Ces`aro Summability of Multi-dimensional Trigonometric-Fourier Series

The d-dimensional classical Hardy spaces Hp Td . are introduced and it is shown that the maximal operator of the Ces`aro means of a distribution is bounded from Hp Td.to Lp Td. 2 dq1.r 2 dq2.-pF`.and is of weak type 1, 1.provided that the supremum in the maximal operator is taken over a positive cone. As a consequence we obtain the summability result due to Marcinkievicz and Zygmund, more exactly, the Ces`aro means of a function fgL1 Td . converge a.e. to the function in question, provided again that the limit is taken over a positive cone. Similar results for the C, b.summability are also formulated