arXiv subjects

Tanausú Aguilar-Hernández

Publications and source records attributed to Tanausú Aguilar-Hernández.

6 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↗

arXiv2024

Maximal theorems for weighted analytic tent and mixed norm spaces

Let $ω$ be a radial weight, $0<p,q<\infty$ and $Γ(ξ)=\left\{z\in\mathbb{D}:|\arg z-\argξ|<(|ξ|-|z|)\right\}$ for $ξ\in\overline{\mathbb{D}}$ . The average radial integrability space $L^q_p(ω)$ consists of complex-valued measurable functions $f$ on the unit disc $\mathbb{D}$ such that $$\|f\|^q_{L^q_p(ω)}=\frac{1}{2π}\int_{0}^{2π}\left(\int_{0}^{1}|f(re^{iθ})|^pω(r)r\,dr\right)^{\frac{q}{p}}dθ<\infty,$$ and the tent space $T^q_p(ω)$ is the set of those $f$ for which $$\|f\|^q_{T_{p}^{q}(ω)}=\frac{1}{2π}\int_{\partial{\mathbb{D}}}\left(\int_{Γ(ξ)}|f(z)|^pω(z)\frac{dA(z)}{1-|z|}\right)^{\frac{q}{p}}\,|dξ|<\infty.$$ Let $\mathcal{H}(\mathbb{D})$ denote the space of analytic functions in $\mathbb{D}$. It is shown that the non-tangential maximal operator $$f\mapsto N(f)(ξ)=\sup_{z\inΓ(ξ)}|f(z)|,\quad ξ\in \mathbb{D},$$ is bounded from $AL^q_p(ω)=L^q_p(ω)\cap\mathcal{H}(\mathbb{D})$ and $AT^q_p(ω)=T^q_p(ω)\cap\mathcal{H}(\mathbb{D})$ to $L^q_p(ω)$ and $T^q_p(ω)$, respectively. These pivotal inequalities are used to establish further results such as the density of polynomials in $AL^q_p(ω)$ and $AT^q_p(ω)$, and the identity $AL^q_p(ω)=AT^q_p(ω)$ for weights admitting a one-sided integral doubling condition. It is also shown that the boundedness of the classical Bergman projection $P_γ$, induced by the standard weight $(γ+1)(1-|z|^2)^γ$, on $L^q_p(ω)$ and $T^q_p(ω)$ with $1<q,p<\infty$ is independent of $q$, and is described by a Bekollé-Bonami type condition.

math.CV↗

arXiv2023

Inequalities on Tent Spaces and Closed Range Integration Operators on spaces of Average Radial Integrability

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.

math.CV↗

arXiv2023

Average radial integrability spaces, tent spaces and integration operators

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)$.

math.FA↗

arXiv2022

Mixed norm spaces and RM(p,q) spaces

In this paper we present the containment relationship between the spaces of analytic functions with average radial integrability RM(p,q) and a family of mixed norm spaces.

math.FA↗

arXiv2020

Integration operators in average radial integrability spaces of analytic functions

In this paper we characterize the boundedness, compactness, and weak compactness of the integration operators \begin{align*} T_g (f)(z)=\int_{0}^{z} f(w)g'(w)\ dw \end{align*} acting on the average radial integrability spaces $RM(p,q)$. For these purposes, we develop different tools such as a description of the bidual of $RM(p,0)$ and estimates of the norm of these spaces using the derivative of the functions, a family of results that we call Littlewood-Paley type inequalities.

math.FA↗