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Let $\Omega\subset\mathbb{C}^m$ be a bounded pseudoconvex domain with smoothboundary. For each $k\in\mathbb{N}$, we give a sufficient condition to estimatethe $\bar\partial$-Neumann operator in the Sobolev space $W^k(\Omega)$. The keyfeature of our results is a precise formula for $k$ in terms of the geometry ofthe boundary of $\Omega$.

The $\bar{\partial}$-Neumann operator with the Sobolev norm of integer orders