Weyl type bounds for harmonic functions
Let \((M^n,g)\) be a complete Riemannian manifold having the doubling property and satisfying the scale invariant Neumann-Poincaré inequality. Let \({\mathcal H}_d\), \(d\geq 1\), be the linear space of harmonic functions \(u\) on the open manifold \(m^n\) (which can be defined by the Laplacian or more generally by means of a quasi-uniformly elliptic operator) such that \(| u|\leq c(1+r^d)\) holds, where \(r\) is the distance from a fixed point. There is a nice estimate here for the dimension of \({\mathcal H}_d\). Moreover, similar to the Weyl formula for eigenfunctions on compact domains in the Euclidean space, in this interesting article the authors present an asymptotic formula and a uniform estimate on the remainder for harmonic functions with polynomial growth on open manifolds with nonnegative Ricci curvature.
harmonic functions with polynomial growth
scientific article; zbMATH DE number 1145860