Restricted summability of multi-dimensional
Vilenkin–Fourier series

Abstract. It is proved that the maximal operator of the \((C,\alpha)\) \((\alpha=(\alpha_1,\ldots,\alpha_d))\) and Riesz means of a multi-dimensional Vilenkin–Fourier series is bounded from \(H_p\) to \(L_p\) \((1/({\alpha}_k+1) < p < \infty)\) and is of weak type \((1,1)\), provided that the supremum in the maximal operator is taken over a
cone-like set. As a consequence we obtain the a.e. convergence of the summability means of a function \(f \in L_1\) to \(f\).