Bounded solutions of the equation $\Delta u=pu$ on a Riemannian manifold

Abstract:

Given a nonnegative ${C^1}$-function $p(x)$ on a Riemannian manifold $R$, denote by ${B_p}(R)$ the Banach space of all bounded ${C^2}$-solutions of $\Delta u = pu$ with the sup-norm. The purpose of this paper is to give a unified treatment of ${B_p}(R)$ 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 ${B_p}(R)$ for various densities $p(x)$. Typically, let $\beta (p)$ be the set of all $p$-potential nondensity points in the Wiener harmonic boundary $\Delta$, and ${C_p}(\Delta )$ the space of bounded continuous functions $f$ on $\Delta$ with $f|\Delta - \beta (p) \equiv 0$. Theorem. The spaces ${B_p}(R)$ and ${C_p}(\Delta )$ are isometrically isomorphic with respect to the sup-norm.