## Real Analysis Exchange

### 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 $I^2$.

#### Article information

Source
Real Anal. Exchange, Volume 31, Number 2 (2005), 447-464.

Dates
First available in Project Euclid: 10 July 2007

https://projecteuclid.org/euclid.rae/1184104037

Mathematical Reviews number (MathSciNet)
MR2265786

Zentralblatt MATH identifier
1103.42017

#### Citation

Gát, György; Goginava, Ushangi. Maximal convergence space of a subsequence of the logarithmic means of rectangular partial sums of double Walsh-Fourier series. Real Anal. Exchange 31 (2005), no. 2, 447--464. https://projecteuclid.org/euclid.rae/1184104037

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