Weak type inequalities for the Walsh and bounded Ciesielski systems

We investigate both the Walsh system and the bounded Ciesielski systems, which can be obtained from the spline systems of order (m,k) in the same way as the Walsh system can be obtained from the Haar system. We show that the maximal operator of the Fejér means of the Walsh-- and Ciesielski--Fourier series is bounded from the Hardy space H _1/2 to the space weak L_1/2if m -1, |k| m+1. As a consequence, we obtain a new proof for the fact that the Fejér means of the Walsh-- or Ciesielski--Fourier series of a function f L ₁ converge to f a.e.