The d-dimensional classical Hardy spaces Hp(Td) are introduced and it is shown that the maximal operator of the Cesàro means of a distribution is bounded from Hp(Td) to Lp (Td) ((2d + 1)/(2d + 2) < p ≤ ° ∞) 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 Cesàro means of a function f ∈ L1(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, β) summability are also formulated.
ASJC Scopus subject areas
- Applied Mathematics