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
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
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
