The maximal Riesz operator of two-dimensional Fourier transforms and Fourier series on Hp(R x R) and Hp(T x T)

It is proved that the maximal operator of the two-parameter Riesz means wit h parameters alpha, beta less than or equal to 1 is bounded from L-p(R-2) t o L-p(R-2) (1 < p < infinity). The two-dimensional classical Hardy spaces H -p(R x R) are introduced and it is shown that the maximal Riesz operator of a tempered distribution is also bounded from H-p (R x R) to L-p(R-2) (max{ 1/(alpha + 1), 1/(beta + 1)} < p less than or equal to infinity) and is of weak type i (H-1(#)(R x R), L-1(R-2)) where the Hardy space H-1(#)(R x R) i s defined by the hybrid maximal function. As a consequence we obtain that t he Riesz means of a function f epsilon H-1(#)(R x R) superset of L log L(R- 2) converge a.e. to the function in question. Moreover, wt prove that the R iesz means are uniformly bounded on the spaces H-p(R x R) whenever max { 1/ (x + 1), 1/(beta + 1)} < p < infinity. Thus, in case f epsilon H-p(R x R), the Riesz means converge to f in H-p(R x R) norm. The same results are prov ed for the conjugate: Riesz means and Tor two-parameter Fourier series, too . (C) 2000 Academic Press.