The maximal (C, α, β) operator of two-parameter walsh-fourier series

Abstract

The two-parameter dyadic martingale Hardy spacesH p are introduced and it is proved that the maximal operator of the (C, α, β) means of a two-dimensional Walsh-Fourier series is bounded from Hp to Lp (1/(α+1), 1/(β+1)<p<∞) and is of weak type (H 1 # , L1), where the Hardy space H 1 # is defined by the hybrid maximal function. As a consequence, we obtain that the (C, α, β) means of a function f∈H 1 # converge a.e. to the function in question. Moreover, we prove that the (C, α, β) means are uniformly bounded on Hp whenever 1/(α+1), 1/(β+1)<p<∞. Thus in case f∈Hp, the (C, α, β) means converge to f in Hp norm. The same results are proved for the conjugate (C, α, β) means, too.