An inverse Sidon type inequality

S. Fridli

Displaying similar documents to “An inverse Sidon type inequality”

On a variant of the Hardy inequality between weighted Orlicz spaces

Agnieszka Kałamajska, Katarzyna Pietruska-Pałuba (2009)

Studia Mathematica

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Let M be an N-function satisfying the Δ₂-condition, and let ω, φ be two other functions, with ω ≥ 0. We study Hardy-type inequalities $\int_{ℝ ₊} M (ω (x) | u (x) |) e x p (- φ (x)) d x \leq C \int_{ℝ ₊} M (| u^{'} (x) |) e x p (- φ (x)) d x$ , where u belongs to some set of locally absolutely continuous functions containing $C ₀^{\infty} (ℝ ₊)$ . We give sufficient conditions on the triple (ω,φ,M) for such inequalities to be valid for all u from a given set . The set may be smaller than the set of Hardy transforms. Bounds for constants are also given, yielding classical Hardy inequalities with best constants. ...

An inclusion operator in Hardy spaces on the unit ball in $C^{N}$

M. Jevtić (1988)

Matematički Vesnik

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Monge-Ampère measures and Poletsky-Stessin Hardy spaces on bounded hyperconvex domains

Sibel Şahin (2015)

Banach Center Publications

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Poletsky-Stessin Hardy (PS-Hardy) spaces are the natural generalizations of classical Hardy spaces of the unit disc to general bounded, hyperconvex domains. On a bounded hyperconvex domain Ω, the PS-Hardy space $H_{u}^{p} (Ω)$ is generated by a continuous, negative, plurisubharmonic exhaustion function u of the domain. Poletsky and Stessin considered the general properties of these spaces and mainly concentrated on the spaces $H_{u}^{p} (Ω)$ where the Monge-Ampère measure $(d d^{c} u) ⁿ$ has compact support for the associated...

Doubly commuting submodules of the Hardy module over polydiscs

Jaydeb Sarkar, Amol Sasane, Brett D. Wick (2013)

Studia Mathematica

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In this note we establish a vector-valued version of Beurling’s theorem (the Lax-Halmos theorem) for the polydisc. As an application of the main result, we provide necessary and sufficient conditions for the “weak” completion problem in $H^{\infty} (ⁿ)$ .

Extending Hardy fields by non- $^{\infty}$ -germs

Krzysztof Grelowski (2008)

Annales Polonici Mathematici

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For a large class of Hardy fields their extensions containing non- $^{\infty}$ -germs are constructed. Hardy fields composed of only non- $^{\infty}$ -germs, apart from constants, are also considered.

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...

Fejér means of two-dimensional Fourier transforms on $H_{p} (ℝ \times ℝ)$

Ferenc Weisz (1999)

Colloquium Mathematicae

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The two-dimensional classical Hardy spaces $H_{p} (ℝ \times ℝ)$ are introduced and it is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from $H_{p} (ℝ \times ℝ)$ to $L_{p} (ℝ^{2})$ (1/2 < p ≤ ∞) and is of weak type $(H_{1}^{} (ℝ \times ℝ), L_{1} (ℝ^{2}))$ where the Hardy space $H_{1} (ℝ \times ℝ)$ is defined by the hybrid maximal function. As a consequence we deduce that the Fejér means of a function f ∈ $H_{1}^{(} ℝ \times ℝ)$ ⊃ $L l o g L (ℝ^{2})$ converge to f a.e. Moreover, we prove that the Fejér means are uniformly bounded on $H_{p} (ℝ \times ℝ)$ whenever 1/2 < p < ∞. Thus, in case f ∈ $H_{p} (ℝ \times ℝ)$ , the...

On $n$ th order differential equations over Hardy fields

A. Ramayyan (1994)

Kybernetika

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A rigidity phenomenon for the Hardy-Littlewood maximal function

Stefan Steinerberger (2015)

Studia Mathematica

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The Hardy-Littlewood maximal function ℳ and the trigonometric function sin x are two central objects in harmonic analysis. We prove that ℳ characterizes sin x in the following way: Let $f \in C^{α} (ℝ, ℝ)$ be a periodic function and α > 1/2. If there exists a real number 0 < γ < ∞ such that the averaging operator $(A_{x} f) (r) = 1 / 2 r \int_{x - r}^{x + r} f (z) d z$ has a critical point at r = γ for every x ∈ ℝ, then f(x) = a + bsin(cx+d) for some a,b,c,d ∈ ℝ. This statement can be used to derive a characterization of trigonometric functions as...

Factorization of sequences in discrete Hardy spaces

Santiago Boza (2012)

Studia Mathematica

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The purpose of this paper is to obtain a discrete version for the Hardy spaces $H^{p} (ℤ)$ of the weak factorization results obtained for the real Hardy spaces $H^{p} (ℝ ⁿ)$ by Coifman, Rochberg and Weiss for p > n/(n+1), and by Miyachi for p ≤ n/(n+1). It represents an extension, in the one-dimensional case, of the corresponding result by A. Uchiyama who obtained a factorization theorem in the general context of spaces X of homogeneous type, but with some restrictions on the measure that exclude the case...

Gradient regularity via rearrangements for $p$ -Laplacian type elliptic boundary value problems

Andrea Cianchi, Vladimir G. Maz'ya (2014)

Journal of the European Mathematical Society

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A sharp estimate for the decreasing rearrangement of the length of the gradient of solutions to a class of nonlinear Dirichlet and Neumann elliptic boundary value problems is established under weak regularity assumptions on the domain. As a consequence, the problem of gradient bounds in norms depending on global integrability properties is reduced to one-dimensional Hardy-type inequalities. Applications to gradient estimates in Lebesgue, Lorentz, Zygmund, and Orlicz spaces are presented. ...

On unitary equivalence of invariant subspaces of the Dirichlet space

Kunyu Guo, Liankuo Zhao (2010)

Studia Mathematica

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It is shown that in the Dirichlet space , two invariant subspaces ℳ ₁, ℳ ₂ of the Dirichlet shift $M_{z}$ are unitarily equivalent only if ℳ ₁ = ℳ ₂.

Estimates for the Hardy-Littlewood maximal function on the Heisenberg group

Jacek Zienkiewicz (2005)

Colloquium Mathematicae

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We prove the dimension free estimates of the $L^{p} \to L^{p}$ , 1< p ≤ ∞, norms of the Hardy-Littlewood maximal operator related to the optimal control balls on the Heisenberg group ℍⁿ.

On the spectrum of the p-biharmonic operator involving p-Hardy's inequality

Abdelouahed El Khalil, My Driss Morchid Alaoui, Abdelfattah Touzani (2014)

Applicationes Mathematicae

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In this paper, we study the spectrum for the following eigenvalue problem with the p-biharmonic operator involving the Hardy term: ${Δ (| Δ u |}^{p - 2} {Δ u) = λ (| u |}^{p - 2} u) / (δ {(x)}^{2 p})$ in Ω, $u \in W ₀^{2, p} (Ω)$ . By using the variational technique and the Hardy-Rellich inequality, we prove that the above problem has at least one increasing sequence of positive eigenvalues.

A note on rearrangements of Fourier coefficients

Hugh L. Montgomery (1976)

Annales de l'institut Fourier

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Let $f (x) \sim Σ a_{n} e^{2 π i n x}, f * (x) \sim \sum_{n = 0}^{\infty} a *_{n} \cos 2 π n x$ , where the $a *_{n}$ are the numbers $| a_{n} |$ rearranged so that $a_{n}^{*} ↘ 0$ . Then for any convex increasing $ψ$ , $∥ ψ (| f |^{2} ∥_{1} \leq ∥ ψ (20 | f * |^{2} ∥_{1}$ . The special case $ψ (t) = t^{q / 2}$ , $q \geq 2$ , gives $∥ f ∥_{q} \leq 5 ∥ f * ∥_{q}$ an equivalent of Littlewood.

Calderón couples of rearrangement invariant spaces

N. Kalton (1993)

Studia Mathematica

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We examine conditions under which a pair of rearrangement invariant function spaces on [0,1] or [0,∞) form a Calderón couple. A very general criterion is developed to determine whether such a pair is a Calderón couple, with numerous applications. We give, for example, a complete classification of those spaces X which form a Calderón couple with $L_{\infty} .$ We specialize our results to Orlicz spaces and are able to give necessary and sufficient conditions on an Orlicz function F so that the pair...

Growth estimates for generalized factors of $H^{p}$ spaces

Joseph A. Cima, Angeliki Kazas, Michael I. Stessin (2003)

Studia Mathematica

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With φ an inner function and $M_{φ}$ the multiplication operator on a given Hardy space it is known that for any given function f in the Hardy space we may use the Wold decomposition to obtain a factorization of the given f (not the Riesz factorization). This new factorization has been shown to be useful in the study of commutants of Toeplitz operators. We study the smoothness of each factor of this factorization. We show in some cases that the factors lie in the same Hardy space (or smoothness...

The canonical injection of the Hardy-Orlicz space $H^{Ψ}$ into the Bergman-Orlicz space $^{Ψ}$

Pascal Lefèvre, Daniel Li, Hervé Queffélec, Luis Rodríguez-Piazza (2011)

Studia Mathematica

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We study the canonical injection from the Hardy-Orlicz space $H^{Ψ}$ into the Bergman-Orlicz space $^{Ψ}$ .

The maximal theorem for weighted grand Lebesgue spaces

Alberto Fiorenza, Babita Gupta, Pankaj Jain (2008)

Studia Mathematica

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We study the Hardy inequality and derive the maximal theorem of Hardy and Littlewood in the context of grand Lebesgue spaces, considered when the underlying measure space is the interval (0,1) ⊂ ℝ, and the maximal function is localized in (0,1). Moreover, we prove that the inequality ${| | M f | |}_{p), w} {\leq c | | f | |}_{p), w}$ holds with some c independent of f iff w belongs to the well known Muckenhoupt class $A_{p}$ , and therefore iff ${| | M f | |}_{p, w} {\leq c | | f | |}_{p, w}$ for some c independent of f. Some results of similar type are discussed for the case of small...

Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces

Viktor I. Burenkov, Huseyn V. Guliyev (2004)

Studia Mathematica

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The problem of boundedness of the Hardy-Littewood maximal operator in local and global Morrey-type spaces is reduced to the problem of boundedness of the Hardy operator in weighted $L_{p}$ -spaces on the cone of non-negative non-increasing functions. This allows obtaining sufficient conditions for boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions are also necessary.