Weak Type Inequality for Logarithmic Means of Walsh-Kaczmarz-Fourier Series

Abstract

The main aim of this paper is to prove that the Nörlund logarithmic means $t_{n}^{\kappa }f$ of one-dimensional Walsh-Kaczmarz-Fourier series is weak type (1,1), and this fact implies that $t_{n}^{\kappa }f$ converges in measure on $I$ for every function $f\in L(I)$ and $t_{n,m}^{\kappa }f$ converges in measure on $I^{2}$ for every function $f\in L\ln ^{+}L(I^{2}).$ Moreover, the maximal Orlich space such that Nörlund logarithmic means of two-dimensional Walsh-Kaczmarz-Fourier series for the functions from this space converge in two-dimensional measure is found.