Optimal $L^ p$ and Hölder estimates for the Kohn solution of the $\overline \partial$-equation on strongly pseudoconvex domains

Abstract:

Let $\Omega$ be an open, relatively compact subset in ${{\mathbf {C}}^{n + 1}}$, and assume the boundary of $\Omega$, $\partial \Omega$, is smooth and strongly pseudoconvex. Let $\operatorname {Op}(K)$ be an integral operator with mixed type homogeneities defined on $\overline \Omega$: i.e., $K$ has the form as follows: \[ \sum \limits _{k,l \geq 0} {{E_k}{H_l},} \] where ${E_k}$ is a homogeneous kernel of degree $- k$ in the Euclidean sense and ${H_l}$ is homogeneous of degree $- l$ in the Heisenberg sense. In this paper, we study the optimal ${L^p}$ and Hölder estimates for the kernel $K$. We also use Lieb-Range’s method to construct the integral kernel for the Kohn solution $\overline {{\partial ^\ast }} {\mathbf {N}}$ of the Cauchy-Riemann equation on the Siegel upper-half space and then apply our results to $\overline {{\partial ^\ast }} {\mathbf {N}}$. 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 $\Omega$.