Maximal convergence space of a subsequence of the logarithmic means of rectangular partial sums of double Walsh-Fourier series.

Abstract

The main aim of this paper is to prove that the maximal operator of the logarithmic means of rectangular partial sums of double Walsh-Fourier series is of type $(H^{\#},L_{1})$ provided that the supremum in the maximal operator is taken over some special indices. The set of Walsh polynomials is dense in $ H^{\#}$, so by the well-known density argument we have that $t_{2^{n},2^{m}}f\left( x^{1},x^{2}\right) \rightarrow f\left( x^{1},x^{2}\right) $ a. e. as $m,n\rightarrow \infty$ for all $f\in H^{\#} (\supset L\log^{+} L$). We also prove the sharpness of this result. Namely, For all measurable function $\delta :[0,+\infty) \to [0,+\infty) , \, \lim_{t\to\infty}\delta(t)=0$ we have a function $f$ such as $f\in \llogld (L)$ and the two-dimensional Nörlund logarithmic means do not converge to $f$ a.e. (in the Pringsheim sense) on $I^2$.