Rough singular integral operators on Hardy-Sobolev spaces

The authors study the singular integral operator defined on all test functions f, where b is a bounded function, a>0, Θ (y′) is an integrable function on the unit sphere S ⁿ⁻¹ satisfying certain cancellation conditions. It is proved that, for n/(n+a)<p<∞, TΩ,α is a bounded operator from the Hardy-Sobolev space H _a ^p to the Hardy space H ^p. The results and its applications improve some theorems in a previous paper of the author and they are extensions of the main theorems in Wheeden’s paper (1969). The proof is based on a new atomic decomposition of the space H _a ^p by Han, Paluszynski and Weiss(1995). By using the same proof, the singular integral operators with variable kernels are also studied. Supported by 973 project (G1999075105), NSFZJ (RC97017) and RFDP (20030335019).