  We prove several inequalities related to the $\bar\partial$-operator on
bounded domains in $\mathbb{C}^n$, which can be viewed as a
$\bar\partial$-version of the classical Poincar\'e inequality and its various
generalizations, and apply them to derive a generalization of Sobolev
Inequality with Trace in $\mathbb{R}^n$. As applications to complex analysis,
we get an integral form of Maximum Modulus Principle for holomorphic functions,
and an improvement of H\"ormander's $L^2$-estimate for $\bar\partial$ on
bounded strictly pseudoconvex domains.
