Boundedness of the Cesàro operator in Hardy spaces (Q1882642)

It is proved in the paper that the Cesàro operator \(C_\alpha\) is bounded in the (real) Hardy space \(H^p\) for every \(0<p\leq1\) and every \(\alpha>0.\) This is a full generalization of an earlier \textit{Y. Kanjin's} result in [Studia Math. 148, No. 1, 37--45 (2001; Zbl 1001.47018)], restricted to \(2/(2\alpha+1)<p\leq1.\) The modified atomic decomposition of \(H^p\) due to the author (see [Pac. J. Math. 115, 165--175 (1984; Zbl 0503.46036)]) is applied to prove the result.