Cesàro Summability of Two-Parameter Walsh–Fourier Series

Abstract

A new atomic decomposition of the two-parameter dyadic martingale Hardy spacesHpdefined by the quadratic variation is given. We introduceHp-quasi-local operators and prove that if a sublinear operatorVisHp-quasi-local and bounded fromL2toL2then it is also bounded fromHptoLp(0<p⩽1). By an interpolation theorem we get thatVis of weak type (H#1, L1) where the Hardy spaceH#1is defined by the hybrid maximal function. As an application it is shown that the maximal operator of the Cesàro means of a two-parameter martingale is bounded fromHptoLp(4/5<p⩽∞) and is of weak type (H#1, L1). So we obtain that the Cesàro means of a functionf∈H#1converge a.e. to the function in question. Finally, it is verified that if the supremum is taken over all two-powers, only, then the maximal operator of the Cesàro means is bounded fromHptoLpfor every 2/3<p⩽∞.