On the maximal operator of Marcinkiewicz-Fejer means of the two-dimensional Walsh-Kaczmarz system

In 2009, Goginava showed that sigma(kappa,#) f := sup(A is an element of P)vertical bar sigma(kappa)(2A) f vertical bar, where sigma(K)(2A) f is the 2(A)th Marcinkiewicz Fejer mean of the Walsh-Kaczmarz-Fourier series, is not bounded from the Hardy space H-1/2 to the space L-1/2. The main aim of this paper is to show that the maximal operator (sigma) over tilde (kappa,#) f := sup(A is an element of P)vertical bar sigma(kappa)(2A) f vertical bar/(log(2) 2(A)) is bounded from the Hardy space H-1/2 to the space L-1/2. Moreover, it is proved that the order of deviant behavior of the 2(A)th Walsh-Kacmarz-MarcinIdewicz-Fejer mean is exactly log(2) 2(A).