Atomic decomposition of vector Hardy spaces

We study Banach-valued Hardy spaces h X p ( R + n + 1 ) of harmonic functions in the upper half space of R n + 1 defined in terms of maximal functions and the corresponding space of distributional boundary limits H X p ( R n ) , where X is an arbitrary real or complex Banach space. For p > 1 the elements of h X p ( R + n + 1 ) are the Poisson transform of Borel measures with p -bounded variation and values in X . For p ≤ 1 we prove the existence of atomic decomposition of elements in H X p ( R n ) where the atoms are vector measures with certain size and cancellation properties that generalize the atoms in the real valued Hardy spaces.