Differentiation in lacunary directions and an extension of the Marcinkiewicz multiplier theorem

Anthony Carbery

Displaying similar documents to “Differentiation in lacunary directions and an extension of the Marcinkiewicz multiplier theorem”

Two results on the Dunkl maximal operator

Luc Deleaval (2011)

Studia Mathematica

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In this article, we first improve the scalar maximal theorem for the Dunkl maximal operator by giving some precisions on the behavior of the constants of this theorem for a general reflection group. Next we complete the vector-valued theorem for the Dunkl-type Fefferman-Stein operator in the $ℤ ₂^{d}$ case by establishing a result of exponential integrability corresponding to the case p = +∞.

Fourier multipliers for Hölder continuous functions and maximal regularity

Wolfgang Arendt, Charles Batty, Shangquan Bu (2004)

Studia Mathematica

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Two operator-valued Fourier multiplier theorems for Hölder spaces are proved, one periodic, the other on the line. In contrast to the $L^{p}$ -situation they hold for arbitrary Banach spaces. As a consequence, maximal regularity in the sense of Hölder can be characterized by simple resolvent estimates of the underlying operator.

Spherical summation : a problem of E.M. Stein

Antonio Cordoba, B. Lopez-Melero (1981)

Annales de l'institut Fourier

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Writing $(T_{R}^{λ} f) \hat{} (ξ) = (1 - | ξ |^{2} / R^{2})_{+}^{λ} \hat{f} (ξ)$ . E. Stein conjectured $∥ {(\sum_{j} | T_{R_{j}}^{λ} f_{i} |^{2})}^{1 / 2} ∥_{p} \leq C ∥ {(\sum_{j} | f_{j} |^{2})}^{1 / 2} ∥_{p}$ for $λ > 0$ , $\frac{4}{3} \leq p \leq 4$ and $C = C_{λ, p}$ . We prove this conjecture. We prove also $f (x) = {lim}_{j \to \infty} T_{2^{j}}^{λ} f (x)$ a.e. We only assume $\frac{4}{3 + 2 λ} < p < \frac{4}{1 - 2 λ}$ .

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. ...

On the maximal Fejér operator for double Fourier series of functions in Hardy spaces

Ferenc Móricz (1995)

Studia Mathematica

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We consider the Fejér (or first arithmetic) means of double Fourier series of functions belonging to one of the Hardy spaces $H^{(1, 0)} (^{2})$ , $H^{(0, 1)} (^{2})$ , or $H^{(1, 1)} (^{2})$ . We prove that the maximal Fejér operator is bounded from $H^{(1, 0)} (^{2})$ or $H^{(0, 1)} (^{2})$ into weak- $L^{1} (^{2})$ , and also bounded from $H^{(1, 1)} (^{2})$ into $L^{1} (^{2})$ . These results extend those by Jessen, Marcinkiewicz, and Zygmund, which involve the function spaces $L^{1} l o g^{+} L (^{2})$ , $L^{1} {(l o g^{+} L)}^{2} (^{2})$ , and $L^{μ} (^{2})$ with 0 < μ < 1, respectively. We establish analogous results for the maximal conjugate Fejér operators. On closing, we formulate...

A Marcinkiewicz type multiplier theorem for H¹ spaces on product domains

Michał Wojciechowski (2000)

Studia Mathematica

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It is proved that if $m : ℝ^{d} \to ℂ$ satisfies a suitable integral condition of Marcinkiewicz type then m is a Fourier multiplier on the $H^{1}$ space on the product domain $ℝ^{d_{1}} \times . . . \times ℝ^{d_{k}}$ . This implies an estimate of the norm $N (m, L^{p} (ℝ^{d})$ of the multiplier transformation of m on $L^{p} (ℝ^{d})$ as p→1. Precisely we get $N (m, L^{p} (ℝ^{d})) ≲ {(p - 1)}^{- k}$ . This bound is the best possible in general.

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...

The Marcinkiewicz multiplier condition for bilinear operators

Loukas Grafakos, Nigel J. Kalton (2001)

Studia Mathematica

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This article is concerned with the question of whether Marcinkiewicz multipliers on $ℝ^{2 n}$ give rise to bilinear multipliers on ℝⁿ × ℝⁿ. We show that this is not always the case. Moreover, we find necessary and sufficient conditions for such bilinear multipliers to be bounded. These conditions in particular imply that a slight logarithmic modification of the Marcinkiewicz condition gives multipliers for which the corresponding bilinear operators are bounded on products of Lebesgue and Hardy...

Fourier coefficients of continuous functions and a class of multipliers

Serguei V. Kislyakov (1988)

Annales de l'institut Fourier

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If $x$ is a bounded function on $Z$ , the multiplier with symbol $x$ (denoted by $M_{x})$ is defined by $(M_{x} f) \hat{} = x \hat{f}$ , $f \in L^{2} (T)$ . We give some conditions on $x$ ensuring the “interpolation inequality” $∥ M_{x} f ∥_{L^{p}} \leq C ∥ f ∥_{L^{1}}^{α} ∥ M_{x} f ∥_{L^{q}}^{1 - α}$ (here $1 < p < q$ and $α = α (p, q, x)$ is between 0 and 1). In most cases considered $M_{x}$ fails to have stronger $L^{1}$ -regularity properties (e.g. fails to be of weak type (1,1)). The results are applied to prove that for many sets $E \subset Z$ every positive sequence in $ℓ^{2} (E)$ can be majorized by the sequence ${$ $| \hat{f} (n) |}_{n \in E}$ for some continuous funtion $f$ with spectrum...

Spaces with maximal projection constants

Hermann König, Nicole Tomczak-Jaegermann (2003)

Studia Mathematica

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We show that n-dimensional spaces with maximal projection constants exist not only as subspaces of $l_{\infty}$ but also as subspaces of l₁. They are characterized by a rigid set of vector conditions. Nevertheless, we show that, in general, there are many non-isometric spaces with maximal projection constants. Several examples are discussed in detail.

Une inégalité maximale sous-gaussienne sur les espaces de tentes

E. Labeye-Voisin (2003)

Studia Mathematica

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We introduce a maximal function (denoted by π̅ ) on the tent spaces $T^{p} (ℝ ₊^{n + 1})$ , 0 < p < ∞, of Coifman, Meyer and Stein [8]. We prove a good-λ estimate of subgaussian type for this maximal function and for the square function of tent spaces, leading to integrability results for π̅. We deduce convergence results for the singular integral defining π.

Multiplier transformations on $H^{p}$ spaces

Daning Chen, Dashan Fan (1998)

Studia Mathematica

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The authors obtain some multiplier theorems on $H^{p}$ spaces analogous to the classical $L^{p}$ multiplier theorems of de Leeuw. The main result is that a multiplier operator ${(T f)}^{(} x) = λ (x) f ̂ (x)$ $(λ \in C (ℝ^{n}))$ is bounded on $H^{p} (ℝ^{n})$ if and only if the restriction ${λ (ε m)}_{m \in Λ}$ is an $H^{p} (T^{n})$ bounded multiplier uniformly for ε>0, where Λ is the integer lattice in $ℝ^{n}$ .

Convergence a.e. of spherical partial Fourier integrals on weighted spaces for radial functions: endpoint estimates

María J. Carro, Elena Prestini (2009)

Studia Mathematica

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We prove some extrapolation results for operators bounded on radial $L^{p}$ functions with p ∈ (p₀,p₁) and deduce some endpoint estimates. We apply our results to prove the almost everywhere convergence of the spherical partial Fourier integrals and to obtain estimates on maximal Bochner-Riesz type operators acting on radial functions in several weighted spaces.

A remark on the $H^{1}$ -BMO duality in product domains

J. Alvarez (1989)

Colloquium Mathematicae

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Carleson's theorem with quadratic phase functions

Michael T. Lacey (2002)

Studia Mathematica

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It is shown that the operator below maps $L^{p}$ into itself for 1 < p < ∞. $C f (x) : = s u p_{a, b} | p . v . \int f (x - y) e^{i (a y ² + b y)} d y / y |$ . The supremum over b alone gives the famous theorem of L. Carleson [2] on the pointwise convergence of Fourier series. The supremum over a alone is an observation of E. M. Stein [12]. The method of proof builds upon Stein’s observation and an approach to Carleson’s theorem jointly developed by the author and C. M. Thiele [7].

Multipliers of the Hardy space H¹ and power bounded operators

Gilles Pisier (2001)

Colloquium Mathematicae

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We study the space of functions φ: ℕ → ℂ such that there is a Hilbert space H, a power bounded operator T in B(H) and vectors ξ, η in H such that φ(n) = ⟨Tⁿξ,η⟩. This implies that the matrix ${(φ (i + j))}_{i, j \geq 0}$ is a Schur multiplier of B(ℓ₂) or equivalently is in the space (ℓ₁ ⊗̌ ℓ₁)*. We show that the converse does not hold, which answers a question raised by Peller [Pe]. Our approach makes use of a new class of Fourier multipliers of H¹ which we call “shift-bounded”. We show that there is a φ which...

A note on rare maximal functions

Paul Alton Hagelstein (2003)

Colloquium Mathematicae

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A necessary and sufficient condition is given on the basis of a rare maximal function $M_{l}$ such that $M_{l} f \in L ¹ ([0, 1])$ implies f ∈ L log L([0,1]).