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

Abstract

The properties of the maximal operator of the $(C,\alpha )$-means ($\alpha =({\alpha }_1,\dots ,{\alpha }_d)$) 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 $p_0<p$ ($p_0=\max 1/(1+{\alpha }_k):k=1,\dots ,d$) 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 Fejér means of integrable functions. At the endpoint case $p=p_0$, we show that the maximal operator ${\sigma }_L^{\kappa ,\alpha ,\ast }$ is not bounded from the Hardy space $H_{p_0}^{\gamma }$ to the space $L_{p_0}$.