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Bounded Solutions of the Equation Δ u = pu on a Riemannian Manifold

Young K. Kwon
Proceedings of the American Mathematical Society
Vol. 45, No. 3 (Sep., 1974), pp. 377-382
DOI: 10.2307/2039961
Stable URL: http://www.jstor.org/stable/2039961
Page Count: 6
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Bounded Solutions of the Equation Δ u = pu on a Riemannian Manifold
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Abstract

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 Δ 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 Δ, and Cp(Δ) the space of bounded continuous functions f on Δ with $f\mid\Delta - \beta(p) \equiv 0$. Theorem. The spaces Bp(R) and Cp(Δ) are isometrically isomorphic with respect to the sup-norm.

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