Shusen Ding

Inequalities for differential forms

Hardy-Littlewood Inequalities.- Norm Comparison Theorems.- Poincare-type inequalities.- Caccioppoli Inequalities.- Imbedding Theorems.- Reverse Holder Inequaltiies.- Inequalities for Operators.- Estimates for Jacobians.- Lipschitz and BMO norms.- References.- Index

Ls(μ)-averaging domains

Abstract In this paper we first introduce L s ( μ )-averaging domains which are generalizations of L s -averaging domains introduced by S.G. Staples. We characterize L s ( μ )-averaging domains using the quasihyperbolic metric. As applications, we prove norm inequalities for conjugate A -harmonic tensors in L s ( μ )-averaging domains which can be considered as generalizations of the Hardy and Littlewood theorem for conjugate harmonic functions. Finally, we give applications to quasiconformal and quasiregular mappings.

Reverse Hölder inequalities

In this chapter, we will present various versions of the reverse Holder inequality which serve as powerful tools in mathematical analysis. The original study of the reverse Holder inequality can be traced back in Muckenhoupt–s work in [145]. During recent years, different versions of the reverse Holder inequality have been established for different classes of functions, such as eigenfunctions of linear second-order elliptic operators [281], functions with discrete-time variable [282], and continuous exponential martingales [119].

Inequalities for Green's operator applied to the minimizers

In this paper, we prove both the local and global Lφ -norm inequalities for Greens operator applied to minimizers for functionals defined on differential forms in Lφ -averaging domains. Our results are extensions of Lp norm inequalities for Greens operator and can be used to estimate the norms of other operators applied to differential forms.2000 Mathematics Subject Classification: Primary: 35J60; Secondary 31B05, 58A10, 46E35.

Global Caccioppoli-Type and Poincaré Inequalities with Orlicz Norms

We obtain global weighted Caccioppoli-type and Poincaré inequalities in terms of Orlicz norms for solutions to the nonhomogeneous -harmonic equation .

Norm Comparison Inequalities for the Composite Operator

We establish norm comparison inequalities with the Lipschitz norm and the BMO norm for the composition of the homotopy operator and the projection operator applied to differential forms satisfying the A-harmonic equation. Based on these results, we obtain the two-weight estimates for Lipschitz and BMO norms of the composite operator in terms of the -norm.

Norm inequalities for composition of the Dirac and Green's operators

We first prove a norm inequality for the composition of the Dirac operator and Green’s operator. Then, we estimate for the Lipschitz and BMO norms of the composite operator in terms of the Ls norm of a differential form.MSC:26B10, 30C65, 31B10, 46E35.

Inequalities for the composition of Green’s operator and the potential operator

We first establish the Lp-norm inequalities for the composition of Green’s operator and the potential operator. Then we develop the Lφ-norm inequalities for the composition in the Lφ-averaging domains. Finally, we display some examples for applications.MSC:35J60, 31B05, 58A10, 46E35.

Singular integrals of the compositions of Laplace-Beltrami and Green's operators

We establish the Poincaré-type inequalities for the composition of the Laplace-Beltrami operator and the Greens operator applied to the solutions of the non-homogeneous A-harmonic equation in the John domain. We also obtain some estimates for the integrals of the composite operator with a singular density.

Orlicz norm inequalities for the composite operator and applications

In this article, we first prove Orlicz norm inequalities for the composition of the homotopy operator and the projection operator acting on solutions of the nonhomogeneous A-harmonic equation. Then we develop these estimates to Lφ(µ)-averaging domains. Finally, we give some specific examples of Young functions and apply them to the norm inequality for the composite operator.2000 Mathematics Subject Classification: Primary 26B10; Secondary 30C65, 31B10, 46E35.