A weak type inequality for the maximal operator of (C, α)-means of Fourier series with respect to the Walsh-Kaczmarz system

Gy Gát; U. Goginava

doi:10.1007/s10474-009-8217-8

A weak type inequality for the maximal operator of (C, α)-means of Fourier series with respect to the Walsh-Kaczmarz system

Gy Gát, U. Goginava

Research output: Contribution to journal › Article › peer-review

25 Citations (Scopus)

Abstract

Simon [12] proved that the maximal operator of (C, α)-means of Fourier series with respect to the Walsh-Kaczmarz system is bounded from the martingale Hardy space H_p to the space L_p for p > 1/(1 + α). In this paper we prove that this boundedness result does not hold if p ≦ 1/(1 + α). However, in the endpoint case p = 1/(1 + α) the maximal operator σ_*^α,k is bounded from the martingale Hardy space H_1/(1+α) to the space weak-L_1/(1+α).

Original language	English
Pages (from-to)	65-83
Number of pages	19
Journal	Acta Mathematica Hungarica
Volume	125
Issue number	1
DOIs	https://doi.org/10.1007/s10474-009-8217-8
Publication status	Published - 2009
Externally published	Yes

Keywords

Cesàro means
Hardy space
Walsh-Kaczmarz system

ASJC Scopus subject areas

General Mathematics

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10.1007/s10474-009-8217-8

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abstract = "Simon [12] proved that the maximal operator of (C, α)-means of Fourier series with respect to the Walsh-Kaczmarz system is bounded from the martingale Hardy space Hp to the space Lp for p > 1/(1 + α). In this paper we prove that this boundedness result does not hold if p ≦ 1/(1 + α). However, in the endpoint case p = 1/(1 + α) the maximal operator σ*α,k is bounded from the martingale Hardy space H1/(1+α) to the space weak-L1/(1+α).",

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AU - Gát, Gy

AU - Goginava, U.

PY - 2009

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N2 - Simon [12] proved that the maximal operator of (C, α)-means of Fourier series with respect to the Walsh-Kaczmarz system is bounded from the martingale Hardy space Hp to the space Lp for p > 1/(1 + α). In this paper we prove that this boundedness result does not hold if p ≦ 1/(1 + α). However, in the endpoint case p = 1/(1 + α) the maximal operator σ*α,k is bounded from the martingale Hardy space H1/(1+α) to the space weak-L1/(1+α).

AB - Simon [12] proved that the maximal operator of (C, α)-means of Fourier series with respect to the Walsh-Kaczmarz system is bounded from the martingale Hardy space Hp to the space Lp for p > 1/(1 + α). In this paper we prove that this boundedness result does not hold if p ≦ 1/(1 + α). However, in the endpoint case p = 1/(1 + α) the maximal operator σ*α,k is bounded from the martingale Hardy space H1/(1+α) to the space weak-L1/(1+α).

KW - Cesàro means

KW - Hardy space

KW - Walsh-Kaczmarz system

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A weak type inequality for the maximal operator of (C, α)-means of Fourier series with respect to the Walsh-Kaczmarz system

Abstract

Keywords

ASJC Scopus subject areas

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