Counting massive sets and dimensions of harmonic functions.
Let \(M\) be a complete, non-compact Riemannian manifold, and let \(\Delta\) be the Laplace-Beltrami operator on \(M\). Let \(p\in M\) be a fixed point in \(M\) and \(r_p(x)\) the geodesic distance from \(x\in M\) to the point \(p\). For each non-negative number \(d\) we denote by \[ {\mathcal H}_d(M)=\biggl\{f \mid \Delta f=0 \text{ and } \bigl| f(x)\bigr| =O\bigl(r_p^d(x)\bigr)\biggr\} \] the space of polynomial growth harmonic functions of degree at most \(d\). Let \(h_d(M):=\dim{\mathcal H}_d(M)\). In a previous paper of the authors [J. Differ. Geom. 48, 497--530 (1998; Zbl 0926.58011)], they introduced the notion of \(d\)-massive sets. Let \(m_d(M)\) be the maximum number of disjoint \(d\)-massive sets on \(M\). When \(d=0\) it is known that \(m_0(M)=h_0(M)\), but the relationship \(m_d(M)=h_d(M)\) is not established for \(d>0\). The purpose of this paper is to give sharp estimates on both of these quantities in various situations.
scientific article; zbMATH DE number 1782624