The maximal conjugate Fej�r operator is bounded from L 1 to weak- L 1

Abstract

We consider the Fejer (or first arithmetic) means of the conjugate series to the Fourier series of a periodic function f integrable in Lebesgue's sense on the torus \({\Bbb T }:= [-\pi , \pi )\). A classical theorem of A. Zygmund says that the maximal conjugate Fejer operator \(\widetilde \sigma _*(f)\) is bounded from \(L^1({\Bbb T })\) to \(L^p({\Bbb T })\) for any \(0\le p \le 1\). We sharpen this result by proving that \(\widetilde \sigma _*(f)\) is bounded from \(L^1({\Bbb T })\) to weak-\(L^1({\Bbb T })\). We prove an analogous result also for the Fejer means (or Riesz means of first order) of the conjugate integral to the Fourier integral of a function f integrable in Lebesgue's sense on the whole real line \({\Bbb R}:= (-\infty , \infty )\).