We prove several Sobolev-type 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 Sobolev inequality and its various generalizations, and apply them to derive a generalization of the 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.
Categories: math.CV