Norm Estimates Of A-harmonic Tensor And Potential Operator

Differential forms of A-harmonic equation is a special kind of nonlinear ellipticpartial differential equations with profound physics and mechanics background. Thecorresponding conclusions have been being applied in quasi-conformal mapping,elasitic theory and nonlinear potential theory. Many classes of operators play animportant role in PDEs researches. Especially in the study of weighted Sobolev spacesand Riesz potentials, it’s necessary to establish the norm estimates for fractional integraloperator, fractional maximal operator, and potential operator. In fact potential operatoris a sort of generous operator, when the kernel function takes some special functions orsatisfies certain conditions, it includes the usual fractional integral operator, Calderon-Zygmund operator and commutator. Hence studying potential operator is of greatsignificance in PDEs quantum mechanics and other fields. The main work of thedissertation can be summarized as follows:First of all, the paper introduces some relevant background about potential operatorand reviews the basic theory of the A harmonic equation and differential forms.In the second part, based on Pérez’s research result, when the kernels satisfy theweak growth condition, we establish the strong type weighted inequalities for potentialoperator on differential forms. Then obtain local strong type（p,q） inequality on anygiven balls when take the particular weight function that is the ball’ characteristicfunction.The main work of the next part is giving Caccioppoli inequality for potentialoperator P onA harmonic tensor. We also consider the Caccioppoli inequality foroperator Pwhich is an integral type operator with the domain of its integration in abounded open set. Then extend the conclusion to A_r（Ω）and A_r、λ（Ω）, weightedversions.Finally, on the basis of strong type （p,q）estimates for potential operator ondifferential forms, we establish Poincaré type inequality for potential operatorcombining with the decomposition of differential forms under the homotopy operator.Then, we give the comparison among the potential operator’s BMO norm, Lipschitznorm and L^pnorm.