On the Marcinkiewicz-Fejér means of double Fourier series with respect to the Walsh-Kaczmarz system

Abstract

The main aim of this paper is to prove that the maximal operator of Marcinkiewicz-Fejér means of double Fourier series with respect to the Walsh-Kaczmarz system is bounded from the dyadic Hardy-Lorentz space $H_{pq}$ into Lorentz space $L_{pq}$ for every $p>2/3$ and $0<q\leq \infty $ . As a consequence, we obtain the a.e. convergence of Marcinkiewicz-Fejér means of double Fourier series with respect to the Walsh-Kaczmarz system. That is, $\sigma _{n}(f,x^{1},x^{2}\mathbf{)}\rightarrow f(x^{1},x^{2})$ a.e. as $n\to \infty .$