Remarks on Hardy spaces defined by non-smooth approximate identity

We study in this paper some relations between Hardy spaces Hϕ1 which are defined by non-smooth approximate identity ϕ(x), and the end-point Triebel-Lizorkin spaces F˙10,q (1⩽q⩽∞). First, we prove that H1(Rn)⊂Hϕ1(Rn) for compact ϕ which satisfies a slightly weaker condition than Fefferman and Stein's condition. Then we prove that non-trivial Hardy space Hϕ1(R) defined by approximate identity ϕ must contain Besov space B˙10,1(R). Thirdly, we construct certain functions ϕ(x)∈B10,1∩Log012([−1,1]) and a function b(x)∈⋂q>1F˙10,q such that Daubechies wavelet function ψ∈Hϕ1 but bϕ⁎∉L1.