Bounded Solutions of the Equation ?U = Pu on a Riemannian Manifold

• Main Author: Kwon, Young Koan
• Format: text
• Language: English
• Published: Institutional Knowledge at Singapore Management University 1974
• Subjects: Accounting
• Online Access: https://ink.library.smu.edu.sg/soa_research/667
• Institution: Singapore Management University

Given a nonnegative C1-function p(x) on a Riemannian manifold R, denote by Bp(R) the Banach space of all bounded C2-solutions of Î u = pu with the sup-norm. The purpose of this paper is to give a unified treatment of Bp(R) on the Wiener compactification for all densities p(x). This approach not only generalizes classical results in the harmonic case $(p \equiv 0)$ , but it also enables one, for example, to easily compare the Banach space structure of the spaces Bp(R) for various densities p(x). Typically, let β(p) be the set of all p-potential nondensity points in the Wiener harmonic boundary Î, and Cp(Î) the space of bounded continuous functions f on Î with $f\mid\Delta, \beta(p) \equiv 0$ . Theorem. The spaces Bp(R) and Cp(Î) are isometrically isomorphic with respect to the sup-norm.