We prove several Sobolev-type inequalities related to the \({\bar \partial}\) -operator on bounded domains in ℂn, 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 ℝn. As applications to complex analysis, we get an integral form of maximum modulus principle for holomorphic functions, and an improvement of Hörmander’s L2-estimate for \({\bar \partial}\) on bounded strictly pseudoconvex domains.
