Displaying similar documents to “On the maximal operator of Walsh-Kaczmarz-Fejér means”

Walsh-Marcinkiewicz means and Hardy spaces

Károly Nagy, George Tephnadze (2014)

Open Mathematics

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The main aim of this paper is to investigate the Walsh-Marcinkiewicz means on the Hardy space H p, when 0 < p < 2/3. We define a weighted maximal operator of Walsh-Marcinkiewicz means and establish some of its properties. With its aid we provide a necessary and sufficient condition for convergence of the Walsh-Marcinkiewicz means in terms of modulus of continuity on the Hardy space H p, and prove a strong convergence theorem for the Walsh-Marcinkiewicz means.

Cesàro summability of one- and two-dimensional trigonometric-Fourier series

Ferenc Weisz (1997)

Colloquium Mathematicae

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We introduce p-quasilocal operators and prove that if a sublinear operator T is p-quasilocal and bounded from L ∞ to L ∞ then it is also bounded from the classical Hardy space H p ( T ) to L p (0 < p ≤ 1). As an application it is shown that the maximal operator of the one-parameter Cesàro means of a distribution is bounded from H p ( T ) to L p (3/4 < p ≤ ∞) and is of weak type ( L 1 , L 1 ) . We define the two-dimensional dyadic hybrid Hardy space H 1 ♯ ( T 2 ) and verify that the maximal operator of the Cesàro means of a two-dimensional...

Fejér means of two-dimensional Fourier transforms on H p ( ℝ × ℝ )

Ferenc Weisz (1999)

Colloquium Mathematicae

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The two-dimensional classical Hardy spaces H p ( ℝ × 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 × ℝ ) to L p ( ℝ 2 ) (1/2 < p ≤ ∞) and is of weak type ( H 1 ( ℝ × ℝ ) , L 1 ( ℝ 2 ) ) where the Hardy space H 1 ( ℝ × ℝ ) is defined by the hybrid maximal function. As a consequence we deduce that the Fejér means of a function f ∈ H 1 ( ℝ × 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 × ℝ ) whenever 1/2 < p < ∞. Thus, in case f ∈ H p ( ℝ × ℝ ) , the...

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...

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 ∞ -norm, that is, by averaging over cubes, the result extends to block decreasing functions of bounded variation,...