Yuming Xing

A new weight class and Poincaré inequalities with the Radon measure

We first introduce and study a new family of weights, the A(α, β, γ; E)-class which contains the well-known Ar(E)-weight as a proper subset. Then, as applications of the A(α, β, γ ;E)-class, we prove the local and global Poincaré inequalities with the Radon measure for the solutions of the non-homogeneous A-harmonic equation which belongs to a kind of the nonlinear partial differential equations.2000 Mathematics Subject Classification: Primary 26D10; Secondary 35J60; 31B05; 58A10; 46E35.

Poincaré Inequalities with Luxemburg Norms in -Averaging Domains

We prove both local and global Poincaré inequalities with Luxemburg norms for differential forms in -averaging domains, which can be considered as generalizations of the existing versions of Poincaré inequalities.

Weighted Poincaré-type estimates for conjugate-harmonic tensors

We prove Poincaré-type estimates involving the Hodge codifferential operator and Greens operator acting on conjugate-harmonic tensors.

Some Caccioppoli Estimates for Differential Forms

We prove the global Caccioppoli estimate for the solution to the nonhomogeneous -harmonic equation , which is the generalization of the quasilinear equation . We will also give some examples to see that not all properties of functions may be deduced to differential forms.

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 Comparison Estimates for the Composite Operator

This paper obtains the Lipschitz and BMO norm estimates for the composite operator applied to differential forms. Here, is the Hardy-Littlewood maximal operator, and is the potential operator. As applications, we obtain the norm estimates for the Jacobian subdeterminant and the generalized solution of the quasilinear elliptic equation.

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.

Lipschitz and BMO norm inequalities for the composite operator on differential forms

In this paper, we obtain Poincaré-type inequalities for the composite operator acting on differential forms and establish the Lp

Imbedding inequalities with Lφ-norms for composite operators

L^{p}

BMO and Lipschitz norm estimates for the composition of Green's operator and the potential operator

, Lipschitz, and BMO norm estimates. We also give the weighted versions of the comparison theorems for the Lp