Generalized Calderón–Zygmund operators on homogeneous groups and applications

Abstract

There are two folds of this article. The first part is concentrating on estimates for generalized Calderón–Zygmund operators acting on Hardy spaces H ^p (G). Here G is a simply connected homogeneous Lie group. We also obtained estimates on the spaces L ^∞(G) and BMO ^p (G). The second part of this article is applications of results from the first part to the -Neumann problem on bounded, smoothly pseudoconvex domains in C ⁿ⁺¹. We obtain H ^p estimates for the Calderón operator when G = H _n , the n-dimensional Heisenberg group.

Keywords:

• Calderón–Zygmund operator
• homogeneous group
• Hardy spaces
• BMO
• \bar \partial -Neumann problem
• Heisenberg group
• Calderón operator
• pseudo-differential operators of Poisson type

2000 Mathematics Subject Classifications:

• Primary: 42B35
• Secondary: 46E30
• 47B38

Acknowledgements

D.-C. Chang is partially supported by a research grant from United States Army Research Office and a Hong Kong RGC competitive earmarked research grant #600607. M.-Y. Lee is partially supported by a research grant from NSC of ROC.