Cesàro summability of double Walsh-Fourier series

Móricz, F.; Schipp, F.; Wade, W. R.

doi:10.1090/S0002-9947-1992-1030510-8

Cesàro summability of double Walsh-Fourier series
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by F. Móricz, F. Schipp and W. R. Wade PDF

Trans. Amer. Math. Soc. 329 (1992), 131-140 Request permission

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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On the summability of double Fourier series

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