Remarks on Hardy spaces defined by non-smooth approximate identity

We study in this paper some relations between Hardy spaces H ϕ 1 which are defined by non-smooth approximate identity ϕ ( x ) , and the end-point Triebel–Lizorkin spaces F ˙ 1 0 , q ( 1 ⩽ q ⩽ ∞ ). First, we prove that H 1 ( R n ) ⊂ H ϕ 1 ( R n ) for compact ϕ which satisfies a slightly weaker condition than Fefferman and Steinʹs condition. Then we prove that non-trivial Hardy space H ϕ 1 ( R ) defined by approximate identity ϕ must contain Besov space B ˙ 1 0 , 1 ( R ) . Thirdly, we construct certain functions ϕ ( x ) ∈ B 1 0 , 1 ∩ Log 0 1 2 ( [ − 1 , 1 ] ) and a function b ( x ) ∈ ⋂ q > 1 F ˙ 1 0 , q such that Daubechies wavelet function ψ ∈ H ϕ 1 but b ϕ ⁎ ∉ L 1 .