Abstract. We consider the Riemann means of Fourier series of functions belonging to the Hardy space $H^1({\msbm T})$ or $L^1({\msbm T})$, respectively. We prove that the maximal Riemann operator as well as the maximal conjugate Riemann operator are bounded from $H^1({\msbm T})$ into $L^1({\msbm T})$. It is also true that this operator is bounded from $L^1({\msbm T})$ into weak-$L^1({\msbm T})$. On closing, we formulate a conjecture about the maximal conjugate Riemann operator of a function in $L^1({\msbm T})$.

AMS Subject Classification (1991): 47B38

Received February 29, 1996 and in revised form July 11, 1996. (Registered under 6135/2009.)