Optimal l^pand holder estimates for the kohn solution of the ∂-equation on strongly pseudoconvex domains

Let Ω be an open, relatively compact subset in Cⁿ⁺¹, and assume the boundary of Ω, ∂Ω, is smooth and strongly pseudoconvex. Let Op(K) be an integral operator with mixed type homogeneities defined on Ω: i.e., K has the form as follows: where E_kis a homogeneous kernel of degree -K: in the Euclidean sense and H₁is homogeneous of degree -1 in the Heisenberg sense. In this paper, we study the optimal U and Holder estimates for the kernel K. We also use Lieb- Range's method to construct the integral kernel for the Kohn solution ∂N of the Cauchy-Riemann equation on the Siegel upper-half space and then apply our results to ∂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 fi.