Almost everywhere convergence of a subsequence of the NɆrlund logarithmic means of Walsh-Kaczmarz-Fourier series

Abstract

The main aim of this paper is to prove that the maximal operator of a subsequence of the (one-dimensional) logarithmic means of Walsh-Kaczmarz-Fourier series is of weak type (1,1) . Moreover, we prove that the maximal operator of the logarithmic means of quadratical partial sums of double Walsh-Kaczmarz-Fourier series is of weak type (1,1) , provided that the supremum in the maximal operator is taken over special indices. The set of Walsh-Kaczmarz polynomials is dense in L1 , so by the well-known density argument the logarithmic means tκ 2n ( f ) converge a.e. to f for all integrable function f . Mathematics subject classification (2000): 42C10.