On weighted Hardy spaces on the unit disk

Evgeny A. Poletsky; Khim R. Shrestha

Displaying similar documents to “On weighted Hardy spaces on the unit disk”

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 ≤ ∞.

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

C. J. Neugebauer (2009)

Studia Mathematica

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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.

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).

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 ${BMO}_{L, w} (ℝ^{n} \times ℝ^{n})$ .

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.

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

M. Mateljević (1979)

Matematički Vesnik

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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.

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...

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 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 $L_{1 / 2}$ .

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.

One-sided discrete square function

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

Studia Mathematica

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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 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 $ℂ$ .

Lipschitz continuity in Muckenhoupt 𝓐₁ weighted function spaces

Dorothee D. Haroske (2011)

Banach Center Publications

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We study continuity envelopes of function spaces $B_{p, q}^{s} (ℝ ⁿ, w)$ and $F_{p, q}^{s} (ℝ ⁿ, w)$ where the weight belongs to the Muckenhoupt class ₁. This essentially extends partial forerunners in [13, 14]. We also indicate some applications of these results.

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

Yuichi Kanjin (2001)

Studia Mathematica

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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 $H^{p} (ℝ)$ .

Embeddings between weighted Copson and Cesàro function spaces

Amiran Gogatishvili, Rza Mustafayev, Tuğçe Ünver (2017)

Czechoslovak Mathematical Journal

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In this paper, characterizations of the embeddings between weighted Copson function spaces ${Cop}_{p_{1}, q_{1}} (u_{1}, v_{1})$ and weighted Cesàro function spaces ${Ces}_{p_{2}, q_{2}} (u_{2}, v_{2})$ are given. In particular, two-sided estimates of the optimal constant $c$ in the inequality $\begin{matrix} d (\int_{0}^{\infty} & {(\int_{0}^{t} f {(τ)}^{p_{2}} v_{2} (τ) d τ)}^{q_{2} / p_{2}} u_{2} (t) d t)^{1 / q_{2}} \\ \leq c {(\int_{0}^{\infty} {(\int_{t}^{\infty} f {(τ)}^{p_{1}} v_{1} (τ) d τ)}^{q_{1} / p_{1}} u_{1} (t) d t)}^{1 / q_{1}}, \end{matrix} d$ where $p_{1}, p_{2}, q_{1}, q_{2} \in (0, \infty)$ , $p_{2} \leq q_{2}$ and $u_{1}$ , $u_{2}$ , $v_{1}$ , $v_{2}$ are weights on $(0, \infty)$ , are obtained. The most innovative part consists of the fact that possibly different parameters $p_{1}$ and $p_{2}$ and possibly different inner weights $v_{1}$ and $v_{2}$ are allowed. The proof is based on the combination of duality techniques...

Some Banach spaces of Dirichlet series

Maxime Bailleul, Pascal Lefèvre (2015)

Studia Mathematica

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The Hardy spaces of Dirichlet series, denoted by $^{p}$ (p ≥ 1), have been studied by Hedenmalm et al. (1997) when p = 2 and by Bayart (2002) in the general case. In this paper we study some $L^{p}$ -generalizations of spaces of Dirichlet series, particularly two families of Bergman spaces, denoted $^{p}$ and $ℬ^{p}$ . Each could appear as a “natural” way to generalize the classical case of the unit disk. We recover classical properties of spaces of analytic functions: boundedness of point evaluation, embeddings...

The continuity of pseudo-differential operators on weighted local Hardy spaces

Ming-Yi Lee, Chin-Cheng Lin, Ying-Chieh Lin (2010)

Studia Mathematica

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We first show that a linear operator which is bounded on $L ²_{w}$ with w ∈ A₁ can be extended to a bounded operator on the weighted local Hardy space $h ¹_{w}$ if and only if this operator is uniformly bounded on all $h ¹_{w}$ -atoms. As an application, we show that every pseudo-differential operator of order zero has a bounded extension to $h ¹_{w}$ .

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...

Some weighted norm inequalities for a one-sided version of $g *_{λ}$

L. de Rosa, C. Segovia (2006)

Studia Mathematica

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We study the boundedness of the one-sided operator $g ⁺_{λ, φ}$ between the weighted spaces $L^{p} (M ¯ w)$ and $L^{p} (w)$ for every weight w. If λ = 2/p whenever 1 < p < 2, and in the case p = 1 for λ > 2, we prove the weak type of $g ⁺_{λ, φ}$ . For every λ > 1 and p = 2, or λ > 2/p and 1 < p < 2, the boundedness of this operator is obtained. For p > 2 and λ > 1, we obtain the boundedness of $g ⁺_{λ, φ}$ from $L^{p} ({(M ¯)}^{[p / 2] + 1} w)$ to $L^{p} (w)$ , where ${(M ¯)}^{k}$ denotes the operator M¯ iterated k times.

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...