Boundedness of commutators on Hardy type spaces

Let [b, T] be the commutator of the function b is an element of Lip(beta) (R-n) (0 < beta less than or equal to 1) and the Calderon-Zygmund singular integral operator T. The authors study the boundedness properties of [b, T] on the classical Hardy spaces and the Herz-type Hardy spaces in non-extreme cases. For the boundedness of these commutators in extreme cases, some characterizations are also given. Moreover, the authors prove that these commutators are bounded from Hardy type spaces to the weak Lebesgue or Herz spaces in extreme cases.