arXiv subjects

Elena de la Rosa

Publications and source records attributed to Elena de la Rosa.

7 recordsLinked to original sources

arXiv2026

The Hilbert matrix on analytic tent spaces

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 $μ$ 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.

math.CV↗

arXiv2026

Hilbert-type operator induced by radial weight

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 from $H^\infty$ to the Bloch space if and only if $ω$ belongs to the class $\widehat{\mathcal{D}}$, which consists of radial weights $ω$ satisfying the doubling condition $\sup_{0\le r<1} \frac{\int_r^1 ω(s)\,ds}{\int_{\frac{1+r}{2}}^1ω(s)\,ds}<\infty$. Further, we describe the weights $ω\in \widehat{\mathcal{D}}$ such that $H_ω$ is bounded on the Hardy space $H^1$, and we show that for any $ω\in \widehat{\mathcal{D}}$ and $p\in (1,\infty)$, $H_ω:\,L^p_{[0,1)} \to H^p$ is bounded if and only if the Muckenhoupt type condition \begin{equation*} \sup\limits_{0<r<1}\left(1+\int_0^r \frac{1}{\widehatω(t)^p} dt\right)^{\frac{1}{p}} \left(\int_r^1 ω(t)^{p'}\,dt\right)^{\frac{1}{p'}} <\infty, \end{equation*} holds. Moreover, we address the analogous question about the action of $H_ω$ on weighted Bergman spaces $A^p_ν$.

math.CV↗

arXiv2026

Littlewood-Paley inequalities for fractional derivative on Bergman spaces

For any pair $(n,p)$, $n\in\mathbb{N}$ and $0 1$ such that $\inf_{0\le r<1} \frac{\int_{r}^1ω(s)\,ds}{\int_{1-\frac{1-r}{k}}^1 ω(s)\,ds}>1$. In this paper we extend this result to the setting of fractional derivatives. Being precise, for an analytic function $f(z)=\sum_{n=0}^\infty \widehat{f}(n) z^n$ we consider the fractional derivative $ D^μ(f)(z)=\sum\limits_{n=0}^{\infty} \frac{\widehat{f}(n)}{μ_{2n+1}} z^n $ induced by a radial weight $μ\in \mathcal{D}$, where $μ_{2n+1}=\int_0^1 r^{2n+1}μ(r)\,dr$. Then, we prove that for any $p\in (0,\infty)$, the Littlewood-Paley equivalence $$\int_{\mathbb{D}} |f(z)|^p ω(z)\,dA(z)\asymp \int_{\mathbb{D}}|D^μ(f)(z)|^p\left[\int_{|z|}^1μ(s)\,ds\right]^pω(z)\,dA(z)$$ holds for any analytic function $f$ in $\mathbb{D}$ if and only if $ω\in\mathcal{D}$. We also prove that for any $p\in (0,\infty)$, the inequality $$\int_{\mathbb{D}} |D^μ(f)(z)|^p \left[\int_{|z|}^1μ(s)\,ds\right]^pω(z)\,dA(z) \lesssim \int_{\mathbb{D}} |f(z)|^p ω(z)\,dA(z) $$ holds for any analytic function $f$ in $\mathbb{D}$ if and only if $ω\in \widehat{\mathcal{D}}$.

math.CV↗

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↗

arXiv2023

Fractional derivative description of the Bloch space

We establish new characterizations of the Bloch space $\mathcal{B}$ which include descriptions in terms of classical fractional derivatives. Being precise, for an analytic function $f(z)=\sum_{n=0}^\infty \widehat{f}(n) z^n$ in the unit disc $\mathbb{D}$, we define the fractional derivative $ D^μ(f)(z)=\sum\limits_{n=0}^{\infty} \frac{\widehat{f}(n)}{μ_{2n+1}} z^n $ induced by a radial weight $μ$, where $μ_{2n+1}=\int_0^1 r^{2n+1}μ(r)\,dr$ are the odd moments of $μ$. Then, we consider the space $ \mathcal{B}^μ$ of analytic functions $f$ in $\mathbb{D}$ such that $\|f\|_{\mathcal{B}^μ}=\sup_{z\in \mathbb{D}} \widehatμ(z)|D^μ(f)(z)|<\infty$, where $\widehatμ(z)=\int_{|z|}^1 μ(s)\,ds$. We prove that $\mathcal{B}^μ$ is continously embedded in $\mathcal{B}$ for any radial weight $μ$, and $\mathcal{B}=\mathcal{B}^μ$ if and only if $μ\in \mathcal{D}=\widehat{\mathcal{D}}\cap\check{\mathcal{D}}$. A radial weight $μ\in \widehat{\mathcal{D}}$ if $\sup_{0\le r <1}\frac{\widehatμ(r)}{\widehatμ\left(\frac{1+r}{2}\right)}<\infty$ and a radial weight $μ\in \check{\mathcal{D}}$ if there exist $K=K(μ)>1$ such that $\inf_{0\le r<1}\frac{\widehatμ(r)}{ \widehatμ\left(1-\frac{1-r}{K}\right)}>1.$

math.CV↗

arXiv2023

Bergman projection on Lebesgue space induced by doubling weight

Let $ω$ and $ν$ be radial weights on the unit disc of the complex plane, and denote $σ=ω^{p'}ν^{-\frac{p'}p}$ and $ω_x =\int_0^1 s^x ω(s)\,ds$ for all $1\le x<\infty$. Consider the one-weight inequality \begin{equation}\label{ab1} \|P_ω(f)\|_{L^p_ν}\le C\|f\|_{L^p_ν},\quad 1 1$. In addition, an analogous result for the one weight inequality $ \|P_ω(f)\|_{D^p_{ν,k}}\le C\|f\|_{L^p_ν}, $ where $$ \Vert f \Vert_{D^p_{ν, k}}^p =\sum\limits_{j=0}^{k-1}| f^{(j)}(0)|^p+ \int_{\mathbb{D}} \vert f^{(k)}(z)\vert^p (1-|z| )^{kp}ν(z)\,dA(z)<\infty, \quad k\in \mathbb{N}, $$ is established. The inequality \eqref{ab1} is further studied by using the necessary condition $D_p(ω,ν)<\infty$ in the case of the exponential type weights $ν(r)=\exp \left(-\fracα{(1-r^l)^β} \right)$ and $ω(r)= \exp \left(-\frac{\widetildeα}{(1-r^{\widetilde{l}})^{\widetildeβ}} \right)$, where $0<α, \, \widetildeα, \, l, \, \widetilde{l}<\infty$ and $0<β, \, \widetildeβ\le 1$.

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↗