$\bar{\mathrm{\partial }}$ Sobolev-type inequality and an improved $L^2$-estimate of $\bar{\mathrm{\partial }}$ on bounded strictly pseudoconvex domains

We prove several Sobolev-type inequalities related to the $\bar{\mathrm{\partial }}$-operator on bounded domains in ${\mathbb{C}}^n$, which can be viewed as a $\bar{\mathrm{\partial }}$-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 Hörmander’s $L^2$-estimate for $\bar{\mathrm{\partial }}$ on bounded strictly pseudoconvex domains.