Atomic decomposition of vector Hardy spaces

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