It is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from thed-dimensional Hardy spaceHp(R×···×R) toLp(Rd) (1/2<p<∞) and is of weak type (H1♯i
,L1) (i=1,…,d), where the Hardy spaceH1♯i
is defined by a hybrid maximal function. As a consequence, we obtain that the Fejér means of a functionf ∈H1♯i
⊃L(logL)d−1 converge a.e. to the function in question. Moreover, we prove that the Fejér means are uniformly bounded onHp(R×···×R) whenever 1/2<p<∞. Thus, in casef ∈Hp(R×···×R) the Fejér means converge tof inHp(R×···×R) norm. The same results are proved for the conjugate Fejér means, too.