A Maximal Inequality for H 1 -Functions on a Generalized Walsh-Paley Group

Fujii, Nobuhiko

doi:10.2307/2042726

Cited by 43 publications

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“…Note that, in the one-parameter case, (10) was proved by Fujii [6] for p = q = 1 (see also Schipp, Simon [12]) and (11) was shown by Schipp [11]. For double trigonometric-Fourier series the analogous theorem is verified by the author [16].…”

Section: Cesàro Summability Of Double Walsh-fourier Seriesmentioning

confidence: 70%

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Cesàro Summability of Two-dimensional Walsh-Fourier Series

Weisz¹

1996

Trans. Amer. Math. Soc.

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Abstract. We introduce p-quasi-local operators and the two-dimensional dyadic Hardy spaces Hp defined by the dyadic squares. It is proved that, if a sublinear operator T is p-quasi-local and bounded from L∞ to L∞, then it is also bounded from Hp to Lp (0 < p ≤ 1). As an application it is shown that the maximal operator of the Cesàro means of a martingale is bounded from Hp to Lp (1/2 < p ≤ ∞) and is of weak type (1,1) provided that the supremum in the maximal operator is taken over a positive cone. So we obtain the dyadic analogue of a summability result with respect to two-dimensional trigonometric Fourier series due to Marcinkievicz and Zygmund; more exactly, the Cesàro means of a function f ∈ L 1 converge a.e. to the function in question, provided again that the limit is taken over a positive cone. Finally, it is verified that if we take the supremum in a cone, but for two-powers, only, then the maximal operator of the Cesàro means is bounded from Hp to Lp for every 0 < p ≤ ∞.

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Section: Cesàro Summability Of Double Walsh-fourier Seriesmentioning

confidence: 70%

“…|σ 2 n ,2 m f |. In the one-dimensional case it is known that σ * is bounded from H 1 to L 1 and is of weak type (1,1) (see Fujii [6] and Schipp [11]). It will be shown, in the two-parameter case, that σ * is p-quasi-local for each 1/2 < p ≤ 1.…”

Section: Introductionmentioning

confidence: 99%

Cesàro Summability of Two-dimensional Walsh-Fourier Series

Weisz¹

1996

Trans. Amer. Math. Soc.

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“…Also, for Walsh-Fourier series, the boundedness of the operator sup n∈N |σ n | from H p to L p was shown by Fujii [12] (p = 1) and by Weisz [21] (1/2 < p ≤ 1).…”

Section: Introduction It Can Be Found In Zygmundmentioning

confidence: 94%

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

Weisz

1997

Colloq. Math.

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“…This result was generalized later by Schipp [16] for Walsh series and Pál and Simon [15] for bounded Vilenkin series by showing that the corresponding maximal operators σ * satisfy the weak type (1 1) inequality. Fujii [5] and Simon [18] verified for Walsh and Vilenkin series that σ * is bounded from H 1 to L 1 . Weisz [28] generalized this result for Walsh series and proved the boundedness of maximal operator from the martingale space H to the space L for > 1/2.…”

Section: Introductionmentioning

confidence: 99%

Walsh-Marcinkiewicz means and Hardy spaces

Nagy

Tephnadze²

2014

Open Mathematics

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Abstract:The main aim of this paper is to investigate the Walsh-Marcinkiewicz means on the Hardy space H , when 0 < < 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 WalshMarcinkiewicz means in terms of modulus of continuity on the Hardy space H , and prove a strong convergence theorem for the Walsh-Marcinkiewicz means. MSC: 42C10 Keywords

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A Maximal Inequality for H 1 -Functions on a Generalized Walsh-Paley Group

Cited by 43 publications

References 0 publications

Cesàro Summability of Two-dimensional Walsh-Fourier Series

Cesàro Summability of Two-dimensional Walsh-Fourier Series

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

Walsh-Marcinkiewicz means and Hardy spaces

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