The maximal conjugate Fejer operator is bounded from L-1 to weak-L-1

We consider the Fejer (or first arithmetic) means of the conjugate series t o the Fourier series of a periodic function f integrable in Lebesgue's sens e on the torus T := [-pi, pi). A classical theorem of A. Zygmund says that the maximal conjugate Fejer operator <(sigma)over tilde>, (f) is bounded fr om L-1(T) to L-p(T) for any 0 < p < 1. We sharpen this result by proving th at <(sigma)over tilde>(*)(f) is bounded from L-1(T) to weak-L-1 (TT). We pr ove an analogous result also for the Fejer means (or Riesz means of first o rder) of the conjugate integral to the Fourier integral of a function f int egrable in Lebesgue's sense on the whole real line R := (-infinity, infinit y).