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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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

Similarity:

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.

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

R. Demazeux (2011)

Studia Mathematica

Similarity:

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

Similarity:

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

Similarity:

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. ...

On contractive projections in Hardy spaces

Florence Lancien, Beata Randrianantoanina, Eric Ricard (2005)

Studia Mathematica

Similarity:

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

Similarity:

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 < ∞.

Optimal estimates for the fractional Hardy operator

Yoshihiro Mizuta, Aleš Nekvinda, Tetsu Shimomura (2015)

Studia Mathematica

Similarity:

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

Similarity:

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

Similarity:

Generalized Hilbert Operators on Bergman and Dirichlet Spaces of Analytic Functions

Sunanda Naik, Karabi Rajbangshi (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

Similarity:

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 $_{a, b}$ ....

Boundedness of Stein's square functions and Bochner-Riesz means associated to operators on Hardy spaces

Xuefang Yan (2015)

Czechoslovak Mathematical Journal

Similarity:

Let $(X, d, μ)$ be a metric measure space endowed with a distance $d$ and a nonnegative Borel doubling measure $μ$ . Let $L$ be a non-negative self-adjoint operator of order $m$ on $L^{2} (X)$ . Assume that the semigroup $e^{- t L}$ generated by $L$ satisfies the Davies-Gaffney estimate of order $m$ and $L$ satisfies the Plancherel type estimate. Let $H_{L}^{p} (X)$ be the Hardy space associated with $L .$ We show the boundedness of Stein’s square function $𝒢_{δ} (L)$ arising from Bochner-Riesz means associated to $L$ from Hardy spaces $H_{L}^{p} (X)$ to $L^{p} (X)$ , and also study...

The Hausdorff operators on the real Hardy spaces $H^{p} (ℝ)$

Yuichi Kanjin (2001)

Studia Mathematica

Similarity:

We prove that the Hausdorff operator generated by a function ϕ is bounded on the real Hardy space $H^{p} (ℝ)$ , 0 < p ≤ 1, if the Fourier transform ϕ̂ of ϕ satisfies certain smoothness conditions. As a special case, we obtain the boundedness of the Cesàro operator of order α on $H^{p} (ℝ)$ , 2/(2α+1) < p ≤ 1. Our proof is based on the atomic decomposition and molecular characterization of $H^{p} (ℝ)$ .

Order bounded composition operators on the Hardy spaces and the Nevanlinna class

Nizar Jaoua (1999)

Studia Mathematica

Similarity:

We study the order boundedness of composition operators induced by holomorphic self-maps of the open unit disc D. We consider these operators first on the Hardy spaces $H^{p}$ 0 < p < ∞ and then on the Nevanlinna class N. Given a non-negative increasing function h on [0,∞[, a composition operator is said to be X,Lh-order bounded (we write (X,Lh)-ob) with $X = H^{p}$ or X = N if its composition with the map f ↦ f*, where f* denotes the radial limit of f, is order bounded from X into $L_{h}$ . We give...

One-sided discrete square function

A. de la Torre, J. L. Torrea (2003)

Studia Mathematica

Similarity:

Let f be a measurable function defined on ℝ. For each n ∈ ℤ we consider the average $A ₙ f (x) = 2^{- n} \int_{x}^{x + 2 ⁿ} f$ . The square function is defined as $S f (x) = (\sum_{n = - \infty}^{\infty} | A ₙ f (x) - A_{n - 1} f (x) {| ²)}^{1 / 2}$ . The local version of this operator, namely the operator $S ₁ f (x) = (\sum_{n = - \infty}^{0} | A ₙ f (x) - A_{n - 1} f (x) {| ²)}^{1 / 2}$ , is of interest in ergodic theory and it has been extensively studied. In particular it has been proved [3] that it is of weak type (1,1), maps $L^{p}$ into itself (p > 1) and $L^{\infty}$ into BMO. We prove that the operator S not only maps $L^{\infty}$ into BMO but it also maps BMO into BMO. We also prove that the $L^{p}$ boundedness...

A Hardy type inequality for $W_{0}^{m, 1} (Ω)$ functions

Hernán Castro, Juan Dávila, Hui Wang (2013)

Journal of the European Mathematical Society

Similarity:

We consider functions $u \in W_{0}^{m, 1} (Ω)$ , where $Ω \subset ℝ^{N}$ is a smooth bounded domain, and $m \geq 2$ is an integer. For all $j \geq 0, 1 \leq k \leq m - 1$ , such that $1 \leq j + k \leq m$ , we prove that $\frac{\partial^{i} u (x)}{d {(x)}^{m - j - k}} \in W_{0}^{k, 1} (Ω)$ with $∥\partial^{k} {(\frac{\partial^{i} u (x)}{d {(x)}^{m - j - k}})}_{L^{1} (Ω)} \leq C∥ u ∥_{W^{m, 1} (Ω)}$ , where $d$ is a smooth positive function which coincides with dist $(x, \partial Ω)$ near $\partial Ω$ , and $\partial^{l}$ denotes any partial differential operator of order $l$ .

Weighted variable $L^{p}$ integral inequalities for the maximal operator on non-increasing functions

C. J. Neugebauer (2009)

Studia Mathematica

Similarity:

Let $B_{p}$ be the Ariõ-Muckenhoupt weight class which controls the weighted $L^{p}$ -norm inequalities for the Hardy operator on non-increasing functions. We replace the constant p by a function p(x) and examine the associated $L^{p (x)}$ -norm inequalities of the Hardy operator.

The comparison principle and Dirichlet problem in the class $_{p} (f)$ , p > 0

Pham Hoang Hiep (2006)

Annales Polonici Mathematici

Similarity:

We establish the comparison principle in the class $_{p} (f)$ . The result obtained is applied to the Dirichlet problem in $_{p} (f)$ .

Isometric composition operators on weighted Dirichlet space

Shi-An Han, Ze-Hua Zhou (2016)

Czechoslovak Mathematical Journal

Similarity:

We investigate isometric composition operators on the weighted Dirichlet space $𝒟_{α}$ with standard weights ${(1 - | z |}^{2})^{α}$ , $α > - 1$ . The main technique used comes from Martín and Vukotić who completely characterized the isometric composition operators on the classical Dirichlet space $𝒟$ . We solve some of these but not in general. We also investigate the situation when $𝒟_{α}$ is equipped with another equivalent norm.