The maximal operator of the (C, \(\alpha\)) means of the Walsh-Fourier series. (English) Zbl 1121.42020

F. Weisz [Anal. Math. 27, 141–156 (2001; Zbl 0992.42016)] proved the boundedness of the maximal operator \(\sigma_*^{\alpha}\) for \((C,\alpha)\) summability of Walsh - Fourier series in the Hardy - Lorentz spaces \(H_{p,q}\) of dyadic martingales on the unit interval for \(0<\alpha<1,1/(\alpha+1)<p<\infty\) and \(0<q\leq\infty.\) In the same paper he guessed that this assertion is not true for \(p\leq 1/(\alpha+1).\) Here the author proves that hypothesis for the case \(q=p.\)


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


Zbl 0992.42016