Some Banach spaces of Dirichlet series

Maxime Bailleul; Pascal Lefèvre

Displaying similar documents to “Some Banach spaces of Dirichlet series”

Bounded evaluation operators from $H^{p}$ into $ℓ^{q}$

Martin Smith (2007)

Studia Mathematica

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Given 0 < p,q < ∞ and any sequence z = zₙ in the unit disc , we define an operator from functions on to sequences by $T_{z, p} (f) = {(1 - | z ₙ | ²)}^{1 / p} f (z ₙ)$ . Necessary and sufficient conditions on zₙ are given such that $T_{z, p}$ maps the Hardy space $H^{p}$ boundedly into the sequence space $ℓ^{q}$ . A corresponding result for Bergman spaces is also stated.

On weighted Hardy spaces on the unit disk

Evgeny A. Poletsky, Khim R. Shrestha (2015)

Banach Center Publications

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In this paper we completely characterize those weighted Hardy spaces that are Poletsky-Stessin Hardy spaces $H_{u}^{p}$ . We also provide a reduction of $H^{\infty}$ problems to $H_{u}^{p}$ problems and demonstrate how such a reduction can be used to make shortcuts in the proofs of the interpolation theorem and corona problem.

Hardy-Rogers-type fixed point theorems for $α$ - $G F$ -contractions

Muhammad Arshad, Eskandar Ameer, Aftab Hussain (2015)

Archivum Mathematicum

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The aim of this paper is to introduce some new fixed point results of Hardy-Rogers-type for $α$ - $η$ - $G F$ -contraction in a complete metric space. We extend the concept of $F$ -contraction into an $α$ - $η$ - $G F$ -contraction of Hardy-Rogers-type. An example has been constructed to demonstrate the novelty of our results.

Translations of functions iv vector Hardy classes on the unit disk

Michalak Artur

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AbstractThe paper contains studies of relationships between properties of the “translation” mappings $T_{F}$ and the topological and geometric structure of spaces X and Hardy classes $h^{p} (, X)$ of X-valued harmonic functions on the open unit disk in ℂ (X is a Banach space). The mapping $T_{F}$ transforming the unit circle of ℂ into $h^{p} (, X)$ is associated with a function $F \in h^{p} (, X)$ by the formula $T_{F} (t) = F \circ ϕ ₜ$ , where ϕₜ is the rotation of through t.AcknowledgmentsThis work is based in part on the author’s doctoral thesis written at...

The Bohr inequality for ordinary Dirichlet series

R. Balasubramanian, B. Calado, H. Queffélec (2006)

Studia Mathematica

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We extend to the setting of Dirichlet series previous results of H. Bohr for Taylor series in one variable, themselves generalized by V. I. Paulsen, G. Popescu and D. Singh or extended to several variables by L. Aizenberg, R. P. Boas and D. Khavinson. We show in particular that, if $f (s) = \sum_{n = 1}^{\infty} a ₙ n^{- s}$ with ${| | f | |}_{\infty} : = s u p_{ℜ s > 0} | f (s) | < \infty$ , then $\sum_{n = 1}^{\infty} | a ₙ | n^{- 2} {\leq | | f | |}_{\infty}$ and even slightly better, and $\sum_{n = 1}^{\infty} | a ₙ | n^{- 1 / 2} {\leq C | | f | |}_{\infty}$ , C being an absolute constant.

Boundedness of Littlewood-Paley operators relative to non-isotropic dilations

Shuichi Sato (2019)

Czechoslovak Mathematical Journal

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We consider Littlewood-Paley functions associated with a non-isotropic dilation group on $ℝ^{n}$ . We prove that certain Littlewood-Paley functions defined by kernels with no regularity concerning smoothness are bounded on weighted $L^{p}$ spaces, $1 < p < \infty$ , with weights of the Muckenhoupt class. This, in particular, generalizes a result of N. Rivière (1971).

Essential norms of weighted composition operators between Hardy spaces $H^{p}$ and $H^{q}$ for 1 ≤ p,q ≤ ∞

R. Demazeux (2011)

Studia Mathematica

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We complete the different cases remaining in the estimation of the essential norm of a weighted composition operator acting between the Hardy spaces $H^{p}$ and $H^{q}$ for 1 ≤ p,q ≤ ∞. In particular we give some estimates for the cases 1 = p ≤ q ≤ ∞ and 1 ≤ q < p ≤ ∞.

The weak type inequality for the Walsh system

Ushangi Goginava (2008)

Studia Mathematica

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The main aim of this paper is to prove that the maximal operator $σ^{}$ is bounded from the Hardy space $H_{1 / 2}$ to weak- $L_{1 / 2}$ and is not bounded from $H_{1 / 2}$ to $L_{1 / 2}$ .

A Hankel matrix acting on Hardy and Bergman spaces

Petros Galanopoulos, José Ángel Peláez (2010)

Studia Mathematica

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Let μ be a finite positive Borel measure on [0,1). Let $ℋ_{μ} = {(μ_{n, k})}_{n, k \geq 0}$ be the Hankel matrix with entries $μ_{n, k} = \int_{[0, 1)} t^{n + k} d μ (t)$ . The matrix $_{μ}$ induces formally an operator on the space of all analytic functions in the unit disc by the fomula $ℋ_{μ} (f) (z) = \sum_{n = 0}^{\infty} i (\sum_{k = 0}^{\infty} μ_{n, k} a_{k}) z ⁿ$ , z ∈ , where $f (z) = \sum_{n = 0}^{\infty} a ₙ z ⁿ$ is an analytic function in . We characterize those positive Borel measures on [0,1) such that $ℋ_{μ} (f) (z) = \int_{[0, 1)} f (t) / (1 - t z) d μ (t)$ for all f in the Hardy space H¹, and among them we describe those for which $ℋ_{μ}$ is bounded and compact on H¹. We also study the analogous problem for the Bergman space A². ...

A generalization to the Hardy-Sobolev spaces $H^{k, p}$ of an $L^{p}$ - $L^{1}$ logarithmic type estimate

Imed Feki, Ameni Massoudi, Houda Nfata (2018)

Czechoslovak Mathematical Journal

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The main purpose of this article is to give a generalization of the logarithmic-type estimate in the Hardy-Sobolev spaces $H^{k, p} (G)$ ; $k \in ℕ^{*}$ , $1 \leq p \leq \infty$ and $G$ is the open unit disk or the annulus of the complex space $ℂ$ .

On contractive projections in Hardy spaces

Florence Lancien, Beata Randrianantoanina, Eric Ricard (2005)

Studia Mathematica

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We prove a conjecture of Wojtaszczyk that for 1 ≤ p < ∞, p ≠ 2, $H_{p} ()$ does not admit any norm one projections with dimension of the range finite and greater than 1. This implies in particular that for 1 ≤ p < ∞, p ≠ 2, $H_{p}$ does not admit a Schauder basis with constant one.

Generalized Cesàro operators on certain function spaces

Sunanda Naik (2010)

Annales Polonici Mathematici

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Motivated by some recent results by Li and Stević, in this paper we prove that a two-parameter family of Cesàro averaging operators $^{b, c}$ is bounded on the Dirichlet spaces $_{p, a}$ . We also give a short and direct proof of boundedness of $^{b, c}$ on the Hardy space $H^{p}$ for 1 < p < ∞.

The weighted Hardy spaces associated to self-adjoint operators and their duality on product spaces

Suying Liu, Minghua Yang (2018)

Czechoslovak Mathematical Journal

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Let $L$ be a non-negative self-adjoint operator acting on $L^{2} (ℝ^{n})$ satisfying a pointwise Gaussian estimate for its heat kernel. Let $w$ be an $A_{r}$ weight on $ℝ^{n} \times ℝ^{n}$ , $1 < r < \infty$ . In this article we obtain a weighted atomic decomposition for the weighted Hardy space $H_{L, w}^{p} (ℝ^{n} \times ℝ^{n})$ , $0 < p \leq 1$ associated to $L$ . Based on the atomic decomposition, we show the dual relationship between $H_{L, w}^{1} (ℝ^{n} \times ℝ^{n})$ and ${BMO}_{L, w} (ℝ^{n} \times ℝ^{n})$ .

Optimal estimates for the fractional Hardy operator

Yoshihiro Mizuta, Aleš Nekvinda, Tetsu Shimomura (2015)

Studia Mathematica

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Let $A_{α} f (x) = {| B (0, | x |) |}^{- α / n} \int_{B (0, | x |)} f (t) d t$ be the n-dimensional fractional Hardy operator, where 0 < α ≤ n. It is well-known that $A_{α}$ is bounded from $L^{p}$ to $L^{p_{α}}$ with $p_{α} = n p / (α p - n p + n)$ when n(1-1/p) < α ≤ n. We improve this result within the framework of Banach function spaces, for instance, weighted Lebesgue spaces and Lorentz spaces. We in fact find a ’source’ space $S_{α, Y}$ , which is strictly larger than X, and a ’target’ space $T_{Y}$ , which is strictly smaller than Y, under the assumption that $A_{α}$ is bounded from X into Y and the Hardy-Littlewood...

On linear extension for interpolating sequences

Eric Amar (2008)

Studia Mathematica

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Let A be a uniform algebra on X and σ a probability measure on X. We define the Hardy spaces $H^{p} (σ)$ and the $H^{p} (σ)$ interpolating sequences S in the p-spectrum $ℳ_{p}$ of σ. We prove, under some structural hypotheses on A and σ, that if S is a “dual bounded” Carleson sequence, then S is $H^{s} (σ)$ -interpolating with a linear extension operator for s < p, provided that either p = ∞ or p ≤ 2. In the case of the unit ball of ℂⁿ we find, for instance, that if S is dual bounded in $H^{\infty} ()$ then S is $H^{p} ()$ -interpolating with...

The isoperimetric inequality in the Hardy class $H^{1}$

M. Mateljević (1979)

Matematički Vesnik

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Some Hölder-logarithmic estimates on Hardy-Sobolev spaces

Imed Feki, Ameni Massoudi (2024)

Czechoslovak Mathematical Journal

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We prove some optimal estimates of Hölder-logarithmic type in the Hardy-Sobolev spaces $H^{k, p} (G)$ , where $k \in ℕ^{*}$ , $1 \leq p \leq \infty$ and $G$ is either the open unit disk $𝔻$ or the annular domain $G_{s}$ , $0 < s < 1$ of the complex space $ℂ$ . More precisely, we study the behavior on the interior of $G$ of any function $f$ belonging to the unit ball of the Hardy-Sobolev spaces $H^{k, p} (G)$ from its behavior on any open connected subset $I$ of the boundary $\partial G$ of $G$ with respect to the $L^{1}$ -norm. Our results can be viewed as an improvement and generalization of...

Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions

Sunanda Naik, Karabi Rajbangshi (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let f be an analytic function on the unit disk . We define a generalized Hilbert-type operator $_{a, b}$ by $_{a, b} (f) (z) = Γ (a + 1) / Γ (b + 1) \int_{0}^{1} (f (t) {(1 - t)}^{b}) / ({(1 - t z)}^{a + 1}) d t$ , where a and b are non-negative real numbers. In particular, for a = b = β, $_{a, b}$ becomes the generalized Hilbert operator $_{β}$ , and β = 0 gives the classical Hilbert operator . In this article, we find conditions on a and b such that $_{a, b}$ is bounded on Dirichlet-type spaces $S^{p}$ , 0 < p < 2, and on Bergman spaces $A^{p}$ , 2 < p < ∞. Also we find an upper bound for the norm of the operator $_{a, b}$ ....