The maximal conjugate Fejér operator is bounded from L1 to weak-L1

Abstract.

We consider the Fejér (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 Fejér 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 Fejér 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 ${\Bbb R}:= (-\infty , \infty )$.