Riemann summability of multi-dimensional trigonometric-fourier series

The d-dimensional classical Hardy spaces H_p(T ^d) are introduced and it is shown that the maximal operator of the Riemann sums of a distribution is bounded from H_p(T ^d) to L_p(T ²) (d/(d+1)<p≤∞) and is of weak type (1,1) provided that the supremum in the maximal operator is taken over a positive cone. The same is proved for the conjugate Riemann sums. As a consequence we obtain that every function f∈L₁(T ^d) is a. e. Riemann summable to f, provided again that the limit is taken over a positive cone. This research was partly supported by the Hungarian Scientific Research Funds (OTKA) No F019633.