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Transactions of the American Mathematical Society

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Cesàro summability of double Walsh-Fourier series

Authors: F. Móricz, F. Schipp and W. R. Wade
Journal: Trans. Amer. Math. Soc. 329 (1992), 131-140
MSC: Primary 42C10; Secondary 42B08
MathSciNet review: 1030510
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Abstract: We introduce quasi-local operators (these include operators of Calderón-Zygmund type), a hybrid Hardy space $ {{\mathbf{H}}^\sharp }$ of functions of two variables, and we obtain sufficient conditions for a quasi-local maximal operator to be of weak type $ (\sharp ,1)$. As an application, we show that Cesàro means of the double Walsh-Fourier series of a function $ f$ converge a.e. when $ f$ belongs to $ {{\mathbf{H}}^\sharp }$. We also obtain the dyadic analogue of a summability result of Marcienkiewicz and Zygmund valid for all $ f \in {L^1}$ provided summability takes place in some positive cone.

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Keywords: Walsh functions, Cesàro summability, quasi-local operators, operators of Calderón-Zygmund type, dyadic Hardy spaces
Article copyright: © Copyright 1992 American Mathematical Society

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