Abstract. In the one- and two-dimensional cases it has been proved (see [1], [2]) that the dyadic Cesàro operator is bounded on the spaces $L^p[0,1)$, $L^p([0,1) \times[0,1))$ $(1\le p< \infty )$ and on the dyadic Hardy spaces $H^1[0,1)$, $H^1([0,1)\times[0,1)$), but it is not bounded on the spaces VMO$[0,1)$, $L^\infty[0,1)$ and $L^\infty([0,1)\times[0,1)$). It is also proved that the continuous variant of the dyadic Cesàro operator is bounded on $L^p[0,\infty )$ (see [3]). In this paper we prove that the operator is bounded on the Hardy spaces $H^p[0,1)$ ($1/2< p\le1$) wich gives a new proof for boundedness of the Cesàro operator on the Hardy space $H^1[0,1)$.

AMS Subject Classification (1991): 42B30, 42C10, 42B05

Received November 6, 2000, and in revised form February 12, 2001. (Registered under 2837/2009.)