The two-dimensional classical Hardy spaces H-p(TxT) on the bidisc are introduced and it is shown that the maximal operator of the Cesaro means of a distribution is bounded from H-p(TxT) to L-p(T-2) (3/4<p less than or equal to infinity) and is of weak type (H-1 not equal(TxT), L-1(T-2)) where the Hardy space H-1 not equal(TxT) is defined by the hybrid maximal function. As a consequence we obtain that the Cesaro means of a function integral is an element of H-1 not equal(TxT)superset of LlogL(T-2) converge a.e. to the Function in question. (C) 1997 Academic Press.
