A remark on the centered $n$-dimensional Hardy-Littlewood maximal function

J. M. Aldaz

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

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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.

A sharp estimate for the Hardy-Littlewood maximal function

Loukas Grafakos, Stephen Montgomery-Smith, Olexei Motrunich (1999)

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The best constant in the usual $L^{p}$ norm inequality for the centered Hardy-Littlewood maximal function on $ℝ^{1}$ is obtained for the class of all “peak-shaped” functions. A function on the line is called peak-shaped if it is positive and convex except at one point. The techniques we use include variational methods.

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

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

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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.

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 $L^{1, \infty} (d μ)$ .

Outer measures and weak type $(1, 1)$ estimates of Hardy-Littlewood maximal operators.

Terasawa, Yutaka (2006)

Journal of Inequalities and Applications [electronic only]

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Regularity of the Hardy-Littlewood maximal operator on block decreasing functions

J. M. Aldaz, F. J. Pérez Lázaro (2009)

Studia Mathematica

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We study the Hardy-Littlewood maximal operator defined via an unconditional norm, acting on block decreasing functions. We show that the uncentered maximal operator maps block decreasing functions of special bounded variation to functions with integrable distributional derivatives, thus improving their regularity. In the special case of the maximal operator defined by the $ℓ_{\infty}$ -norm, that is, by averaging over cubes, the result extends to block decreasing functions of bounded variation,...

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...

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...

Weak type estimates for the Hardy-Littlewood maximal functions

Luis Caffarelli, Calixto Calderón (1974)

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

M. Jevtić (1988)

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On pointwise estimates for maximal and singular integral operators

A. Lerner (2000)

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We prove two pointwise estimates relating some classical maximal and singular integral operators. In particular, these estimates imply well-known rearrangement inequalities, $L_{ω}^{p}$ and BLO-norm inequalities

Eigenfunctions of the Hardy-Littlewood maximal operator

Leonardo Colzani, Javier Pérez Lázaro (2010)

Colloquium Mathematicae

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We prove that peak shaped eigenfunctions of the one-dimensional uncentered Hardy-Littlewood maximal operator are symmetric and homogeneous. This implies that the norms of the maximal operator on L(p) spaces are not attained.

CLO spaces and central maximal operators

Martha Guzmán-Partida (2013)

Archivum Mathematicum

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We consider central versions of the space $BLO$ studied by Coifman and Rochberg and later by Bennett, as well as some natural relations with a central version of a maximal operator.

Displaying similar documents to “A remark on the centered n -dimensional Hardy-Littlewood maximal function”

Displaying similar documents to “A remark on the centered $n$ -dimensional Hardy-Littlewood maximal function”