On compactness of the $\bar{\mathrm{\partial }}$-Neumann operator on Hartogs domains

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