Cesàro Summability of Two-dimensional Walsh-Fourier Series

Abstract:

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