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José Ángel Peláez

Publications and source records attributed to José Ángel Peláez.

At least 37 records · Page 2Linked to original sources
arXiv2019

Harmonic conjugates on Bergman spaces induced by doubling weights

A radial weight $ω$ belongs to the class $\widehat{\mathcal{D}}$ if there exists $C=C(ω)\ge 1$ such that $\int_r^1 ω(s)\,ds\le C\int_{\frac{1+r}{2}}^1ω(s)\,ds$ for all $0\le r<1$. Write $ω\in\check{\mathcal{D}}$ if there exist constants $K=K(ω)>1$ and $C=C(ω)>1$ such that $\widehatω(r)\ge C\widehatω\left(1-\frac{1-r}{K}\right)$ for all $0\le r<1$. In a recent paper, we have recently prove that these classes of radial weights arise naturally in the operator theory of Bergman spaces induced by radial weights. Classical results by Hardy and Littlewood, and Shields and Williams, show that the weighted Bergman space of harmonic functions is not closed by harmonic conjugation if $ω\in\widehat{\mathcal{D}}\setminus \check{\mathcal{D}}$ and $0<p\le 1$. In this paper we establish sharp estimates for the norm of the analytic Bergman space $A^p_ω$, with $ω\in\widehat{\mathcal{D}}\setminus \check{\mathcal{D}}$ and $0<p<\infty$, in terms of quantities depending on the real part of the function. It is also shown that these quantities result equivalent norms for certain classes of radial weights.

math.CV↗
arXiv2018

Hankel operators induced by radial Bekollé-Bonami weights on Bergman spaces

We study big Hankel operators $H_f^ν:A^p_ω\to L^q_ν$ generated by radial Bekollé-Bonami weights $ν$, when $1<p\leq q<\infty$. Here the radial weight $ω$ is assumed to satisfy a two-sided doubling condition, and $A^p_ω$ denotes the corresponding weighted Bergman space. A characterization for simultaneous boundedness of $H_f^ν$ and $H_{\overline{f}}^ν$ is provided in terms of a general weighted mean oscillation. Compared to the case of standard weights that was recently obtained by Pau, Zhao and Zhu (Indiana Univ. Math. J. 2016), the respective spaces depend on the weights $ω$ and $ν$ in an essentially stronger sense. This makes our analysis deviate from the blueprint of this more classical setting. As a consequence of our main result, we also study the case of anti-analytic symbols.

math.CV↗
arXiv2018

Radial two weight inequality for maximal Bergman projection induced by a regular weight

It is shown in quantitative terms that the maximal Bergman projection \begin{equation*} P^{+}_ω(f)(z)=\int_\mathbb{D} f(ζ)|B^ω_z(ζ)|ω(ζ)\,dA(ζ), \end{equation*} is bounded from $L^p_ν$ to $L^p_η$ if and only if \begin{equation*} \sup_{0<r<1}\left(\int_0^r\frac{η(s)}{\left(\int_{s}^1ω(t)\,dt\right)^p}\,ds\right)^{\frac{1}{p}} \left(\int_r^1\left(\frac{ω(s)}{ν(s)^\frac{1}{p}}\right)^{p'}ds\right)^{\frac{1}{p'}}<\infty, \end{equation*} provided $ω,ν,η$ are radial regular weights. A radial weight $σ$ is regular if it satisfies $σ(r)\asymp\int_{r}^1σ(t)\,dt/(1-r)$ for all $0\leq r<1$. It is also shown that under an appropriate additional hypothesis involving $ω$ and $η$, the Bergman projection $P_ω$ and $P^+_ω$ are simultaneously bounded.

math.CV↗
arXiv2017

Small Hankel operators on generalized Fock spaces

We consider Fock spaces $F^{p,\ell}_α$ of entire functions on ${\mathbb C}$ associated to the weights $e^{-α|z|^{2\ell}}$, where $α>0$ and $\ell$ is a positive integer. We compute explicitly the corresponding Bergman kernel associated to $F^{2,\ell}_α$ and, using an adequate factorization of this kernel, we characterize the boundedness and the compactness of the small Hankel operator $\mathfrak{h}^{\ell}_{b,α}$ on $F^{p,\ell}_α$. Moreover, we also determine when $\mathfrak{h}^{\ell}_{b,α}$ is a Hilbert-Schmidt operator on $F^{2,\ell}_α$.

math.CV↗
arXiv2017

Atomic decomposition and Carleson measures for weighted mixed norm spaces

The purpose of this paper is to establish an atomic decomposition for functions in the weighted mixed norm space $A^{p,q}_ω$ induced by a radial weight $ω$ in the unit disc admitting a two-sided doubling condition. The obtained decomposition is further applied to characterize Carleson measures for $A^{p,q}_ω$, and bounded differentiation operators $D^{(n)}(f)=f^{(n)}$ acting from $A^{p,q}_ω$ to $L^p_μ$, induced by a positive Borel measure $μ$, on the full range of parameters $0<p,q,s<\infty$.

math.CV↗
arXiv2016

Berezin transform and Toeplitz operators on weighted Bergman spaces induced by regular weights

Given a regular weight $ω$ and a positive Borel measure $μ$ on the unit disc $\mathbb{D}$, the Toeplitz operator associated with $μ$ is $$ \mathcal{T}_μ(f)(z)=\int_{\mathbb{D}} f(ζ)\bar{B_z^ω(ζ)}\,dμ(ζ), $$ where $B^ω_{z}$ are the reproducing kernels of the weighted Bergman space $A^2_ω$. We describe bounded and compact Toeplitz operators $\mathcal{T}_μ:A^p_ω\to A^q_ω$, $1<q,p<\infty$, in terms of Carleson measures and the Berezin transform $$ \widetilde{\mathcal{T}_μ}(z)=\frac{\langle\mathcal{T}_μ(B^ω_{z}), B^ω_{z} \rangle_{A^2_ω}}{\|B_z^ω\|^2_{A^2_ω}}. $$ We also characterize Schatten class Toeplitz operators in terms of the Berezin transform and apply this result to study Schatten class composition operators.

math.FA↗
arXiv2015

Schatten classes of generalized Hilbert operators

Let $\mathcal{D}_v$ denote the Dirichlet type space in the unit disc induced by a radial weight $v$ for which $\widehat{v}(r)=\int_r^1 v(s)\,ds$ satisfies the doubling property $\int_r^1 v(s)\,ds\le C \int_{\frac{1+r}{2}}^1 v(s)\,ds.$ In this paper, we characterize the Schatten classes $S_p(\mathcal{D}_v)$ of the generalized Hilbert operators \begin{equation*} \mathcal{H}_g(f)(z)=\int_0^1f(t)g'(tz)\,dt \end{equation*} acting on $\mathcal{D}_v$, where $v$ satisfies the Muckenhoupt-type conditions $$ \sup_{0<r<1}\left(\int_r^1 \frac{\widehat{v}(s)}{(1-s)^2} \,ds\right)^{1/2} \left(\int_0^r \frac{1}{\widehat{v}(s)} \,ds\right)^{1/2}<\infty $$ and $$\sup_{0< r<1}\left(\int_{0}^r \frac{\widehat{v}(s)}{(1-s)^4}\,ds\right)^{\frac{1}{2}} \left(\int_{r}^1\frac{(1-s)^2}{\widehat{v}(s)}\,ds\right)^\frac{1}{2}<\infty. $$ For $p\ge 1$, it is proved that $\mathcal{H}_{g}\in S_p(\mathcal{D}_v)$ if and only if \begin{equation*} \int_0^1 \left((1-r)\int_{-π}^π|g'(re^{iθ})|^2\,dθ\right)^{\frac{p}{2}}\frac{dr}{1-r} <\infty. \end{equation*}

math.CV↗
arXiv2015

Small weighted Bergman spaces

This paper is based on the course \lq\lq Weighted Hardy-Bergman spaces\rq\rq\, I delivered in the Summer School \lq\lq Complex and Harmonic Analysis and Related Topics\rq\rq at the Mekrijärvi research station of University of Eastern Finland, June $2014$. The main purpose of this survey is to present recent progress on the theory of Bergman spaces $A^p_ω$, induced by radial weights $ω$ satisfying the doubling property $\int_r^1ω(s)\,ds\le C\int_{\frac{1+r}{2}}^1ω(s)\,ds$.

math.CV↗
arXiv2015

Embedding Bergman spaces into tent spaces

Let $A^p_ω$ denote the Bergman space in the unit disc $\mathbb{D}$ of the complex plane induced by a radial weight $ω$ with the doubling property $\int_{r}^1ω(s)\,ds\le C\int_{\frac{1+r}{2}}^1ω(s)\,ds$. The tent space $T^q_s(ν,ω)$ consists of functions such that \begin{equation*} \begin{split} \|f\|_{T^q_s(ν,ω)}^q =\int_{\mathbb{D}}\left(\int_{Γ(ζ)}|f(z)|^s\,dν(z)\right)^\frac{q}sω(ζ)\,dA(ζ) <\infty,\quad 0 0$, by considering a generalized area operator. The results are provided in terms of Carleson measures for $A^p_ω$.

math.CV↗
arXiv2015

On the boundedness of Bergman projection

The main purpose of this survey is to gather results on the boundedness of the Bergman projection. First, we shall go over some equivalent norms on weighted Bergman spaces $A^p_ω$ which are useful in the study of this question. In particular, we shall focus on a decomposition norm theorem for radial weights~$ω$ with the doubling property $\int_{r}^1ω(s)\,ds\le C\int_{\frac{1+r}{2}}^1ω(s)\,ds$.

math.CV↗
arXiv2014

Trace class criteria for Toeplitz and composition operators on small Bergman spaces

We characterize the Schatten class Toeplitz operators induced by a positive Borel measure on the unit disc and the reproducing kernel of the Bergman space $A^2_ω$, where $ω$ is a radial weight satisfying the doubling property $\int_r^1ω(s)\,ds\le C\int_{\frac{1+r}{2}}^1ω(s)\,ds$. By using this, we describe the Schatten class composition operators. We also discuss basic properties of composition operators acting from $A^p_ω$ to $A^q_v$.

math.FA↗
arXiv2014

Two weight inequality for Bergman projection

The motivation of this paper comes from the two weight inequality $$\|P_ω(f)\|_{L^p_v}\le C\|f\|_{L^p_v},\quad f\in L^p_v,$$ for the Bergman projection $P_ω$ in the unit disc. We show that the boundedness of $P_ω$ on $L^p_v$ is characterized in terms of self-improving Muckenhoupt and Bekollé-Bonami type conditions when the radial weights $v$ and $ω$ admit certain smoothness. En route to the proof we describe the asymptotic behavior of the $L^p$-means and the $L^p_v$-integrability of the reproducing kernels of the weighted Bergman space $A^2_ω$.

math.FA↗
arXiv2014

Embedding theorems for Bergman spaces via harmonic analysis

Let $A^p_ω$ denote the Bergman space in the unit disc induced by a radial weight~$ω$ with the doubling property $\int_{r}^1ω(s)\,ds\le C\int_{\frac{1+r}{2}}^1ω(s)\,ds$. The positive Borel measures such that the differentiation operator of order $n\in\mathbb{N}\cup\{0\}$ is bounded from $A^p_ω$ into $L^q(μ)$ are characterized in terms of geometric conditions when $0<p,q<\infty$. En route to the proof a theory of tent spaces for weighted Bergman spaces is built.

math.CV↗
arXiv2013

Integral operators, embedding theorems and a Littlewood-Paley formula on weighted Fock spaces

We obtain a complete characterization of the entire functions $g$ such that the integral operator $(T_ g f)(z)=\int_{0}^{z}f(ζ)\,g'(ζ)\,dζ$ is bounded or compact, on a large class of Fock spaces $\mathcal{F}^ϕ_p$, induced by smooth radial weights that decay faster than the classical Gaussian one. In some respects, these spaces turn out to be significantly different than the classical Fock spaces. Descriptions of Schatten class integral operators are also provided. En route, we prove a Littlewood-Paley formula for $||\cdot||_{\mathcal{F}^ϕ_p}$ and we characterize the positive Borel measures for which $\mathcal{F}^ϕ_p\subset L^q(μ)$, $0<p,q<\infty$. In addition, we also address the question of describing the subspaces of $\mathcal{F}^ϕ_p$ that are invariant under the classical Volterra integral operator.

math.FA↗
arXiv2013

Generalized Hilbert operators on weighted Bergman spaces

The main purpose of this paper is to study the generalized Hilbert operator {equation*} \mathcal{H}_g(f)(z)=\int_0^1f(t)g'(tz)\,dt {equation*} acting on the weighted Bergman space $A^p_\om$, where the weight function $\om$ belongs to the class $\R$ of regular radial weights and satisfies the Muckenhoupt type condition {equation}\label{Mpconditionaabstract} \sup_{0\le r<1}\bigg(\int_{r}^1(\int_t^1\om(s)ds)^{-\frac{p'}{p}}\,dt\bigg)^\frac{p}{p'} \int_{0}^r(1-t)^{-p}(\int_t^1\om(s)ds)\,dt<\infty. \tag† {equation} If $q=p$, the condition on $g$ that characterizes the boundedness (or the compactness) of $\hg: A^p_\om\to A^q_\om$ depends on $p$ only, but the situation is completely different in the case $q\ne p$ in which the inducing weight $\om$ plays a crucial role. The results obtained also reveal a natural connection to the Muckenhoupt type condition \eqref{Mpconditionaabstract}. Indeed, it is shown that the classical Hilbert operator (the case $g(z)=\log\frac{1}{1-z}$ of $\H_g$) is bounded from $L^p_{\int_{t}^1\om(s)\,ds}([0,1))$ (the natural restriction of $A^p_\om$ to functions defined on $[0,1)$) to $A^p_\om$ if and only if $\om$ satisfies the condition \eqref{Mpconditionaabstract}. On the way to these results decomposition norms for the weighted Bergman space $A^p_\om$ are established.

math.CV↗
arXiv2013

Schatten classes of integration operators on Dirichlet spaces

We address the question of describing the membership to Schatten-Von Neumann ideals $\mathcal{S}_ p$ of integration operators $(T_ g f)(z)=\int_{0}^{z}f(ζ)\,g'(ζ)\,dζ$ acting on Dirichlet type spaces. We also study this problem for multiplication, Hankel and Toeplitz operators. In particular, we provide an extension of Luecking's result on Toeplitz operators.

math.FA↗