Abstract
Let Q be an open, relatively compact subset in Cn+l , and assume the boundary of Q, aQ, is smooth and strongly pseudoconvex. Let Op(K) be an integral operator with mixed type homogeneities defined on Q: i.e., K has the form as follows: EL Ek Hl, k,l>O where Ek is a homogeneous kernel of degree -k in the Euclidean sense and H1 is homogeneous of degree -1 in the Heisenberg sense. In this paper, we study the optimal LIP and H6lder estimates for the kernel K . We also use LiebRange's method to construct the integral kernel for the Kohn solution 0*N of the Cauchy-Riemann equation on the Siegel upper-half space and then apply our results to o*N. On the other hand, we prove Lieb-Range's kernel gains 1 in good directions (hence gains 1/2 in all directions) via Phong-Stein's theory. We also discuss the transferred kernel from the Siegel upper-half space to Q.
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