Abstract. We consider the Riemann means of double Fourier series of functions belonging to one of the Hardy spaces $H^{(1,0)} ({\msbm T}^2)$, $H^{(0,1)} ({\msbm T}^2)$, or $H^{(1,1)} ({\msbm T}^2)$, where ${\msbm T}^2 := [-\pi, \pi ) \times[-\pi, \pi )$ is the two-dimensional torus. We prove that the maximal Riemann operator is bounded from $H^{(1,0)} ({\msbm T}^2)$ or $H^{(0,1)} ({\msbm T}^2)$ into weak-$L^1({\msbm T}^2)$, and conjecture that it is bounded from $H^{(1,1)}({\msbm T}^2)$ into $L^1({\msbm T}^2)$. As corollaries, we obtain analogous results on the maximal conjugate Riemann operators, as well as we deduce the pointwise convergence of both the Riemann and the conjugate Riemann means almost everywhere.

AMS Subject Classification (1991): 47B38, 42A50, 42B08

Received March 24, 1997 and in revised form December 3, 1997. (Registered under 2663/2009.)