sequences of zeros of certain spaces of analytic functions in the unit disc $\mathbb D$ to study the question of characterizing the weighted superposition operators which map one of these spaces into another. We also prove that for a large class of Banach spaces of analytic functions in $\mathbb D$, $Y$, we have that if the superposition operator $S_φ$ associated to the entire function $φ$ is a bounded operator from $X$, a certain Banach space of analytic functions in $\mathbb D$, into $Y$, then the superposition operator $S_{φ^\prime }$ maps $X$ into $Y$.
We use the Baernstein star-function to investigate several questions about the integral means of the convolution of two analytic functions in the unit disc. The theory of univalent functions plays a basic role in our work.
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.
For $0 -1$ the space of Dirichlet type $\mathcal D^p_α$ consists of those functions $f$ which are analytic in the unit disc $\mathbb D$ and satisfy $\int_{\mathbb D}(1-| z| )^α| f^\prime (z)| ^p\,dA(z)<\infty $. The space $\Dp$ is the closest one to the Hardy space $H^p$ among all the $\mathcal D^p_α$. Our main object in this paper is studying similarities and differences between the spaces $H^p$ and $\Dp$ ($0<p<\infty $) regarding superposition operators. Namely, for $0<p<\infty $ and $0<s<\infty $, we characterize the entire functions $φ$ such that the superposition operator $S_φ$ with symbol $φ$ maps the conformally invariant space $Q_s$ into the space $\Dp$, and, also, those which map $\Dp$ into $Q_s$ and we compare these results with the corresponding ones with $H^p$ in the place of $\Dp$. We also study the more general question of characterizing the superposition operators mapping $\mathcal D^p_α$ into $Q_s$ and $Q_s$ into $\mathcal D^p_α$, for any admissible triplet of numbers $(p, α, s)$.
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.
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.
If $μ$ is a positive Borel measure on the interval $[0, 1)$, the Hankel matrix $\mathcal H_μ=(μ_{n,k})_{n,k\ge 0}$ with entries $μ_{n,k}=\int_{[0,1)}t^{n+k}\,dμ(t)$ 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} $. In this paper we describe those measures $μ$ for which $\mathcal{H}_μ$ is a bounded (compact) operator from $H^p$ into $H^q$, $0<p,q<\infty $. We also characterize the measures $μ$ for which $\mathcal H_μ$ lies in the Schatten class $S_p(H^2)$, $1<p<\infty$.
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 $.
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 $\mathcal D^p_α$.