On approach regions for the conjugate Poisson integral and singular integrals

S. Ferrando; R. Jones; K. Reinhold

Displaying similar documents to “On approach regions for the conjugate Poisson integral and singular integrals”

On the $A$ -integrability of singular integral transforms

Shobha Madan (1984)

Annales de l'institut Fourier

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In this article we study the weak type Hardy space of harmonic functions in the upper half plane $R_{+}^{n + 1}$ and we prove the $A$ -integrability of singular integral transforms defined by Calderón-Zygmund kernels. This generalizes the corresponding result for Riesz transforms proved by Alexandrov.

On convergence for the square root of the Poisson kernel in symmetric spaces of rank 1

Jan-Olav Rönning (1997)

Studia Mathematica

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Let P(z,β) be the Poisson kernel in the unit disk , and let $P_{λ} f (z) = ʃ_{\partial} P {(z, φ)}^{1 / 2 + λ} f (φ) d φ$ be the λ -Poisson integral of f, where $f \in L^{p} (\partial)$ . We let $P_{λ} f$ be the normalization $P_{λ} f / P_{λ} 1$ . If λ >0, we know that the best (regular) regions where $P_{λ} f$ converges to f for a.a. points on ∂ are of nontangential type. If λ =0 the situation is different. In a previous paper, we proved a result concerning the convergence of $P_{0} f$ toward f in an $L^{p}$ weakly tangential region, if $f \in L^{p} (\partial)$ and p > 1. In the present paper we will extend the result to symmetric...

Singular integrals on $C_{p}^{α}$

Robert Sharpley, Yong-Sun Shim (1989)

Studia Mathematica

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Singular integrals with highly oscillating kernels on product spaces

Elena Prestini (2000)

Colloquium Mathematicae

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We prove the $L^{2} (^{2})$ boundedness of the oscillatory singular integrals $P_{0} f (x, y) = \int_{D_{x}} e^{i (M_{2} (x) y^{'} + M_{1} (x) x^{'})} ο v e r x^{'} y^{'} f (x - x^{'}, y - y^{'}) d x^{'} d y^{'}$ for arbitrary real-valued $L^{\infty}$ functions $M_{1} (x), M_{2} (x)$ and for rather general domains $D_{x} \subseteq^{2}$ whose dependence upon x satisfies no regularity assumptions.

Estimates for maximal singular integrals

Loukas Grafakos (2003)

Colloquium Mathematicae

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It is shown that maximal truncations of nonconvolution L²-bounded singular integral operators with kernels satisfying Hörmander’s condition are weak type (1,1) and $L^{p}$ -bounded for 1 < p< ∞. Under stronger smoothness conditions, such estimates can be obtained using a generalization of Cotlar’s inequality. This inequality is not applicable here and the point of this article is to treat the boundedness of such maximal singular integral operators in an alternative way.

Some weighted inequalities for general one-sided maximal operators

F. Martín-Reyes, A. de la Torre (1997)

Studia Mathematica

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We characterize the pairs of weights on ℝ for which the operators $M_{h, k}^{+} f (x) = s u p_{c > x} h (x, c) ʃ_{x}^{c} f (s) k (x, s, c) d s$ are of weak type (p,q), or of restricted weak type (p,q), 1 ≤ p < q < ∞, between the Lebesgue spaces with the coresponding weights. The functions h and k are positive, h is defined on $(x, c) : x < c$ , while k is defined on $(x, s, c) : x < s < c$ . If $h (x, c) = {(c - x)}^{- β}$ , $k (x, s, c) = {(c - s)}^{α - 1}$ , 0 ≤ β ≤ α ≤ 1, we obtain the operator $M_{α, β}^{+} f = s u p_{c > x} 1 / {(c - x)}^{β} ʃ_{x}^{c} f (s) / {(c - s)}^{1 - α} d s$ . For this operator, under the assumption 1/p - 1/q = α - β, we extend the weak type characterization to the case p = q and prove that in the case of equal...

Mapping properties of fundamental operators in harmonic analysis related to Bessel operators

Jorge J. Betancor, Eleonor Harboure, Adam Nowak, Beatriz Viviani (2010)

Studia Mathematica

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We obtain sharp power-weighted $L^{p}$ , weak type and restricted weak type inequalities for the heat and Poisson integral maximal operators, Riesz transform and a Littlewood-Paley type square function, emerging naturally in the harmonic analysis related to Bessel operators.

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 $ℝ^{n}$ .

On the maximal function for rotation invariant measures in $ℝ^{n}$

Ana Vargas (1994)

Studia Mathematica

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Given a positive measure μ in $ℝ^{n}$ , there is a natural variant of the noncentered Hardy-Littlewood maximal operator $M_{μ} f (x) = s u p_{x \in B} 1 / μ (B) ʃ_{B} | f | d μ$ , where the supremum is taken over all balls containing the point x. In this paper we restrict our attention to rotation invariant, strictly positive measures μ in $ℝ^{n}$ . We give some necessary and sufficient conditions for $M_{μ}$ to be bounded from $L^{1} (d μ)$ to $L^{1, \infty} (d μ)$ .

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...

S'-convolvability with the Poisson kernel in the Euclidean case and the product domain case

Josefina Alvarez, Martha Guzmán-Partida, Urszula Skórnik (2003)

Studia Mathematica

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We obtain real-variable and complex-variable formulas for the integral of an integrable distribution in the n-dimensional case. These formulas involve specific versions of the Cauchy kernel and the Poisson kernel, namely, the Euclidean version and the product domain version. We interpret the real-variable formulas as integrals of S’-convolutions. We characterize those tempered distribution that are S’-convolvable with the Poisson kernel in the Euclidean case and the product domain case....

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.

Mean field limit for the one dimensional Vlasov-Poisson equation

Maxime Hauray (2012-2013)

Séminaire Laurent Schwartz — EDP et applications

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We consider systems of $N$ particles in dimension one, driven by pair Coulombian or gravitational interactions. When the number of particles goes to infinity in the so called mean field scaling, we formally expect convergence towards the Vlasov-Poisson equation. Actually a rigorous proof of that convergence was given by Trocheris in [Tro86]. Here we shall give a simpler proof of this result, and explain why it implies the so-called “Propagation of molecular chaos”. More precisely, both...