MAXIMAL ESTIMATES FOR THE (C,α) MEANS OF d-DIMENSIONAL WALSH-FOURIER SERIES

The d-dimensional dyadic martingale Hardy spaces Hp are introduced and it is proved that the maximal operator of the (C, α) (α = (α1, . . . , αd)) means of a Walsh-Fourier series is bounded from Hp to Lp (1/(αk + 1) < p <∞) and is of weak type (L1, L1), provided that the supremum in the maximal operator is taken over a positive cone. As a consequence we obtain that the (C, α) means of a function f ∈ L1 converge a.e. to the function in question. Moreover, we prove that the (C, α) means are uniformly bounded on Hp whenever 1/(αk + 1) < p < ∞. Thus, in case f ∈ Hp, the (C, α) means converge to f in Hp norm. The same results are proved for the conjugate (C, α) means, too.