Weak type inequalityy for the maximal operator of the \((C,\alpha)\) means of two-dimensional Walsh-Fourier series
It is known that the maximal operator \(\sigma_*^1\) of the Fejér means of the cubical partial sums \(S_{j,j}f\) of the two-dimensional Walsh-Fourier series of \(f\) is bounded from \(H_p\) to \(L_p\) if \(2/3<p<\infty\) (see \textit{F. Weisz} [Approximation Theory Appl. 17, No. ~2, 32--44 (2001; Zbl 0992.42013)]). Moreover, \textit{U. Goginava} [Anal. Math. 33, No. ~4, 263--286 (2007)] proved that the maximal Cesàro operator \(\sigma_*^\alpha\) \((0<\alpha<1)\) is bounded from \(H_p\) to \(L_p\) when \(2/(2+\alpha)<p<\infty\) and it is not bounded when \(0<p\leq 2/(2+\alpha)\). In this paper it is shown for the critical index \(p=2/(2+\alpha)\) that \(\sigma_*^\alpha\) is bounded from \(H_p\) to the weak \(L_p\) space.
scientific article; zbMATH DE number 5931409