Bounded Solutions of the Equation ?U = Pu on a Riemannian Manifold
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...
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| Language: | English |
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Institutional Knowledge at Singapore Management University
1974
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| Online Access: | https://ink.library.smu.edu.sg/soa_research/667 |
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| Institution: | Singapore Management University |
| Language: | English |
| Summary: | 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. |
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