arXiv subjects

José Ángel Peláez

Publications and source records attributed to José Ángel Peláez.

At least 19 recordsLinked to original sources

arXiv2026

The Boundedness Problem for Generalized Hilbert Operators on Hardy Spaces

Let $g$ be analytic in the unit disc and consider the generalized Hilbert operator $$ \mathcal{H}_g(f)(z)=\int_0^1 f(t)g'(tz)\, dt. $$ The boundedness of $\mathcal H_g$ on $H^p$ is characterized by the mean Lipschitz condition $g\inΛ\left(p,\frac{1}{p}\right)$ when $1<p\leq2$, while the problem remains open for $2<p<\infty$. It has been recently proved that the condition $g\inΛ\left(p,\frac{1}{p}\right)$ does not imply the boundedness of $\mathcal H_g$ on $H^p$, $2<p<\infty$ \cite{GuoTang2026}. We show that this condition is far from sufficient in the latter range: for every $2<p<\infty$, there exists a function $g\inΛ\left(p,\frac{1}{p}\right)$ such that $\mathcal H_g$ is not bounded even from $H^p$ into $H^1$. The main ingredient is an exact characterization of the boundedness of $\mathcal{H}_g:H^p\to H^2$ for all $1\leq p\leq\infty$. In particular, when $2<p<\infty$, this mapping is bounded if and only if $g'$ belongs to a certain mixed-norm space. For lacunary symbols, the same mixed-norm condition also characterizes the boundedness of $\mathcal H_g$ on $H^p$, and hence gives a complete solution of the open problem within this class of symbols. We also show that, for $1\leq q\leq\infty$, boundedness of $\mathcal H_g:H^1\to H^q$ is characterized by the condition $g'\in H^q$. We also characterize compactness of $\mathcal H_g$ in the aforementioned cases.

math.CV↗

arXiv2026

Improvement of a Hardy-Littlewood inequality and applications to the boundedness of analytic paraproducts on mixed norm spaces

Let $\mathcal{H}(\mathbb{D})$ denote the space of analytic functions in the unit disc $\mathbb{D}=\{z\in\mathbb{C}:|z|<1\}$. For $0<p<\infty$ and $f\in\mathcal{H}(\mathbb{D})$, let $M_p^p(r,f)=\int_0^{2π}|f(re^{iθ})|^p \frac{dθ}{2π}$ and $M_\infty(r,f) = \sup_{|z|=r}|f(z)|$. For $0<p<q\leq \infty$, Hardy and Littlewood proved the prevalent inequality $$M_q(r,f)\le C(p,q)\frac{M_p(ρ,f)}{(ρ-r)^{\frac{1}{p}-\frac{1}{q}}}$$ for $0\leq r<ρ\leq 1$ and $f\in\mathcal{H}(\mathbb{D})$. In this paper, we obtain an improvement of this well-known inequality which is employed to characterize the symbols $g\in\mathcal{H}(\mathbb{D})$ such that the analytic paraproducts $T_gf(z)=\int_0^z f(ζ)g'(ζ)\,dζ$, $S_gf(z)=\int_0^z f'(ζ)g(ζ)\,dζ$ and $M_gf(z)=f(z)g(z)$, are bounded between two different mixed-norm spaces $A^{p,q}_ω=\{ g\in \mathcal{H}(\mathbb{D}): \int_0^1 M_p^q(r,g) ω(r)\,dr<\infty\}$ induced by a radial doubling weight $ω$. En route to the proof of these characterizations, we consider an open Carleson measure problem posed by Luecking and we solve it in a meaningful particular case.

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↗

arXiv2025

Duality of mixed norm spaces induced by radial one-sided doubling weight

For $0 0$ such that $$\int_r^1ω(s)ds \leq C \int_{\frac{1+r}{2}}^1ω(s)\,ds \,\, \text{for every}\,\, 0\leq r <1.$$ We describe the dual space of $A^{p,q}_ω$ for every $0<p,q<\infty$ and $ω\in\widehat{\mathcal{D}}$. Later on, we apply the obtained description of the dual space of $A^{p,q}_ω$ to prove that the Bergman projection induced by $ω$, $P_ω$, is bounded on $L^{p,q}_ω$ for $1<p,q<\infty$ and $ω\in \widehat{\mathcal{D}}$. Besides, we also prove that $P_ω$ and the corresponding maximal Bergman projection $P_ω^+$ are not simultaneously bounded on $L^{p,q}_ω$ for $1<p,q<\infty$ and $ω\in \widehat{\mathcal{D}}$.

math.CV↗

arXiv2025

Fractional Volterra-type operator induced by radial weight acting on Hardy space

Given a radial doubling weight $μ$ on the unit disc $\mathbb{D}$ of the complex plane and its odd moments $μ_{2n+1}=\int_0^1 s^{2n+1}μ(s)\, ds$, we consider the fractional derivative $$ D^μ(f)(z)=\sum_{n=0}^{\infty} \frac{\widehat{f}(n)}{μ_{2n+1}}z^n, $$ of a function $ f(z)=\sum_{n=0}^{\infty}\widehat{f}(n)z^n$ analytic in $\mathbb{D}$. We also consider the fractional integral operator $I^μ(f)(z)=\sum_{n=0}^{\infty} μ_{2n+1}\widehat{f}(n)z^n$, and the fractional Volterra-type operator $$ V_{μ,g}(f)(z)= I^μ(f\cdot D^μ(g))(z),\quad f\in\mathcal{H}(\mathbb{D}), $$ for any fixed $g\in\mathcal{H}(\mathbb{D})$. We prove that $V_{μ,g}$ is bounded (compact) on a Hardy space $H^p$, $0<p<\infty$, if and only if $g$ belongs to $\text{BMOA}$ ($\text{VMOA}$). Moreover, if $\int_0^1 \frac{\left(\int_r^1 μ(s)\, ds\right)^p}{(1-r)^2}\,dr=+\infty$, we prove that $V_{μ,g}$ belongs to the Schatten class $S_p(H^2)$ if and only if $g=0$. On the other hand, if $\frac{\left(\int_r^1 μ(s)\, ds\right)^p}{(1-r)^2}$ is a radial doubling weight it is proved that $V_{μ,g} \in S_p(H^2)$ if and only if $g$ belongs to the Besov space $B_p$. En route, we obtain descriptions of $H^p$, $\text{BMOA}$, $\text{VMOA}$ and $B_p$ in terms of the fractional derivative $D^μ$.

math.CV↗

arXiv2025

Words of analytic paraproducts on Bergman spaces

For a fixed analytic function g on the unit disc, we consider the analytic paraproducts induced by g, which are formally defined by $T_gf(z)=\int_0^zf(ζ)g'(ζ)dζ$, $S_gf(z)=\int_0^zf'(ζ)g(ζ)dζ$, and $M_gf(z)=g(z)f(z)$. An $N$-letter $g$-word is an operator of the form $L=L_1\cdots L_N$, where each $L_j$ is either $M_g$, $S_g$ or $T_g$. It has been recently proved, in a recent paper by A. Aleman and the authors of this paper, that understanding the boundedness of a $g$-word on classical Hardy and Bergman spaces is a challenging problem due to the potential cancellations involved. Our main result provides a complete quantitative characterization of the boundedness of an arbitrary $g$-word on a weighted Bergman space $A^p_{ω^{p/2}}$, where $ω=e^{-2φ}$ is a smooth rapidly decreasing weight. In particular, it states that any $N$-letter $g$-word such that $\#\{j:L_j=T_g\}=n\ge 1$ is bounded on $A^p_{ω^{p/2}}$ if and only if $g$ satisfies the "fractional" Bloch-type condition \[ \|g\|_{\mathcal{B}^s_φ}^s= \sup_{z\in\mathbb{D}}\frac{s|g(z)|^{s-1}|g'(z)|}{1+φ'(|z|)} <\infty, \] where $s=\frac{N}{n}$, and $\|L\|_{A^p_{ω^{p/2}}}\simeq \|g\|_{\mathcal{B}^s_φ}^N$. The class of smooth rapidly decreasing weights contains the radial weights \begin{equation*} ω_n(z)=e^{-2\exp_{n}(g_{α,c}(|z|))}, \quad\mbox{where}\quad g_{α,c}(r)=\tfrac{c}{(1-r^2)^α}, \quad\mbox{for $c,α>0$,} \end{equation*} $\exp_0(x)=x$ and $\exp_n(x)=e^{\exp_{n-1}(x)}$, for $n\in\mathbb{N}$. Therefore it contains weights which decrease arbitrarily rapidly to zero as $|z|\to 1^-$.

math.CV↗

arXiv2025

Bergman projection induced by radial weight

The question of when the Bergman projection $P_ω$ induced by a radial weight $ω$ on the unit disc is a bounded operator from one space into another is of primordial importance in the theory of Bergman spaces. The long-standing problem of describing the radial weights $ω$ such that $P_ω$ is bounded on the Lebesgue space $L^p_ω$ had been known to experts since decades before it was formally posed by Dostanić in 2004. A natural limit case of this setting is when $P_ω$ acts from $L^\infty$ to the Bloch space. The surjectivity of the operator becomes another relevant question in this limit case. The main findings of this study are shortly listed as follows. We establish characterizations of the radial weights $ω$ on the unit disc such that $P_ω:L^\infty\to\mathcal{B}$ is bounded and/or acts surjectively, or the dual of $A^1_ω$ is isomorphic to the Bloch space $\mathcal{B}$ under the $A^2_ω$-pairing. We also solve the problem posed by Dostanić under a weak regularity hypothesis on the weight involved. With regard to Littlewood-Paley estimates, we describe the radial weights $ω$ such that the norm of any function in $A^p_ω$ is comparable to the norm in $L^p_ω$ of its derivative times the distance from the boundary. This last-mentioned result solves another well-known problem on the area. All characterizations can be given in terms of doubling conditions on moments and/or tail integrals $\int_r^1ω(t)\,dt$ of $ω$, and are therefore easy to interpret. We also make substantial progress about the two weight inequality $$ \|P_ω(f)\|_{L^p_ν}\le C\|f\|_{L^p_ν},\quad f\in L^p_ν, \quad 1<p<\infty. $$ for radial weights $ω$ and $ν$.

math.FA↗

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↗

arXiv2024

Small Hankel operator induced by measurable symbol acting on weighted Bergman spaces

The boundedness of the small Hankel operator $h^ω_{f}(g)=\overline{P_ω}(fg)$ induced by a measurable symbol $f$ and the Bergman projection $P_ω$ associated to a radial weight $ω$ acting from the weighted Bergman space $A^p_ω$ to its conjugate analytic counterpart $\overline{A^p_ω}$ is characterized on the range $1<p<\infty$ when $ω$ belongs to the class $\mathcal{D}$ of radial weights admitting certain two-sided doubling conditions. On the way to the proof a sharp integral estimate for certain modified Bergman kernels is obtained.

math.CV↗

arXiv2024

Bergman projection induced by radial weight acting on growth spaces

Let $ω$ be a radial weight on the unit disc of the complex plane $\mathbb{D}$ and denote $ω_x =\int_0^1 s^x ω(s)\,ds$, $x\ge 0$, for the moments of $ω$ and $\widehatω(r)=\int_r^1 ω(s)\,ds$ for the tail integrals. A radial weight $ω$ belongs to the class $\widehat{\mathcal{D}}$ if satisfies the upper doubling condition $$\sup_{0<r<1}\frac{\widehatω(r)}{\widehatω\left(\frac{1+r}{2}\right)}<\infty.$$ If $ν$ or $ω$ belongs to $\widehat{\mathcal{D}}$, it is described the boundedness of the Bergman projection $P_ω$ induced by $ω$ on the growth space $L^\infty_{\widehatν} =\{ f: \|f\|_{\infty,v}={ esssup}_{z\in\mathbb{D}} |f(z)|\widehatν(z)<\infty\}$ in terms of neat conditions on the moments and/or the tail integrals of $ω$ and $ν$. Moreover, it is solved the analogous problem for $P_ω$ from $L^\infty_{\widehatν}$ to the Bloch type space $B^\infty_{\widehatν}$ of analytic functions such that $\sup_{z\in \mathbb{D}}(1-|z|)\widehatν(z) |f'(z)|<\infty.$ We also study similar questions for exponentially decreasing radial weights.

math.CV↗

arXiv2023

Words of analytic paraproducts on Hardy and weighted Bergman spaces

For a fixed analytic function g on the unit disc, we consider the analytic paraproducts induced by g, which are formally defined by $T_gf(z)=\int_0^zf(ζ)g'(ζ)dζ$, $S_gf(z)=\int_0^zf'(ζ)g(ζ)dζ$, and $M_gf(z)=g(z)f(z)$. We are concerned with the study of the boundedness of operators in the algebra $\mathcal{A}_g$ generated by the above operators acting on Hardy, or standard weighted Bergman spaces on the disc. The general question is certainly very challenging, since operators in $\mathcal{A}_g$ are finite linear combinations of finite products (words) of $T_g,S_g,M_g$ which may involve a large amount of cancellations to be understood. The results in the paper "Composition of analytic paraproducts, J. Math. Pures Appl. 158 (2022) 293--319" show that boundedness of operators in a fairly large subclass of $\mathcal{A}_g$ can be characterized by one of the conditions $g\in H^\infty$, or $g^n$ belongs to $BMOA$ or the Bloch space, for some integer $n>0$. However, it is also proved that there are many operators, even single words in $\mathcal{A}_g$ whose boundedness cannot be described in terms of these conditions. The present paper provides a considerable progress in this direction. Our main result provides a complete quantitative characterization of the boundedness of an arbitrary word in $\mathcal{A}_g$ in terms of a ``fractional power'' of the symbol $g$, that only depends on the number of appearances of each of the letters $T_g,S_g,M_g$ in the given word.

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

Volterra-type operators mapping weighted Dirichlet space into $H^\infty$

The problem of describing the analytic functions $g$ on the unit disc such that the integral operator $T_g(f)(z)=\int_0^zf(ζ)g'(ζ)\,dζ$ is bounded (or compact) from a Banach space (or complete metric space) $X$ of analytic functions to the Hardy space $H^\infty$ is a tough problem and remains unsettled in many cases. For analytic functions $g$ with non-negative Maclaurin coefficients, we describe the boundedness and compactness of $T_g$ acting from a weighted Dirichlet space $D^p_ω$, induced by an upper doubling weight $ω$, to $H^\infty$. We also characterize, in terms of neat conditions on $ω$, the upper doubling weights for which $T_g: D^p_ω\to H^\infty$ is bounded (or compact) only if $g$ is constant.

math.CV↗

arXiv2022

Bergman projection and BMO in hyperbolic metric -- improvement of classical result

The Bergman projection $P_α$, induced by a standard radial weight, is bounded and onto from $L^\infty$ to the Bloch space $\mathcal{B}$. However, $P_α: L^\infty\to \mathcal{B}$ is not a projection. This fact can be emended via the boundedness of the operator $P_α:BMO_2(Δ)\to\mathcal{B}$, where $BMO_2(Δ)$ is the space of functions of bounded mean oscillation in the Bergman metric. We consider the Bergman projection $P_ω$ and the space $BMO_{ω,p}(Δ)$ of functions of bounded mean oscillation induced by $1<p<\infty$ and a radial weight $ω\in\mathcal{M}$. Here $\mathcal{M}$ is a wide class of radial weights defined by means of moments of the weight, and it contains the standard and the exponential-type weights. We describe the weights such that $P_ω:BMO_{ω,p}(Δ)\to\mathcal{B}$ is bounded. They coincide with the weights for which $P_ω: L^\infty \to \mathcal{B}$ is bounded and onto. This result seems to be new even for the standard radial weights when $p\ne2$.

math.CV↗

arXiv2021

One weight inequality for Bergman projection and Calderón operator induced by radial weight

Let $ω$ and $ν$ be radial weights on the unit disc of the complex plane such that $ω$ admits the doubling property $\sup_{0\le r<1}\frac{\int_r^1 ω(s)\,ds}{\int_{\frac{1+r}{2}}^1 ω(s)\,ds}<\infty$. Consider the one weight inequality \begin{equation}\label{ab1} \|P_ω(f)\|_{L^p_ν}\le C\|f\|_{L^p_ν},\quad 1<p<\infty,\tag† \end{equation} for the Bergman projection $P_ω$ induced by $ω$. It is shown that the Muckenhoupt-type condition $$ A_p(ω,ν)=\sup_{0\le r<1}\frac{\left(\int_r^1 sν(s)\,ds \right)^{\frac{1}{p}}\left(\int_r^1 s\left(\frac{ω(s)}{ν(s)^{\frac1p}}\right)^{p'}\,ds \right)^{\frac{1}{p'}}}{\int_r^1 sω(s)\,ds}<\infty, $$ is necessary for \eqref{ab1} to hold, and sufficient if $ν$ is of the form $ν(s)=ω(s)\left(\int_r^1 sω(s)\,ds \right)^α$ for some $-1<α<\infty$. This result extends the classical theorem due to Forelli and Rudin for a much larger class of weights. In addition, it is shown that for any pair $(ω,ν)$ of radial weights the Calderón operator $$ H^\star_ω(f)(z)+H_ω(f)(z) =\int_{0}^{|z|} f\left(s\frac{z}{|z|}\right)\frac{sω(s)\,ds}{\int_s^1 tω(t)\,dt} +\frac{\int_{|z|}^1f\left(s\frac{z}{|z|}\right) sω(s)\,ds}{\int_{|z|}^1 sω(s)\,ds}\,ds $$ is bounded on $L^p_ν$ if and only if $A_p(ω,ν)<\infty$.

math.CV↗

arXiv2021

Integral operators induced by symbols with non-negative Maclaurin coefficients mapping into $H^\infty$

For analytic functions $g$ on the unit disc with non-negative Maclaurin coefficients, we describe the boundedness and compactness of the integral operator $T_g(f)(z)=\int_0^zf(ζ)g'(ζ)\,dζ$ from a space $X$ of analytic functions in the unit disc to $H^\infty$, in terms of neat and useful conditions on the Maclaurin coefficients of $g$. The choices of $X$ that will be considered contain the Hardy and the Hardy-Littlewood spaces, the Dirichlet-type spaces $D^p_{p-1}$, as well as the classical Bloch and BMOA spaces.

math.CV↗