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

Xuexin Li,Yuming Xing,Yong Wang

doi:10.1186/s13660-015-0896-9

Xuexin Li, Yuming Xing + Show 1 more

Open Access

https://doi.org/10.1186/s13660-015-0896-9

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Abstract

In this paper, we obtain Poincare-type inequalities for the composite operator acting on differential forms and establish the $L^{p}$ , Lipschitz, and BMO norm estimates. We also give the weighted versions of the comparison theorems for the $L^{p}$ , Lipschitz, and BMO norms.

Highlights

Differential forms are a generalization of the traditional functions
Differential forms have been widely used in physics systems, differential geometry, and PDEs
Operator theory plays a critical role in investigating the properties of the solutions to partial differential equations

Summary

Introduction

Differential forms are a generalization of the traditional functions. In recent years, differential forms have been widely used in physics systems, differential geometry, and PDEs. M for differential forms u, where φ ∈ C ∞(M) is normalized by M φ(y) dy = , and Ky is the liner operator defined by. In regard to Green’s operator, we need the following results in [ ]: dd∗G(u) s,B + d∗dG(u) s,B + dG(u) s,B + d∗G(u) s,B + G(u) s,B ≤ C(s) u s,B, d∗G(u) s,B = Gd∗(u) s,B, and dG(u) s,B = Gd(u) s,B for any differential form u in M and < s < ∞. If u(x) ∈ D (M, l) satisfies φ(|u|) ∈ L (M, μ) and for any constant c, μ x ∈ M : |u – c| > > , where μ is the Radon measure defined by dμ(x) = ω(x) dx with weight ω(x), for any a > , we have φ a|u| dμ ≤ C φ a|u – uM| dμ, where C is a constant independent of u. Ms DG(u) ∗,M ≤ C Ms DG(u) loc Lipk,M, where k is a constant with ≤ k ≤ , and C is a constant independent of u

Proof From the definition of the BMO norms we obtain

It follows that