Integral Inequality Of Differential Form Under Some Composite Operators

Differential form is an essential tool in modern mathematics and applied science.Differential forms have been widely used in partial differential equations, differentialgeometry, algebraic topology, elasticity and physics and so on. A harmonic equationis a kind of nonlinear elliptic partial differential equation, and it is generalized by theLaplace equation that has important application value. In this paper, we will giveseveral norm estimates and Poincaré inequalities of differential form which satisfiesnonlinear A harmonic equations and be under some composite operators.In the first part, this paper will show related concepts, research background andcurrent situation of the A harmonic equations, differential form and Poincaréinequalities. Then in the second part it will introduce the norm estimates and localPoincaréinequalities for the composition M_s^#sοHο G of sharpmaximal operatorM_s^#、Green operator G and homotopy operator H. Then we consider the Ar weighted and A_r （λ,Ω）, two-weighted results, and generalize the above estimates tothe global situation, establishing the corresponding situationJohn to-John domain.In the third part of this paper, we will consider the compositionM_s^#sοHο H ofsharpmaximal operator, homotopy operator T and projection operator andobtain the norm estimates, Poincaré inequalities, A_r （Ω） weighted andA_r （λ,Ω） two-weighted results and the global situation on-John domain for A harmonictensors under. In the last part of the paper we will study the compositionM_s^#sοHο Tof sharpmaximal operator, homotopy operator and projectionoperator, correspondingly we get the norm estimates, Poincaréinequalities,A_r （Ω） weighted andA_r （λ,Ω） two-weighted results and the global situation on δ-Johndomain for harmonic tensors underM_s^#sοHο T.