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Norm convergence of Fejér means of two-dimensional Walsh-Fourier series. (English) Zbl 1246.42023

Nawrocki, Marek (ed.) et al., Marcinkiewicz centenary volume. Proceedings of the Józef Marcinkiewicz centenary conference, June 28–July 2, 2010. Warszawa: Polish Academy of Sciences, Institute of Mathematics (ISBN 978-83-86806-14-0/pbk). Banach Center Publications 95, 317-324 (2011).
It is known that the maximal operator \(\sigma^*\) of the two-dimensional Fejér means \(\sigma_{n,m}f\) of the Walsh-Fourier series is bounded from the Hardy space \(H_p\) to \(L_p\) if \(p>1/2\); see F. Weisz [Anal. Math. 22, No. 3, 229–242 (1996; Zbl 0866.42020)].
In the paper under review it is proved that this is not true for \(p=1/2\), since there exists \(f\in H_{1/2}\) such that \(\sigma_{n,n}f\) is not uniformly bounded in weak \(L_{1/2}\).
For the entire collection see [Zbl 1234.00021].

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

Citations:

Zbl 0866.42020
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