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It is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from thed-dimensional Hardy spaceH p(R×···×R) toL p(R d) (1/2<p<∞) and is of weak type (H 1 ♯i ,L 1) (i=1,…,d), where the Hardy spaceH 1 ♯i is defined by a hybrid maximal function. As a consequence, we obtain that the Fejér means of a functionfH 1 ♯iL(logL)d−1 converge a.e. to the function in question. Moreover, we prove that the Fejér means are uniformly bounded onH p(R×···×R) whenever 1/2<p<∞. Thus, in casefH p(R×···×R) the Fejér means converge tof inH p(R×···×R) norm. The same results are proved for the conjugate Fejér means, too.

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