ABSTRACT. We introduce Quasi-local operators (these include operators of Calderon-Zygmund type), a hybrid Hardy space H# of functions of two variables, and we obtain sufficient conditions for a Quasi-local maximal operator to be of weak type (", I). As an application, we show that Cesaro means of the double Walsh-Fourier series of a function f converge a.e. when f belongs to H#. We also obtain the dyadic analogue of a summability result of Marcienkiewicz and Zygmund valid for all fELl provided summability takes place in some positive cone.
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