On compactness of the ∂̄-Neumann operator on Hartogs domains

ABSTRACT

We show that Property $\left(P\right)$ of $\mathrm{\partial \Omega }$, compactness of the $\overline{\mathrm{\partial }}$-Neumann operators $N_1$, and compactness of Hankel operator on a smooth bounded pseudoconvex complete Hartogs domain $\mathrm{\Omega }=\left\{\right(z,w_1,w_2,\dots ,w_n)\in {\mathbb{C}}^{n+1}\mid \sum _{k=1}^n|w_k{|}^2<e^{-2\varphi \left(z\right)},z\in D\}$ are equivalent, where D is a smooth bounded connected open set in $\mathbb{C}$.

COMMUNICATED BY:

• H. Boas

KEYWORDS:

• Compactness
• $\overline{\mathrm{\partial }}$-Neumann operator
• property $\left(P_q\right)$
• Hankel operator
• Hartogs domain

1991 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 32W05
• 32W10
• 47B35

Acknowledgments

The author would like to express his gratitude to his advisor, Professor Yuan Yuan, for his guidance and encouragement. The author would like to thank Professor Siqi Fu, Professor Sönmez Şahutoğlu and Professor Yunus Zeytuncu for their useful comments. The author is very grateful to the referees for their very useful suggestions that improve the exposition of the paper tremendously.

Disclosure statement

No potential conflict of interest was reported by the author.

Additional information

Funding

The author is supported in part by National Science Foundation grant DMS-1412384.