MAXIMAL CONVERGENCE SPACE OF A SUBSEQUENCE OF THE LOGARITHMIC MEANS OF RECTANGULAR PARTIAL SUMS OF DOUBLE WALSH-FOURIER SERIES

摘要

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 (x~1, x~2) → f (x~1, x~2) a. e. as m,n → ∞ for all f ∈ H~# (is contained in L log~+ L). We also prove the sharpness of this result. Namely, For all measurable function δ : [0, +∞) → [0, +∞), lim_(t → ∞) δ(t) = 0 we have a function f such as f ∈ L log~+ L δ(L) and the two-dimensional Noerlund logarithmic means do not converge to f a.e. (in the Pringsheim sense) on I~2.