Abstract. The classical Hardy spaces $H_p({\msbm R})$ are introduced and it is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from $H_p({\msbm R})$ to $L_p({\msbm R})$ $(1/2< p< \infty )$ and is of weak type (1,1). As a consequence we obtain that the Fejér means of a function $f \in L_1({\msbm R})$ converge a.e. to $f$. Moreover, we prove that the Fejér means are uniformly bounded on the spaces $H_p({\msbm R})$ whenever $1/2< p< \infty $. Thus, in case $f \in H_p({\msbm R})$, the Fejér means converge to $f$ in $H_p({\msbm R})$ norm $(1/2< p< \infty )$. The same results are proved for the conjugate Fejér means, too.

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

Keyword(s): Hardy spaces, p-atom, interpolation, Fourier transforms, Fejér means

Received September 10, 1997 and in revised form April 3, 1998. (Registered under 3305/2009.)