Some Sobolev-type inequalities for twisted differential forms on real and complex manifolds
Fusheng Deng, Gang Huang, Xiangsen Qin
TL;DR
The paper develops Sobolev-type inequalities for twisted differential forms on real and complex manifolds by constructing integral representations from Green forms of the Hodge-Laplacian and proving sharp pointwise and gradient bounds under nonnegativity of the Weitzenböck curvature $\mathfrak{Ric}_p^E$. These uniform Green-form estimates enable $L^p$ and $L^{q,p}$ bounds for the operators $d$ and $\bar{\partial}$, respectively, and lead to applications such as improved $L^2$-Hörmander estimates on strictly pseudoconvex domains in Kähler manifolds. The main technical contributions include precise bounds for $|G_p(x,y)|$ (and its derivatives) and integral representations that express forms in terms of $\Delta_p f$ and boundary data, yielding Sobolev-type inequalities on manifolds with boundary. Together, these results provide a unified framework for analytic inequalities on differential forms in both the real and complex settings and have potential applications to geometric analysis and several complex variables.
Abstract
We prove certain $L^p$ Sobolev-type inequalities for twisted differential forms on real (and complex) manifolds for the Laplace operator $Δ$, the differential operators $d$ and $d^*$, and the operator $\bar\partial$. 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 $\bar\partial$. Furthermore, we get an improved $L^2$-estimate of Hörmander on a strictly pseudoconvex open subset of a Kähler manifold.