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

María J. Carro; Elena Prestini

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Displaying similar documents to “Convergence a.e. of spherical partial Fourier integrals on weighted spaces for radial functions: endpoint estimates”

Weighted inequalities for rough square functions through extrapolation

Javier Duoandikoetxea, Edurne Seijo (2002)

Studia Mathematica

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Weighted inequalities for some square functions are studied. L² results are proved first using the particular structure of the operator and then extrapolation of weights is applied to extend the results to other $L^{p}$ spaces. In particular, previous results for square functions with rough kernel are obtained in a simpler way and extended to a larger class of weights.

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...

ω-Calderón-Zygmund operators

Sijue Wu (1995)

Studia Mathematica

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We prove a T1 theorem and develop a version of Calderón-Zygmund theory for ω-CZO when $ω \in A_{\infty}$ .

The John-Nirenberg inequality for functions of bounded mean oscillation with bounded negative part

Min Hu, Dinghuai Wang (2022)

Czechoslovak Mathematical Journal

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A version of the John-Nirenberg inequality suitable for the functions $b \in BMO$ with $b^{-} \in L^{\infty}$ is established. Then, equivalent definitions of this space via the norm of weighted Lebesgue space are given. As an application, some characterizations of this function space are given by the weighted boundedness of the commutator with the Hardy-Littlewood maximal operator.

Two-weighted criteria for integral transforms with multiple kernels

Vakhtang Kokilashvili, Alexander Meskhi (2006)

Banach Center Publications

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Necessary and sufficient conditions governing two-weight $L^{p}$ norm estimates for multiple Hardy and potential operators are presented. Two-weight inequalities for potentials defined on nonhomogeneous spaces are also discussed. Sketches of the proofs for most of the results are given.

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 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...

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

Anthony Carbery (1988)

Annales de l'institut Fourier

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We show that the maximal operator associated to the family of rectangles in $R^{3}$ one of whose sides is parallel to $(1, 2^{j}, 2^{k})$ for some j,k $\in H Z$ is bounded on $L^{p}$ , $1 < p < \infty$ . We give an application of this theorem to obtain an extension of the Marcinkiewicz multiplier theorem.

Boundedness of Riesz transforms on weighted Carleson measure spaces

Ming-Yi Lee (2012)

Studia Mathematica

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Let w be in the Muckenhoupt $A_{\infty}$ weight class. We show that the Riesz transforms are bounded on the weighted Carleson measure space $C M O_{w}^{p}$ , the dual of the weighted Hardy space $H_{w}^{p}$ , 0 < p ≤ 1.

Generalized Hörmander conditions and weighted endpoint estimates

María Lorente, José María Martell, Carlos Pérez, María Silvina Riveros (2009)

Studia Mathematica

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We consider two-weight estimates for singular integral operators and their commutators with bounded mean oscillation functions. Hörmander type conditions in the scale of Orlicz spaces are assumed on the kernels. We prove weighted weak-type estimates for pairs of weights (u,Su) where u is an arbitrary nonnegative function and S is a maximal operator depending on the smoothness of the kernel. We also obtain sufficient conditions on a pair of weights (u,v) for the operators to be bounded...

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 = +∞.

Absolute convergence of multiple Fourier integrals

Yurii Kolomoitsev, Elijah Liflyand (2013)

Studia Mathematica

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Various new sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained. The results are given in terms of $L_{p}$ integrability of the function and its partial derivatives, each with a different p. These p are subject to certain relations known earlier only for some particular cases. Sharpness and applications of the results obtained are also discussed.

Multilinear Calderón-Zygmund operators on weighted Hardy spaces

Wenjuan Li, Qingying Xue, Kôzô Yabuta (2010)

Studia Mathematica

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Grafakos-Kalton [Collect. Math. 52 (2001)] discussed the boundedness of multilinear Calderón-Zygmund operators on the product of Hardy spaces. Then Lerner et al. [Adv. Math. 220 (2009)] defined $A_{p ⃗}$ weights and built a theory of weights adapted to multilinear Calderón-Zygmund operators. In this paper, we combine the above results and obtain some estimates for multilinear Calderón-Zygmund operators on weighted Hardy spaces and also obtain a weighted multilinear version of an inequality for...

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. ...

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- $L_{1 / 2}$ .

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 $\frac{4}{3 + 2 λ} < p < \frac{4}{1 - 2 λ}$ .

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].

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...

The linear bound in A₂ for Calderón-Zygmund operators: a survey

Michael Lacey (2011)

Banach Center Publications

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For an L²-bounded Calderón-Zygmund Operator T acting on $L ² (ℝ^{d})$ , and a weight w ∈ A₂, the norm of T on L²(w) is dominated by $C_{T} {| | w | |}_{A ₂}$ . The recent theorem completes a line of investigation initiated by Hunt-Muckenhoupt-Wheeden in 1973 (MR0312139), has been established in different levels of generality by a number of authors over the last few years. It has a subtle proof, whose full implications will unfold over the next few years. This sharp estimate requires that the A₂ character of the weight can...

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

J. Alvarez (1989)

Colloquium Mathematicae

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Sharp Logarithmic Inequalities for Two Hardy-type Operators

Adam Osękowski (2015)

Bulletin of the Polish Academy of Sciences. Mathematics

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For any locally integrable f on ℝⁿ, we consider the operators S and T which average f over balls of radius |x| and center 0 and x, respectively: $S f (x) = 1 / | B (0, | x |) | \int_{B (0, | x |)} f (t) d t$ , $T f (x) = 1 / | B (x, | x |) | \int_{B (x, | x |)} f (t) d t$ for x ∈ ℝⁿ. The purpose of the paper is to establish sharp localized LlogL estimates for S and T. The proof rests on a corresponding one-weight estimate for a martingale maximal function, a result which is of independent interest.

Weighted norm inequalities for maximal singular integrals with nondoubling measures

Guoen Hu, Dachun Yang (2008)

Studia Mathematica

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Let μ be a nonnegative Radon measure on $ℝ^{d}$ which satisfies μ(B(x,r)) ≤ Crⁿ for any $x \in ℝ^{d}$ and r > 0 and some positive constants C and n ∈ (0,d]. In this paper, some weighted norm inequalities with $A_{p}^{ϱ} (μ)$ weights of Muckenhoupt type are obtained for maximal singular integral operators with such a measure μ, via certain weighted estimates with $A_{\infty}^{ϱ} (μ)$ weights of Muckenhoupt type involving the John-Strömberg maximal operator and the John-Strömberg sharp maximal operator, where ϱ,p ∈ [1,∞).

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).

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.