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  • Author or Editor: Ferenc Weisz x
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Abstract  

A general summability method of orthogonal series is given with the help of an integrable function Θ. Under some conditions on Θ we show that if the maximal Fejér operator is bounded from a Banach space X to Y, then the maximal Θ-operator is also bounded. As special cases the trigonometric Fourier, Walsh, Walsh--Kaczmarz, Vilenkin and Ciesielski--Fourier series and the Fourier transforms are considered. It is proved that the maximal operator of the Θ-means of these Fourier series is bounded from H p to L p (1/2<p≤; ∞) and is of weak type (1,1). In the endpoint case p=1/2 a weak type inequality is derived. As a consequence we obtain that the Θ-means of a function fL 1 converge a.e. to f. Some special cases of the Θ-summation are considered, such as the Weierstrass, Picar, Bessel, Riesz, de la Vallée-Poussin, Rogosinski and Riemann summations. Similar results are verified for several-dimensional Fourier series and Hardy spaces.

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Abstract

It is proved that the maximal operator of the triangular Cesàro means of a two-dimensional Fourier series is bounded from the periodic Hardy space to for all 2/(2+α)<p≦∞ and, consequently, is of weak type (1,1). As a consequence we obtain that the triangular Cesàro means of a function converge a.e. to f.

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Abstract  

It is shown that the maximal operator of the one-dimensional dyadic derivative of the dyadic integral is bounded from the dyadic Hardy-Lorentz spaceHp,q toLp,q (1/2<p<∞, 0<q≤∞) and is of weak type (L1,L1). We define the twodimensional dyadic hybrid Hardy spaceH1 and verify that the corresponding maximal operator of a two-dimensional function is of weak type (H1 ,L1). As a consequence, we obtain that the dyadic integral of a two-dimensional functionfεH1LlogL is dyadically differentiable and its derivative is a.e.f.

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It is shown that the maximal operator of the two-parameter dyadic derivative of the dyadic integral is bounded from the two-parameter dyadic Hardy-Lorentz space H p,q to L p,q (1/2 < p < 8, 0 < q L is dyadically differentiable and its derivative is f a.e.

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Abstract  

ВВОДьтсь ДВА НОВых кл АссА пРОстРАНстВBMO, ИМ ЕУЩИх БОльшОЕ жНАЧЕНИЕ В тЕОРИИ ИНтЕРпОлИРО ВАНИь, И ДОкАжыВАУтсь НЕкОтОРыЕ сООтНОшЕНИь ЁкВИВАл ЕНтНОстИ Их И ОБыЧНых пРОстРАН стВBMO. ИжУЧАУтсь сООтВЕтстВУУЩИЕ “sharp”Ф УНкцИИ И ОБОБЩАУтсь НЕкОтОР ыЕ тЕОРЕМы гАРсИИ, И ФЕ ФФЕРМАНА И стЕИНА.

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В работе получено нео бходимое и достаточн ое условие для эквивалентности двупараметрической системы Хаара и некот орых ее специальных перестановок в диади ческом пространствеh 1, а также в ВМО иL p.

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Abstract  

We investigate the Kronecker product of bounded Ciesielski systems, which can be obtained from the spline systems of order (m, k) in the same way as the Walsh system from the Haar system. It is shown that the maximal operator of the Fejér means of the d-dimensional Ciesielski-Fourier series is bounded from the Hardy space H p([0, 1)d 1×…×[0, 1)d l to L p ([0, 1)d) if 1/2<p<∞ and m j≥0, ‖k j‖≤m j+1. By an interpolation theorem, we get that the maximal operator is also of weak type (

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$(H_1^{\# _i } ,L_1 )$$ \end{document}
) (i=1,…,l), where the Hardy space
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$H_1^{\# _i }$$ \end{document}
is defined by a hybrid maximal function and
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$H_1^{\# _i } \supset L(\log L)^{l - 1}$$ \end{document}
. As a consequence, we obtain that the Fejér means of the Ciesielski-Fourier series of a function f converge to f a.e. if f
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$H_1^{\# _i }$$ \end{document}
and converge in a cone if fεL 1.

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Summary The inequality ( n=2 Σn p-2 ^ f(n) │ p )≤C p ││f││ (0<p≤2) is proved for the one- and two-dimensional Ciesielski-Fourier coefficients of functions in Hardy spaces.

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