Abstract. We prove that the maximal Fejér operator is bounded from the (real) Hardy space $H^1({\msbm R})$ into $L^1({\msbm R})$, and is also bounded from $L^1({\msbm R})$ into {\it weak}-$L^1({\msbm R})$. We introduce the (hybrid) Hardy spaces $H^{(1,0)} ({\msbm R}^2)$, $H^{(0,1)} ({\msbm R}^2)$, and $H^{(1,1)} ({\msbm R}^2)$. We prove that the maximal Fejér operator is bounded from $H^{(1,1)} ({\msbm R}^2)$ into $L^1({\msbm R}^2)$, and is also bounded from $H^{(1,0)} ({\msbm R}^2)$ or $H^{(0,1)}({\msbm R}^2)$ into {\it{\rm weak}}-$L^1({\msbm R}^2)$. We establish analogous results for the maximal conjugate Fejér operators, too.

AMS Subject Classification (1991): 47B28

Received February 29, 1996. (Registered under 6134/2009.)