Maximal estimates for the $(C,\alpha )$ means of $d$-dimensional Walsh-Fourier series

Abstract:

The $d$-dimensional dyadic martingale Hardy spaces $H_{p}$ are introduced and it is proved that the maximal operator of the $(C,\alpha )$ $(\alpha =(\alpha _{1},\ldots ,\alpha _{d}))$ means of a Walsh-Fourier series is bounded from $H_{p}$ to $L_{p}$ $(1/(\alpha _{k}+1)<p<\infty )$ and is of weak type $(L_{1},L_{1})$, provided that the supremum in the maximal operator is taken over a positive cone. As a consequence we obtain that the $(C,\alpha )$ means of a function $f \in L_{1}$ converge a.e. to the function in question. Moreover, we prove that the $(C,\alpha )$ means are uniformly bounded on $H_{p}$ whenever $1/(\alpha _{k}+ 1)<p < \infty$. Thus, in case $f \in H_{p}$, the $(C,\alpha )$ means converge to $f$ in $H_{p}$ norm. The same results are proved for the conjugate $(C,\alpha )$ means, too.