Discrete Littlewood-Paley-Stein Characterization and L² Atomic Decomposition of Local Hardy Spaces

Usually, the condition that T is bounded on L²(ℝⁿ) is assumed to prove the boundedness of an operator T on a Hardy space. With this assumption, one only needs to prove the uniformly boundness of T on atoms, since T(f)= ∑_i λ_iT(a_i), provided that f = ∑_i λ_ia_i in L² (ℝⁿ), where a_i is an L² atom of this Hardy space. So far, the L² atomic decomposition of local Hardy spaces h^p(ℝⁿ), 0 > p ≤ 1, hasn't been established. In this paper, we will solve this problem, and also show that h^p(ℝⁿ) can also be characterized by discrete Littlewood-Paley functions.