Restricted summability of the multi-dimensional Ces\`aro means of Walsh-Kaczmarz-Fourier series

Abstract

The properties of the maximal operator of the $(C,\alpha)$-means ($\alpha=(\alpha_1,\ldots,\alpha_d)$) of the multi-dimensional Walsh-Kaczmarz-Fourier series are discussed, where the set of indices is inside a cone-like set. We prove that the maximal operator is bounded from $H_p^\gamma$ to $L_p$ for $p_0< p $ ($p_0=\max{1/(1+\alpha_k): k=1,\ldots, d}$) and is of weak type $(1,1)$. As a corollary we get the theorem of Simon \cite{S1} on the a.e. convergence of cone-restricted two-dimensional Fej\'er means of integrable functions. At the endpoint case $p=p_0$, we show that the maximal operator $\sigma^{\kappa,\alpha,*}_L$ is not bounded from the Hardy space $H_{p_0}^\gamma$ to the space $L_{p_0}$.