Strong convexity for harmonic functions on compact symmetric spaces

Abstract

Let h h be a harmonic function defined on a spherical disk. It is shown that Δ k | h | 2 \Delta ^k |h|^2 is nonnegative for all k ∈ N k\in \mathbb {N} where Δ \Delta is the Laplace-Beltrami operator. This fact is generalized to harmonic functions defined on a disk in a normal homogeneous compact Riemannian manifold, and in particular in a symmetric space of the compact type. This complements a similar property for harmonic functions on R n \mathbb {R}^n discovered by the first two authors and is related to strong convexity of the L 2 L^2 -growth function of harmonic functions.