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Luis Rodríguez-Piazza

Publications and source records attributed to Luis Rodríguez-Piazza.

At least 19 recordsLinked to original sources
arXiv2026

The Bergman and the growth numbers of domains and hyperbolic geometry

In this article we completely characterize the Bergman number of general domains. Namely, we prove that the Bergman number of a hyperbolic unbounded domain can be calculated in terms of the asymptotic behavior of its hyperbolic metric near infinity. To obtain the proof of this result we first work in the setting of growth spaces, defining the growth number of a domain. Later, we prove an equality relating the Bergman number and the growth number of domains. We also provide examples of domains with prescribed Bergman number which have zero Hardy number, solving a question posed previously by Betsakos and Cruz-Zamorano. At the end, a similar idea is treated for the case of Bloch-type spaces.

math.CV↗
arXiv2024

Examples in Discrete Iteration of Arbitrary Intervals of Slopes

Given a compact interval $[a,b] \subset [0,π]$, we construct a parabolic self-map of the upper half-plane whose set of slopes is $[a,b]$. The nature of this construction is completely discrete and explicit: we explicitly construct a self-map and we explicitly show in which way its orbits wander towards the Denjoy-Wolff point. We also analyze some properties of the Herglotz measure corresponding to such example, which yield the regularity of such self-map in its Denjoy-Wolff point.

math.CV↗
arXiv2024

On the Hardy number of Koenigs domains

This work studies the Hardy number for the class of hyperbolic planar domains satisfying Abel's inclusion property, which are usually known as Koenigs domains. More explicitly, we prove that for all regular domains in the above class, the Hardy number is greater or equal than $1/2$, and this lower bound is sharp. In contrast to this result, we provide examples of general domains whose Hardy numbers are arbitrarily small. Additionally, we outline the connection of the aforementioned class of domains with the discrete dynamics of the unit disc and obtain results on the range of Hardy number of Koenigs maps, in the hyperbolic and parabolic case.

math.CV↗
arXiv2024

Hardy Number of Koenigs Domains: Sharp Estimate

Let $Ω$ be a regular Koenigs domain in the complex plane $\mathbb{C}$. We prove that the Hardy number of $Ω$ is greater or equal to $1/2$. That is, every holomorphic function in the unit disc $f \colon \mathbb{D} \to Ω$ belongs to the Hardy space $H^{p}(\mathbb{D})$ for all $p<1/2$.

math.CV↗
arXiv2024

The Slope Problem in Discrete Iteration

The slope problem in holomorphic dynamics in the unit disk goes back to Wolff in 1929. However, there have been several contributions to this problem in the last decade. In this article the problem is revisited, comparing the discrete and continuous cases. Some advances are derived in the discrete parabolic case of zero hyperbolic step, showing that the set of slopes has to be a closed interval which is independent of the initial point. The continuous setting is used to show that any such interval is a possible example. In addition, the set of slopes of a family of parabolic function is discussed, leading to examples of functions with some regularity whose set of slopes is non-trivial.

math.CV↗
arXiv2024

Characterization of the hyperbolic step of parabolic functions

A classical problem in Complex Dynamics on hyperbolic domains is to characterize the hyperbolic step of parabolic functions. This topic has been studied by several authors, leading to different results and providing characterizations that depend on the behaviour of the iterates of such function. In this work we provide new characterizations in terms of intrinsic properties of the function.

math.CV↗
arXiv2024

On some questions about composition operators on weighted Hardy spaces

We first consider some questions raised by N. Zorboska in her thesis. In particular she asked for which sequences $β$ every symbol $φ\colon \mathbb{D} \to \mathbb{D}$ with $φ\in H^2 (β)$ induces a bounded composition operator $C_ϕ$ on the weighted Hardy space $H^2 (β)$. We give partial answers and investigate when $H^2 (β)$ is an algebra. We answer negatively another question in showing that there are a sequence $β$ and $φ\in H^2 (β)$ such that $\| φ\|_\infty < 1$ and the composition operator $C_φ$ is not bounded on $H^2 (β)$. In a second part, we show that for $p \neq 2$, no automorphism of $\mathbb{D}$, except those that fix $0$, induces a bounded composition operator on the Beurling-Sobolev space $\ell^p_A$, and even on any weighted version of this space.

math.FA↗
arXiv2024

On the uniform convergence of continuous semigroups

Let $Ω$ be a region in the complex plane $\mathbb C$ and let $\{Φ_t \}_{t\ge 0}$ be a continuous semigroup of functions on $Ω$; that is, $Φ_t:Ω\toΩ$ is holomorphic for every $t\ge 0$, $Φ_0(z)=z$, for every $z\inΩ$, $Φ_t\circΦ_s=Φ_{s+t}$, for every $s$, $t\ge 0$, and \[ Φ_t(z)\to z\,,\quad t\to0^+, \] uniformly on compact subsets of $Ω$. Despite this definition only requires the uniform convergence on compact subsets, P. Gumenyuk proved in 2014 that, when $Ω$ is the unit disc, the convergence is uniform on the whole $\mathbb D$. In this paper, we enhance Gumenyuk's result by proving that for every continuous semigroup $\{Φ_t\}_{t\ge 0}$ on $\mathbb D$ we have $$ \sup_{z\in\mathbb D}|Φ_t(z)-z|= O(\sqrt t), \ t\to0^+. $$ In addition, we provide an example showing that $O(\sqrt t)$ is the best possible rate of uniform convergence valid for all semigroups on $\mathbb D$. When $Ω$ is the right half-plane $\mathbb C_+$, we consider semigroups $\{Φ_t\}$ with $\infty$ as its Denjoy-Wolff point. It is not difficult to show that Gumenyuk's result is no longer true for these semigroups. Our second result characterises when such continuous semigroups converge uniformly to the identity, as $t\to0^+$, in terms of their infinitesimal generators. Namely, this convergence holds if and only if the infinitesimal generator of the semigroup is bounded in the half-plane $\{z\in \mathbb C:\, \Re z>1\}$. In this case, we can also prove that the rate of convergence is again $O(\sqrt{t})$, as $t\to0^+$. An example of application of this result is when the semigroup is in the Gordon-Hedenmalm class (the one producing bounded composition operators on Hardy spaces of Dirichlet series). An important ingredient in the proofs of these results is harmonic measure, which we have done through a classic result of M. Lavrentiev.

math.CV↗
arXiv2023

Characterization of weighted Hardy spaces on which all composition operators are bounded

We give a complete characterization of the sequences $β= (β_n)$ of positive numbers for which all composition operators on $H^2 (β)$ are bounded, where $H^2 (β)$ is the space of analytic functions $f$ on the unit disk ${\mathbb D}$ such that $\sum_{n = 0}^\infty |a_n|^2 β_n < + \infty$ if $f (z) = \sum_{n = 0}^\infty a_n z^n$. We prove that all composition operators are bounded on $H^2 (β)$ if and only if $β$ is essentially decreasing and slowly oscillating. We also prove that every automorphism of the unit disk induces a bounded composition operator on $H^2 (β)$ if and only if $β$ is slowly oscillating. We give applications of our results.

math.CV↗
arXiv2023

Composition operators on the algebra of Dirichlet series

The algebra of Dirichlet series $\mathcal{A}(\mathcal{C}_{+})$ consists on those Dirichlet series convergent in the right half-plane $\mathcal{C}_{+}$ and which are also uniformly continuous there. This algebra was recently introduced by Aron, Bayart, Gauthier, Maestre, and Nestoridis. We describe the symbols $Φ:\mathcal{C}_{+}\to\mathcal{C}_{+}$ giving rise to bounded composition operators $\mathit{C}_Φ$ in $\mathcal{A}(\mathcal{C}_{+})$ and denote this class by $\mathcal{G}_{\mathcal{A}}$. We also characterise when the operator $\mathit{C}_Φ$ is compact in $\mathcal{A}(\mathit{C}_{+})$. As a byproduct, we show that the weak compactness is equivalent to the compactness for $\mathit{C}_Φ$. Next, the closure under the local uniform convergence of several classes of symbols of composition operators in Banach spaces of Dirichlet series is discussed. We also establish a one-to-one correspondence between continuous semigroups of analytic functions $\{Φ_{t}\}$ in the class $\mathcal{G}_{\mathcal{A}}$ and strongly continuous semigroups of composition operators $\{T_{t}\}$, $T_{t}f=f\circΦ_{t}$, $f\in\mathcal{A}(\mathcal{C}_{+})$. We conclude providing examples showing the differences between the symbols of bounded composition operators in $\mathcal{A}(\mathcal{C}_{+})$ and the Hardy spaces of Dirichlet series $\mathcal{H}^{p}$ and $\mathcal{H}^{\infty}$.

math.FA↗
arXiv2022

Boundedness of composition operators on general weighted Hardy spaces of analytic functions

We characterize the (essentially) decreasing sequences of positive numbers $β$ = ($β$ n) for which all composition operators on H 2 ($β$) are bounded, where H 2 ($β$) is the space of analytic functions f in the unit disk such that $\infty$ n=0 |c n | 2 $β$ n < $\infty$ if f (z) = $\infty$ n=0 c n z n. We also give conditions for the boundedness when $β$ is not assumed essentially decreasing.

math.FA↗
arXiv2022

Semigroups of composition operators on Hardy spaces of Dirichlet series

We consider continuous semigroups of analytic functions $\{Φ_t\}_{t\geq0}$ in the so-called Gordon-Hedenmalm class $\mathcal{G}$, that is, the family of analytic functions $Φ:\mathbb C_+\to \mathbb C_+$ giving rise to bounded composition operators in the Hardy space of Dirichlet series $\mathcal{H}^2$. We show that there is a one-to-one correspondence between continuous semigroups $\{Φ_{t}\}_{t\geq0}$ in the class $\mathcal G$ and strongly continuous semigroups of composition operators $\{T_t\}_{t\geq0}$, where $T_t(f)=f\circΦ_t$, $f\in\mathcal{H}^2$. We extend these results for the range $p\in[1,\infty)$. For the case $p=\infty$, we prove that there is no non-trivial strongly continuous semigroup of composition operators in $\mathcal{H}^\infty$. We characterize the infinitesimal generators of continuous semigroups in the class $\mathcal G$ as those Dirichlet series sending $\mathbb C_{+}$ into its closure. Some dynamical properties of the semigroups are obtained from a description of the Koenigs map of the semigroup.

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↗
arXiv2018

Composition operators with surjective symbol and small approximation numbers

We give a new proof of the existence of a surjective symbol whose associated composition operator on H 2 (D) is in all Schatten classes, with the improvement that its approximation numbers can be, in some sense, arbitrarily small. We show, as an application, that, contrary to the 1-dimensional case, for N $\ge$ 2, the behavior of the approximation numbers a n = a n (C $Φ$), or rather of $β$ -- N = lim inf n$\rightarrow$$\infty$ [a n ] 1/n 1/N or $β$ + N = lim sup n$\rightarrow$$\infty$ [a n ] 1/n 1/N , of composition operators on H 2 (D N) cannot be determined by the image of the symbol. MSC 2010 Primary: 47B33 Secondary: 32A35 ; 46B28

math.FA↗
arXiv2018

Pluricapacity and approximation numbers of composition operators

For suitable bounded hyperconvex sets $Ω$ in $\mathbb{C}^N$, in particular the ball or the polydisk, we give estimates for the approximation numbers of composition operators $C_ϕ\colon H^2 (Ω) \to H^2 (Ω)$ when $ϕ(Ω)$ is relatively compact in $Ω$, involving the Monge-Ampère capacity of $ϕ(Ω)$.

math.FA↗