A note on the boundedness of Calderón–Zygmund operators on Hardy spaces

It was well known that Calderón–Zygmund operators T are bounded on Hp for n n+ < p 1 provided T ∗(1) = 0. A new Hardy space H p b , where b is a para-accretive function, was introduced in [Y. Han, M. Lee, C. Lin, Hardy spaces and the Tb-theorem, J. Geom. Anal. 14 (2004) 291–318] and the authors proved that Calderón–Zygmund operators T are bounded from the classical Hardy space Hp to the new Hardy space H p b if T ∗(b) = 0. In this note, we give a simple and direct proof of the Hp − H p b boundedness of Calderón–Zygmund operators via the vector-valued singular integral operator theory.  2005 Elsevier Inc. All rights reserved.