Electronic Journal of Differential Equations,
Vol. 2019 (2019), No. 48, pp. 1-22.

Title: Compactness of the canonical solution operator on
       Lipschitz q-pseudoconvex boundaries

Author: Sayed Saber (Beni-Suef University, Egypt)
 
Abstract:
 Let $\Omega\subset\mathbb{C}^n$ be a bounded Lipschitz q-pseudoconvex
 domain that  admit good weight functions. We shall prove that the canonical
 solution operator for the $\overline{\partial}$-equation is compact on
 the boundary of $\Omega$  and is bounded in the Sobolev space
 $W^k_{r,s}(\Omega)$ for some values of $k$. Moreover, we show that
 the Bergman projection and the $\overline\partial$-Neumann operator
 are bounded in the Sobolev space $W^k_{r,s}(\Omega)$
 for some values of k.
 If $\Omega$ is smooth, we shall give sufficient conditions for compactness
 of the $\overline\partial$-Neumann operator.

Submitted May 8, 2018. Published April 10, 2019.
Math Subject Classifications: 35J20, 35J25, 35J60.
Key Words: Lipschitz domain; q-pseudoconvex domain.