New real variable methods in H summability of Fourier series

TL;DR

This paper refines real variable methods to study $H_\alpha$ summability of Fourier series for $L^1$ functions, introducing maximal theorems related to Lebesgue measure and $A_1$ weights.

Contribution

It advances the theory of Fourier series summability by refining Marcinkiewicz's methods and incorporating maximal theorems with respect to Lebesgue measure and $A_1$ weights.

Findings

Established refined real variable methods for $H_\alpha$ summability.

Introduced maximal theorems for Lebesgue measure and $A_1$ weights.

Extended summability results to broader classes of functions.

Abstract

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.