arXiv subjects

Álvaro Miguel Moreno

Publications and source records attributed to Álvaro Miguel Moreno.

6 recordsLinked to original sources

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

Maximal subspaces of strong continuity for composition semigroups

Let $(φ_t)_{t\geq 0} $ a semigroup of holomorphic self-maps of the unit disk and $C_t f = f \circ φ_t $ the semigroup of composition operators which corresponds to $φ_t. $ Given a non-separable Banach space of analytic functions $X $ we study the properties of the maximal subspace of $X $ on which the semigroup $C_t $ is strongly continuous. In particular when $X $ contains the polynomials an interesting question is for which semigroups the maximal subspace of strong continuity coincides with the norm closure of the polynomials. This problem has been investigated in several function spaces including $BMOA$, $BMOA_p $, the Bloch space, $Q_s $ space and analytic Morrey spaces. However, in most cases only partial results are available. We offer a unified approach to this problem which encompasses all of the above spaces as particular examples. Moreover, we completely characterize the semigroups for which the maximal subspace of strong continuity coincides with the norm-closure of the polynomials in the space, giving therefore sharp versions of a number of results in the literature.

math.FA↗

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↗

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

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↗