arXiv subjects
Publications and source records attributed to Petros Galanopoulos.
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.
We study for the first time the action of the Hilbert matrix $$\mathcal H=(c_{n,k})_{n,k\geq 0}, \quad c_{n,k}=\frac{1}{n+k+1}$$ on the analytic tent spaces $AT^q_p, 1 1,$ stand among the $AT_{p}^{q}$ and correspond to the case $p=q$. The multiplication of the Hilbert matrix with the column matrix with entries the Taylor coefficients of an $f(z)=\sum_{k\geq 0} a_k z^k $ analytic in $\mathbb D$ introduces the series $$ \mathcal H (f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} \frac{a_k}{n+k+1}\right)z^n\,, \quad z\in \mathbb D\,\, $$ known in the literature as Hilbert operator. We prove that it is a bounded operator on the $AT_{p}^{q}$ when $1/p + 1/q <1,\, p>2$. This is a natural range for the values of the indices $p,q$ compared to what is known in the special case of the Bergman spaces. We confront the question under discussion through a more general point of view by studying an associated integral operator defined with respect to a positive Borel measure $\mu$ on $[0,1)$. Finally, we provide an estimation of the norm of the Hilbert operator. Our work extends in a non-trivially way previous results on the Bergman spaces to the analytic tent spaces.
In this article we address the question of characterizing the sequences of complex numbers $(η)=\{ η_n\}_{n=0}^\infty $ whose associated Rhaly operator $\mathcal R_{(η)}$ is bounded or compact on the Hardy spaces $H^p$ ($1\le p<\infty $), on the Bergman spaces $A^p_α$, and on the Dirichlet spaces $\mathcal D^p_α$ ($1\le p<\infty $, $α>-1$). We give a number of conditions which are either necessary or sufficient for the boundedness (compactness) of $\mathcal R_{(η)}$ on these spaces. These conditions have to do with the membership in certain mean Lipschitz spaces of analytic functions of the function $F_{(η)}$ defined by $F_{(η)}(z)=\sum_{n=0}^\infty η_nz^n$ ($z\in \mathbb D$). \par We prove that if $2\le p<\infty $ and $η_n=\og \left (\frac{1}{n}\right )$, then $\mathcal R_{(η)}$ is bounded on $H^p$. However, there exists a sequence $(η)$ with $η_n=\og \left (\frac{1}{n}\right )$ such that the operator $\mathcal R_{(η)}$ 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 $\mathcal R_{(η)}$ is a bounded operator from $S^p$ into $S^q$ if and only if it is compact and this happens if and only if $F_{(η)}\in S^q$.
If $(η)=\{ η_n\} _{n=0}^\infty $ is a sequence of complex numbers, the Cesàro-type operator $\mathcal C_{(η)}$ 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 $\mathcal C_{(η)}(f)$ is formally defined by $$\mathcal C_{(η)}(f)(z)=\mathcal C_{\{η_n\}}(f)(z)=\sum_{n=0}^\infty η_n\left (\sum_{k=0}^na_k\right )z^n.$$ The operator $\mathcal C_{(η)}$ is a natural generalization of the Cesàro operator. For each $α\in \mathbb R$ we let $\mathcal D^2_α$ be the space of functions $f\in\hol(\mathbb D)$ such that $|a_0|^2+\sum_{n=1}^\infty n^{1-α} |a_n|^2<\infty$ where$f(z)=\sum_{n=0}^\infty a_nz^n$. In this paper we give a complete characterization of the sequences of complex numbers $(η)$ for which the operator $\mathcal C_{(η)}$ is bounded (compact) from $\mathcal D^2_α$ into $\mathcal D^2_β$ for any $α, β\in \mathbb R$.
We deal with a reverse Carleson measure inequality for the tent spaces of analytic functions in the unit disc $\mathbb{D}$ of the complex plane. The tent spaces of measurable functions were introduced by Coifman, Meyer and Stein. Let $1\leq p,q < \infty$ and consider the positive Borel measure $dμ(z) = χ_{G}(z)\frac{dm(z)}{(1-|z|)}$ defined in terms of a measurable set $G \subseteq \mathbb{D}$ and of the area Lebesgue measure $dm(z)$ in $\mathbb{D}$. We prove a necessary and sufficient condition on $G$ in order to exist a constant $K>0$ such that $$ \int_{\mathbb{T}} \left(\int_{Γ(ξ)} |f(z)|^{p}\ dμ(z) \right)^{q/p}\ |dξ|\geq K \,\int_{\mathbb{T}} \left(\int_{Γ(ξ)} |f(z)|^{p}\ \frac{dm(z)}{1-|z|}\right)^{q/p}\ |dξ|, $$ for any $f$ analytic in $\mathbb{D}$ with the property, the right term of the inequality above is finite. Here $\mathbb{T}$ stands for the unit circle and $Γ(ξ)$ is a non-tangential region with vertex at $ξ\in \mathbb{T}$. This work extends the study of D. Luecking on Bergman spaces to the analytic tent spaces. We apply our result to characterize the closed range property of the integration operator known as Pommerenke operator when acting on the average radial integrability spaces. T. Aguilar-Hernández, M. Contreras and L. Rodríguez-Piazza introduced these spaces for the first time in the literature. The Hardy and the Bergman spaces form part of this family.
We prove the nontrivial variant \[ \sum\limits_{m,n=1}^{\infty}\Big(\frac{n}{m}\Big)^{\frac{1}{q}-\frac{1}{p}}\frac{a_mb_n}{m+n-1}\leq\fracπ{\sin\fracπ{p}} \Big( \sum\limits_{m=1}^{\infty}a_m^p\Big)^{\frac 1p}\Big( \sum\limits_{n=1}^{\infty}b_n^q\Big)^{\frac 1q} \] of the well known Hilbert's inequality. Then we use this to determine the exact value $\fracπ{\sin\fracπp}$ of the norm of the Hilbert matrix as an operator acting on the Hardy-Littlewood space $K^p$. This space consists of all functions $f(z)=\sum\limits_{m=0}^{\infty}a_mz^m$ analytic in the unit disc with $\|f\|_{K^p}^p=\sum\limits_{m=0}^{\infty}(m+1)^{p-2}|a_m|^p<+\infty$.
We deal with a Carleson measure type problem for the tent spaces $AT_{p}^{q}(α)$ in the unit disc of the complex plane. They consist of the analytic functions of the tent spaces $T_{p}^{q}(α)$ introduced by Coifman, Meyer and Stein. Well known spaces like the Bergman spaces arise as a special case of this family. Let $s,t,p,q\in (0,\infty)$ and $α>0\,.$ We find necessary and sufficient conditions on a positive Borel measure $μ$ of the unit disc in order to exist a positive constant $C $ such that $$ \int_{\mathbb{T}} \left(\int_{Γ(ξ)} |f(z)|^{t}\ dμ(z)\right)^{s/t}\ |dξ|\leq C \|f\|^s_{T_{p}^{q}(α)} \,,\quad f\in AT_{p}^{q}(α)\,, $$ where $Γ(ξ) = Γ_M (ξ)=\{ z\in \mathbb{D} : |1-\barξ z |< M (1-|z|^2)\},$ $M> 1/2 $ and $ξ$ is a boundary point of the unit disk. This problem was originally posed by D. Luecking. We apply our results to the study of the action of the integration operator $T_g$, also known as Pommerenke operator, between the average integrability spaces $RM(p,q) ,$ for $p,q\in [1,\infty)$. These spaces have appeared recently in the work of the first author with M. D. Contreras and L. Rodríguez-Piazza. We also consider the action from an $RM(p,q)$ to a Hardy space $H^s$, where $ p,q,s \in [1,\infty)$.
We characterize the membership of Hankel operators with general symbols in the Schatten Classes $S^p,\, p\in(0,1),$ of the large Bergman spaces $A^2_ω$. The case $p\geq 1$ was proved by Lin and Rochberg.
The aim of this article is to study semigroups of composition operators on the BMOA-type spaces $BMOA_p$, and on their "little oh" analogues $VMOA_p$. The spaces $BMOA_p$ were introduced by R. Zhao as part of the large family of F(p,q,s) spaces, and are the Möbius invariant subspaces of the Dirichlet spaces $D^p_{p-1}$. We study the maximal subspace of strong continuity, providing a sufficient condition on the infinitesimal generator of $ϕ$, under which $[ϕ_t,BMOA_p]=VMOA_p$, and a related necessary condition in the case where the Denjoy - Wolff point of the semigroup is in $\mathbb{D}$. Further, we characterize those semigroups, for which $[ϕ_t, BMOA_p]=VMOA_p$, in terms of the resolvent operator of the infinitesimal generator of $T_t$. In addition we provide a connection between the maximal subspace of strong continuity and the Volterra-type operators $T_g$. We characterize the symbols g for which $T_g$ acting from $BMOA$ to $BMOA_1$ is bounded or compact, thus extending a related result to the case $p=1$. We also prove that for $1<p<2$ compactness of $T_g$ on $BMOA_p$ is equivalent to weak compactness.
Let $μ$ be a positive Borel measure on the positive real axis. We study the integral operator $$ \mathcal{H}_μ(f)(z)=\int_{0}^{\infty}\frac{1}{t}f\left(\frac{z}{t}\right)\,dμ(t),\quad z\in \mathbb{C}\,, $$ acting on the Fock spaces $F^{p}_α$, $p\in [1,\infty],\,α>0$. Its action is easily seen to be a coefficient multiplication by the moment sequence $$ μ_n= \int_{1}^{\infty}\frac{1}{t^{n+1}}\,dμ(t) . $$ We prove that \begin{equation*} ||\mathcal{H}_μ||_{F^{p}_α\to F^{p}_α}=\sup_{n\in\mathbb{N}}μ_n,\,\,\,\,\,1\leq p\leq \infty\,\,. \end{equation*} A little-o,condition describes the compactness of $\mathcal{H}_μ$ on every $F^{p}_α,\,p\in (1,\infty )$. In addition, we completely characterize the Schatten class membership of $\mathcal{H}_μ$.
We show that the already known results for a composition operator to have closed range on H2 (Cima, Thomson, and Wogen (1974), Zorboska (1994)) can be extended to Hp for p>0 .
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.
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.
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 $(p, α, s)$.
A description of the Bloch functions that can be approximated in the Bloch norm by functions in the Hardy space $H^p$ of the unit ball of $\Cn$ for $0<p<\infty$ is given. When $0<p\leq1$, the result is new even in the case of the unit disk.
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 $\mathcal D^p_α$.
These notes are part of the research seminar with title "Theory of Bergman spaces and related function spaces" that took place in the University of Crete, Department of Mathematics (September 2006- December 2007), in the framework of the research program PYTHAGORAS II(75% European funds--25% Greek national funds).