Some Sobolev-type inequalities for twisted differential forms on real and complex manifolds

We prove certain $L^p$ Sobolev-type inequalities for twisted differential forms on real (and complex) manifolds for the Laplace operator $\mathrm{\Delta }$, the differential operators $d$ and $d^{\ast }$, and the operator $\bar{\mathrm{\partial }}$. A key tool to get such inequalities are integral representations for twisted differential forms. The proofs of the main results involves certain uniform estimate for the Green forms and their differentials and codifferentials, which are also established in the present work. As applications of the uniform estimates, using Hodge theory, we can get an $L^{q,p}$-estimate for the operator $d$ or $\bar{\mathrm{\partial }}$. Furthermore, we get an improved $L^2$-estimate of Hörmander on a strictly pseudoconvex open subset of a Kähler manifold.