Integral Estimates for the Laplace-Beltrami and Green's Operators Applied to Differential Forms on Manifolds

Abstract

We obtain A_r(M)-weighted boundedness for compositions of Green's operator and the Laplace-Beltrami operator applied to differential forms on manifolds. As applications, we also prove A_r(M)-weighted Sobolev-Poincare embedding theorems for Green's operator and norm comparison theorems for solutions of the A-harmonic equation on manifolds. These results can be used in developing the Lp theory of differential forms and the Hodge decomposition.