Electron. J. Differential Equations, Vol. 2019 (2019), No. 48, pp. 1-22.

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

Sayed Saber

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.

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Sayed Saber
Mathematics Department
Faculty of Science
Beni-Suef University, Egypt
email: sayedkay@yahoo.com

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