Researchs On Classical Inequalities For Operators Of Differential Forms

Differential forms can be applied in many mathematical fields, such as partialdifferential equations, differential geometry, algebraic topology, mathematical physicsand general relativity theory. As an important class of nonlinear elliptic partialdifferential equations, theoretical results of A-harmonic equation play an practical andinstructive role in the natural sciences and engineering technology. The purpose of thispaper is to establish some norm estimates and Poincaréinequalities for A-harmonictensors acted by several classes of composite operators.Firstly, we give some concepts and related properties of the A-harmonic equationand nonhomogeneous A-harmonic equation of differential forms. Then we introduce thedefinitions and the related theoretical results of the relevant operators, including thehomotopy operatorT, the projection operator H, Green operator G, exterior differentialoperator d and the Laplace-Beltrami operator. We use the definitions andproperties of Orlicz space to study the local Poincaréinequality of the operator H thatoperate on the space L^p （logL）^α-space, and we discuss the simple A₁（Ω） weightcase that the operator operate on the tensor products of the field of L^p （logL）^α（B）.The third part gives two norm inequalities for nonhomogeneous A-harmonic tensors ofthe composite operator G οd οT and Hο dο T. Moreover, the related weighted casesare discussed. Then, we will prove a local Poincaré inequality for the operatorcomposited by the Sharp maximal operatorM_s^?, Green operator G and homotopyoperator T.Furthermore, the local Poincaréinequality forM_s^?Gο Twith the weightA₁（Ω） is obtained. Finally, we extend the locally weighted Poincaréinequality to theglobal case.