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Hervé Queffélec

Publications and source records attributed to Hervé Queffélec.

At least 19 recordsLinked to original sources
arXiv2026

Khintchin conjecture and related topics

Motivated by Khintchin's 1923 conjecture, refuted by Marstrand in 1970, we study the Khintchin class of functions associated to a given increasing sequence of integers. When the Khintchin class contains L^p(\mathbb{T}), we call the sequence a L^p-Khintchin sequence. We establish basic properties of Khintchin sequences, provide several constructions, and propose open problems for further research. We also initiate the study of Khintchin sequences of group endomorphisms on compact abelian groups. Under a Fourier-tightness assumption, we show that ergodicity (respectively, weakly mixing or strongly mixing) of a skew product of endomorphisms is equivalent to the corresponding property of the base system, supporting the idea that typical fiber orbits in such skew products should form Khintchin sequences.

math.DS↗
arXiv2024

Volterra operator acting on Bergman spaces of Dirichlet series

Since their introduction in 1997, the Hardy spaces of Dirichlet series have been broadly and deeply studied. The increasing interest sparked by these Banach spaces of Dirichlet series motivated the introduction of new such spaces, as the Bergman spaces of Dirichlet series $\mathcal{A}^p_μ$ here considered, where $μ$ is a probability measure on $(0,\infty)$. Similarly, recent lines of research have focused their attention on the study of some classical operators acting on these spaces, as it is the case of the Volterra operator $T_g$. In this work, we introduce a new family of Bloch spaces of Dirichlet series, the $\text{Bloch}_μ$-spaces, and study some of its most essential properties. Using these spaces we are able to provide a sufficient condition for the Volterra operator $T_g$ to act boundedly on the Bergman spaces $\mathcal{A}^p_μ$. We also establish a necessary condition for a specific choice of the probability measures $μ$. Sufficient and necessary conditions for compactness are also proven. The membership in Schatten classes is studied as well. Eventually, a radicality result is established for Bloch spaces of Dirichlet series.

math.FA↗
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

Integration type operators and point evaluation on weighted Bergman spaces of Dirichlet series

The theory of Banach spaces of Dirichlet series has drawn an increasing attention in the recent 25 years. One of the main interest of this new theory is that of defining analogues of the classical spaces of analytic functions on the unit disc. In this sense, Bergman spaces were introduced several years ago contributing to broaden the picture of this theory. In the paper presenting these new family of spaces, some of its most essential questions were considered. Among them, some partial estimates of the norm of the pointwise evaluation functional were given. In this work, we introduce a version of the Riemann-Liouville semigroup acting on these spaces, and, with this new tool, we are able to estimate the norm of this functional.

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

Old and new results on the Furstenberg sets

This paper is a complement to our previous paper [21]. It surveys the works on the Furstenberg set $S=\{2^{m}3^{n}: n\ge 0, m\ge 0\}$ and its random version $T$. We also present some new results. For example, it is proved that $T$ almost surely contains a subset of positive lower density which is $\frac{4}{3}$-Rider. It is also proved that a class of random sets of integers are Sidon sets when Bourgain's condition is not satisfied; this generalizes a result of Kahane-Katznelson. Some open questions about $S$ and $T$ are listed at the end of the paper.

math.FA↗
arXiv2022

The Furstenberg set and its random version

We study some number-theoretic, ergodic and harmonic analysis properties of the Furstenberg set of integers $S=\{2^{m}3^{n}\}$ and compare them to those of its random analogue $T$. In this half-expository work, we show for example that $S$ is "Khinchin distributed", is far from being Hartman-distributed while $T$ is, and that $S$ is a $Λ(p)$ set for all $2<p<\infty$ and that $T$ is a $p$-Rider set for all $p$ such that $4/3<p<2$. Measure-theoretic and probabilistic techniques, notably martingales, play an important role in this work.

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↗
arXiv2021

Compactification and decompactification by weights on Bergman spaces

We characterize the symbols $Φ$ for which there exists a weight w such that the weighted composition operator M w C $Φ$ is compact on the weighted Bergman space B 2 $α$. We also characterize the symbols for which there exists a weight w such that M w C $Φ$ is bounded but not compact. We also investigate when there exists w such that M w C $Φ$ is Hilbert-Schmidt on B 2 $α$.

math.FA↗
arXiv2021

Riesz projection and bounded mean oscillation for Dirichlet series

We prove that the norm of the Riesz projection from $L^\infty(\Bbb{T}^n)$ to $L^p(\Bbb{T}^n)$ is $1$ for all $n\ge 1$ only if $p\le 2$, thus solving a problem posed by Marzo and Seip in 2011. This shows that $H^p(\Bbb{T}^{\infty})$ does not contain the dual space of $H^1(\Bbb{T}^{\infty})$ for any $p>2$. We then note that the dual of $H^1(\Bbb{T}^{\infty})$ contains, via the Bohr lift, the space of Dirichlet series in $\operatorname{BMOA}$ of the right half-plane. We give several conditions showing how this $\operatorname{BMOA}$ space relates to other spaces of Dirichlet series. Finally, relating the partial sum operator for Dirichlet series to Riesz projection on $\Bbb{T}$, we compute its $L^p$ norm when $1<p<\infty$, and we use this result to show that the $L^\infty$ norm of the $N$th partial sum of a bounded Dirichlet series over $d$-smooth numbers is of order $\log\log N$.

math.FA↗
arXiv2020

Comparison of singular numbers of composition operators on different Hilbert spaces of analytic functions

We compare the rate of decay of singular numbers of a given composition operator acting on various Hilbert spaces of analytic functions on the unit disk $\D$. We show that for the Hardy and Bergman spaces, our results are sharp. We also give lower and upper estimates of the singular numbers of the composition operator with symbol the ``cusp map'' and the lens maps, acting on weighted Dirichlet spaces.

math.FA↗
arXiv2019

Compactification, and beyond, of composition operators on Hardy spaces by weights

We study when multiplication by a weight can turn a non-compact composition operator on H 2 into a compact operator, and when it can be in Schatten classes. The q-summing case in H p is considered. We also study when this multiplication can turn a compact composition operator into a non-compact one. MSC 2010 primary: 47B33 ; secondary: 46B28

math.FA↗
arXiv2019

On Bernstein's inequality for polynomials

Bernstein's classical inequality asserts that given a trigonometric polynomial $T$ of degree $n\geq1$, the sup-norm of the derivative of $T$ does not exceed $n$ times the sup-norm of $T$. We present various approaches to prove this inequality and some of its natural extensions/variants, especially when it comes to replacing the sup-norm with the $L^p-norm$.

math.CA↗
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↗