The maximal operator of Marcinkiewicz-Fejér means with respect to Walsh-Kaczmarz-Fourier series

In the paper [4, Theorem 1] Gat, Goginava and the author proved that the maximal operator σκ,∗ of Marcinkiewicz-Fejer means of Walsh-Kaczmarz-Fourier series, is bounded from the dyadic Hardy space Hp into the space Lp for p > 2/3 . Moreover, Goginava and the author showed that σκ,∗ is not bounded from the Hardy space H2/3 to the space L2/3 [6, Theorem 1]. The main aim of this paper is to show that the maximal operator σκ,∗ f := supn∈P |σκ n f | log3/2(n+1) , is bounded from the Hardy space H2/3 into the space L2/3. Moreover, we prove that the order of deviant behavior of the n th Walsh-Kacmarz-Marcinkiewicz-Fejer mean is exactly log3/2(n+1) in the endpoint p = 2/3 . Mathematics subject classification (2010): 42C10.