Norm Estimates Of The A-Harmonic Tensors And Related Operators

Differential forms that have been researched deeply have been widely used in many areas, such as physics, general relativity and differential geometry. The harmonic equations that be very important the nonlinear elliptic partial differential equations have been studied deeply and received many important results recently. These results about the A-A-harmonic equations have important theoretical meaning and reality significance in many branches of natural sciences and engineering technology. Many theories about various of operators have been applied in partial differential equation and geometric analysis. We focus our attention on the Poincaré-type inequalities and norm estimate inequalitys for the composition of the different operators acted on the A-harmonic tensors in this paper.Above all, we briefly introduce the A-harmonic equations and differential forms, recall the research status of the Poincaréinequality. In the second part, we establish the norm estimates of the composition of and G operators acted on the nonhomogeneous harmonic tensors, and generalize it to the weighted and two-weighted results, after that we extend local results to the global results by the Whitney cover Lemma. Next part, we give two effective operators in mathematic field: Green's operator and projection operator , prove the Poincaré-type inequality for the composition of and acted on the nonhomogeneous A-GHHGA-harmonic tensors. Finally, we give the definitions of the BMO norm and Lipschitz norm, then explore the norm comparison inequalities of the composition operater and G acted on smooth differential forms satisfied nonhomogeneous HA-harmonic equation on a bounded domain, then extend to the two- weighted inequality. In the last part, we study the Poincaré-type inequality for the composition of #s and T, and prove rA- weighted and ,rA- two- weighted results respectively, then generalize the local estimate to -John domain.