New real variable methods in H summability of Fourier series

In this paper we shall be concerned with $H_\alpha$ summability, for $0 < \alpha \leq 2$ of the Fourier series of arbitrary $L^1([-\pi,\pi]) $ functions. The methods to be employed here are a refinement of the real variable methods introduced by Marcinkiewicz in \cite{Marcin1}. In addition, we introduce maximal theorems with respect to the Lebesgue measure and $A_1$ weights.