Abstract. We will prove that the maximal Cesàro operator $\sigma_*^\delta$ is bounded from $H^p({{\msbm R}})$ to $L^p({{\msbm R}})$ when $\delta >\delta_p:=1/p-1$, $0< p\leq1$, while $\sigma_*^{\delta_p}$ maps $H^p({{\msbm R}})$ boundedly into {\sl weak}-$L^p({{\msbm R}})$ for $0< p< 1$. The weak type estimate is best possible in the sense that it cannot be strengthened to strong type. The results extend and strengthen those of [7], [11], and [1].

AMS Subject Classification (1991): 42A38, 42A08, 42B30

Keyword(s): Fourier transforms, Cesàro means, Hardy spaces

Received June 23, 1999. (Registered under 2755/2009.)