Restricted summability of the multi-dimensional Cesàro means of Walsh—Kaczmarz—Fourier series

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 dyadic Hardy space $H_p^\gamma$ to Lebesgue space $L_p$ 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 a theorem of Simon on the a.e. convergence of cone-restricted two-dimensional Fejér means of integrable functions. In the endpoint case $p=p_0$, we show that the maximal operator $\sigma^{\kappa,\alpha,*}_L$ is not bounded from the dyadic Hardy space $H_{p_0}^\gamma$ to the Lebesgue space $L_{p_0}$.