On the Maximal Operators of Fejér Means with Respect to the Character System of the Group of 2-Adic Integers in Hardy Spaces

It was a question of Taibleson, open for a long time that the almost everywhere convergence of Fejér (or $(C,1)$) means of Fourier series of integrable functions with respect the character system of the group of $2$-adic integers. This question was answered by Gát in 1997. The aim of this paper is to investigate the maximal operator of the $\sup_n|\sigma_n|$. Among other things, we prove that this operator is bounded from the Hardy space $H_p$ to the Lebesgue space $L_p$ if and only if $1/2 < p < \infty$. The two-dimensional maximal operator is also discussed.