Abstract
The d-dimensional classical Hardy spaces Hp(T d) are introduced and it is shown that the maximal operator of the Riemann sums of a distribution is bounded from Hp(T d) to Lp(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.
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