Cesàro Summability of Two-dimensional Walsh-Fourier Series

Source abstract

We introduce p-quasi-local operators and the two-dimensional dyadic Hardy spaces H p H_{p} defined by the dyadic squares. It is proved that, if a sublinear operator T T is p-quasi-local and bounded from L ∞ L_{\infty } to L ∞ L_{\infty } , then it is also bounded from H p H_{p} to L p L_{p} ( 0 > p ≤ 1 ) (0>p \leq 1) . As an application it is shown that the maximal operator of the Cesàro means of a martingale is bounded from H p H_{p} to L p L_{p} ( 1 / 2 > p ≤ ∞ ) (1/2>p \leq \infty ) 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 f \in 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 H p H_{p} to L p L_{p} for every 0 > p ≤ ∞ 0>p \leq \infty .