Remarks on Hardy spaces defined by non-smooth approximate identity

We study in this paper some relations between Hardy spaces H(phi)(1) which are defined by non-smooth approximate identity 0(x), and the end-point Triebel-Lizorkin spaces (F)over dot(1)(0.q) (1 <= q <= infinity). First, we prove that H(1)(R(n)) subset of H(phi)(1)(R(n)) for compact phi which satisfies a slightly weaker condition than Fefferman and Stein's condition. Then we prove that non-trivial Hardy space H(phi)(1)(R) defined by approximate identity phi must contain Besov space (B) over dot(1)(0.1)(R). Thirdly, we construct certain functions phi(x) is an element of B(1)(0.1) boolean AND Log(0)(1/2)([-1, 1]) and a function b(x) is an element of boolean AND(q>1) (F) over dot(1)(0.q) such that Daubechies wavelet function psi is an element of H(phi)(1) but b(phi)(*) is not an element of L(1). (C) 2010 Elsevier Inc. All rights reserved.