A class of singular integrals on the n-complex unit sphere

摘要

The operators on the n-complex unit sphere under study have three forms: the singular integrals with holomorphic kernels, the bounded and holomorphic Fourier multipliers, and the Cauchy-Dundored bounded and holomoprhic functional calculus of the radial Dirac operator D=∑ ~n_k=1 zа:аz_k. The equivalence between the three forms and the strong-type (p,p), 1<∞, and weak-type (1,1)-boundedness of the operators is proved. The results generalize the work of L. K. Hua, A. Koranyli and S. Vagi, W. Rudin and S. Gong on the Cauchy-Szego kernel and the Cauchy singular integral operator.