Spherical summation : a problem of E.M. Stein

Antonio Cordoba; B. Lopez-Melero

Displaying similar documents to “Spherical summation : a problem of E.M. Stein”

Transference and restriction of maximal multiplier operators on Hardy spaces

Zhixin Liu, Shanzhen Lu (1993)

Studia Mathematica

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The aim of this paper is to establish transference and restriction theorems for maximal operators defined by multipliers on the Hardy spaces $H^{p} (ℝ^{n})$ and $H^{p} (^{n})$ , 0 < p ≤ 1, which generalize the results of Kenig-Tomas for the case p > 1. We prove that under a mild regulation condition, an $L^{\infty} (ℝ^{n})$ function m is a maximal multiplier on $H^{p} (ℝ^{n})$ if and only if it is a maximal multiplier on $H^{p} (^{n})$ . As an application, the restriction of maximal multipliers to lower dimensional Hardy spaces is considered. ...

A multiplier theorem for Fourier series in several variables

Nakhle Asmar, Florence Newberger, Saleem Watson (2006)

Colloquium Mathematicae

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We define a new type of multiplier operators on $L^{p} (^{N})$ , where $^{N}$ is the N-dimensional torus, and use tangent sequences from probability theory to prove that the operator norms of these multipliers are independent of the dimension N. Our construction is motivated by the conjugate function operator on $L^{p} (^{N})$ , to which the theorem applies as a particular example.

The space of multipliers into $l_{1}$

P. Wojtaszczyk (1979)

Annales Polonici Mathematici

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Multilinear Fourier multipliers with minimal Sobolev regularity, I

Loukas Grafakos, Hanh Van Nguyen (2016)

Colloquium Mathematicae

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We find optimal conditions on m-linear Fourier multipliers that give rise to bounded operators from products of Hardy spaces $H^{p_{k}}$ , $0 < p_{k} \leq 1$ , to Lebesgue spaces $L^{p}$ . These conditions are expressed in terms of L²-based Sobolev spaces with sharp indices within the classes of multipliers we consider. Our results extend those obtained in the linear case (m = 1) by Calderón and Torchinsky (1977) and in the bilinear case (m = 2) by Miyachi and Tomita (2013). We also prove a coordinate-type Hörmander integral...

On finitely generated closed ideals in $H^{\infty} (D)$

Jean Bourgain (1985)

Annales de l'institut Fourier

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Assume $f_{1}, ..., f_{N}$ a finite set of functions in $H^{\infty} (D)$ , the space of bounded analytic functions on the open unit disc. We give a sufficient condition on a function $f$ in $H^{\infty} (D)$ to belong to the norm-closure of the ideal $I (f_{1}, ..., f_{N})$ generated by $f_{1}, ..., f_{N}$ , namely the property $| f (z) | \leq α (| f_{1} (z) | + ... + | f_{N} (z) |) for z \in D$ for some function $α$ : $R_{+} \to R_{+}$ satisfying ${lim}_{t \to 0} α (t) / t = 0 .$ The main feature in the proof is an improvement in the contour-construction appearing in L. Carleson’s solution of the corona-problem. It is also shown that the property $| f (z) | \leq C max_{1 \leq j \leq N} | f_{j} (z) | for z \in D$ ...

Multipliers of the space $A^{p}$

I. Jovanović (1986)

Matematički Vesnik

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The Herz-Schur multiplier norm of sets satisfying the Leinert condition

Éric Ricard, Ana-Maria Stan (2011)

Colloquium Mathematicae

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It is well known that in a free group , one has $| | χ_{E} {| |}_{M_{c b} A ()} \leq 2$ , where E is the set of all the generators. We show that the (completely) bounded multiplier norm of any set satisfying the Leinert condition depends only on its cardinality. Consequently, based on a result of Wysoczański, we obtain a formula for $| | χ_{E} {| |}_{M_{c b} A ()}$ .

A counter-example in singular integral theory

Lawrence B. Difiore, Victor L. Shapiro (2012)

Studia Mathematica

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An improvement of a lemma of Calderón and Zygmund involving singular spherical harmonic kernels is obtained and a counter-example is given to show that this result is best possible. In a particular case when the singularity is O(|log r|), let $f \in C ¹ (ℝ^{N} ∖ 0)$ and suppose f vanishes outside of a compact subset of $ℝ^{N}$ , N ≥ 2. Also, let k(x) be a Calderón-Zygmund kernel of spherical harmonic type. Suppose f(x) = O(|log r|) as r → 0 in the $L^{p}$ -sense. Set $F (x) = \int_{ℝ^{N}} k (x - y) f (y) d y \forall x \in ℝ^{N} ∖ 0$ . Then F(x) = O(log²r) as r → 0 in the $L^{p}$ -sense, 1 <...

Pointwise multipliers on martingale Campanato spaces

Eiichi Nakai, Gaku Sadasue (2014)

Studia Mathematica

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We introduce generalized Campanato spaces $_{p, ϕ}$ on a probability space (Ω,ℱ,P), where p ∈ [1,∞) and ϕ: (0,1] → (0,∞). If p = 1 and ϕ ≡ 1, then $_{p, ϕ} = B M O$ . We give a characterization of the set of all pointwise multipliers on $_{p, ϕ}$ .

Variations on Bochner-Riesz multipliers in the plane

Daniele Debertol (2006)

Studia Mathematica

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We consider the multiplier $m_{μ}$ defined for ξ ∈ ℝ by $m_{μ} (ξ) ≐ {((1 - ξ ₁ ² - ξ ₂ ²) / (1 - ξ ₁))}^{μ} 1_{D} (ξ)$ , D denoting the open unit disk in ℝ. Given p ∈ ]1,∞[, we show that the optimal range of μ’s for which $m_{μ}$ is a Fourier multiplier on $L^{p}$ is the same as for Bochner-Riesz means. The key ingredient is a lemma about some modifications of Bochner-Riesz means inside convex regions with smooth boundary and non-vanishing curvature, providing a more flexible version of a result by Iosevich et al. [Publ. Mat. 46 (2002)]. As an application, we show...

(E,F)-Schur multipliers and applications

Fedor Sukochev, Anna Tomskova (2013)

Studia Mathematica

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For two given symmetric sequence spaces E and F we study the (E,F)-multiplier space, that is, the space of all matrices M for which the Schur product M ∗ A maps E into F boundedly whenever A does. We obtain several results asserting continuous embedding of the (E,F)-multiplier space into the classical (p,q)-multiplier space (that is, when $E = l_{p}$ , $F = l_{q}$ ). Furthermore, we present many examples of symmetric sequence spaces E and F whose projective and injective tensor products are not isomorphic...

Regularity properties of commutators and $B M O$ -Triebel-Lizorkin spaces

Abdellah Youssfi (1995)

Annales de l'institut Fourier

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In this paper we consider the regularity problem for the commutators $([b, R_{k}])_{1 \leq k \leq n}$ where $b$ is a locally integrable function and $(R_{j})_{1 \leq j \leq n}$ are the Riesz transforms in the $n$ -dimensional euclidean space $ℝ^{n}$ . More precisely, we prove that these commutators $([b, R_{k}])_{1 \leq k \leq n}$ are bounded from $L^{p}$ into the Besov space ${\dot{B}}_{p}^{s, p}$ for $1 < p < + \infty$ and $0 < s < 1$ if and only if $b$ is in the $B M O$ -Triebel-Lizorkin space ${\dot{F}}_{\infty}^{s, p}$ . The reduction of our result to the case $p = 2$ gives in particular that the commutators $([b, R_{k}])_{1 \leq k \leq n}$ are bounded form $L^{2}$ into the Sobolev space ${\dot{H}}^{s}$ if and only if $b$ ...

Maximal operators of Fejér means of double Vilenkin-Fourier series

István Blahota, György Gát, Ushangi Goginava (2007)

Colloquium Mathematicae

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The main aim of this paper is to prove that the maximal operator $σ ₀ * : = s u p ₙ | σ_{n, n} |$ of the Fejér means of the double Vilenkin-Fourier series is not bounded from the Hardy space $H_{1 / 2}$ to the space weak- $L_{1 / 2}$ .