J. Aust. Math. Soc.  79 (2005), 11-24
The higher order commutators of the fractional integrals on Hardy spaces

Shunchao Long
  Mathematics Department
  Xiangtan University
  Xiangtan, 411105
  P. R. China
  sclong@xtu.edu.cn
and
Jian Wang
  Mathematics Department
  Xiangtan University
  Xiangtan, 411105
  P. R. China
  jwang@hnedu.com


Abstract
In this paper we investigate the boundedness on Hardy spaces for the higher order commutator $T^\tau _{b,m}$ generated by the BMO function b and fractional integral type operator $T^\tau $, and establish the boundedness theorems for $T^\tau _{b,m}$ from $H^{p_1,q_1,s} _{b,m}$ to $L^{p_2}$ and to $H^{p_2}$ $(0<p_1 \leq 1)$, and from $H\dot{K}^{\alpha,p_1,s} _{q_1,b,m}$ to $\dot{K}^{\alpha,p_2} _{q_2}$ and to $H\dot{K}^{\alpha,p_2} _{q_2}$, respectively, for certain ranges of $\alpha$, $p_1$, $q_1$, $p_2$, $q_2$ and $s$.
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