Uniform estimates of Green functions and Sobolev-type inequalities on real and complex manifolds

TL;DR

This paper establishes uniform estimates for Green functions and their gradients on real and complex manifolds, leading to Sobolev and Poincaré inequalities involving key differential operators.

Contribution

It provides new uniform estimates for Green functions on manifolds, enabling the derivation of Sobolev-type inequalities for various differential operators.

Findings

Uniform Green function estimates on manifolds

Sobolev and Poincaré inequalities derived

Integral representations are central to proofs

Abstract

We prove certain $L^p$ Sobolev-type and Poincar\'e-type inequalities for functions on real and complex manifolds for the gradient operator $\nabla$, the Laplace operator $\Delta$, and the operator $\bar\partial$. Integral representations for functions are key to get such inequalities. The proofs of the main results involves certain uniform estimates for the Green functions and their gradients on Riemannian manifolds, which are also established in the present work.