Maximal convergence space of a subsequence of the logarithmic means of rectangular partial sums of double Walsh-Fourier series. (English) Zbl 1103.42017

Gát, György; Goginava, Ushangi

Real Anal. Exch. 31(2005-2006), No. 2, 447-464 (2006).

Summary: The main aim of this paper is to prove that the maximal operator of the logarithmic means of rectangular partial sums of double WaishFourier 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 \(t^{2n,2^m} f (x^1,x^2)\to f(x^1,x^2)\) a.e. as \(m,n\to \infty\) for all \(f\in H^{\#}(\supset L\log^+ L)\). We also prove the sharpness of this result. Namely, for every measurable function \(\delta : [0, +\infty)\to [0, +\infty)\), \(\lim_{t\to\infty}\delta(t) = 0\) we have a function \(f\) such as \(f\in L\log^+ L\delta(L)\) and the two-dimensional Nörlund logarithmic means do not converge to \(f\) a.e. (in the Pringsheim sense) on \(I^2\).

MSC:

42C10	Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42B25	Maximal functions, Littlewood-Paley theory

Keywords:

a.e. convergence and divergence