The Fourier Transform on Harmonic Manifolds of Purely Exponential Volume Growth

Abstract

Let X be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of non-compact harmonic manifolds except for the flat spaces. Denote by $h > 0$ the mean curvature of horospheres in X, and set $\rho = h/2$. Fixing a basepoint $o \in X$, for $\xi \in \partial X$, denote by $B_{\xi }$ the Busemann function at $\xi$ such that $B_{\xi }(o) = 0$. Then for $\lambda \in \mathbb {C}$ the function $e^{(i\lambda - \rho )B_{\xi }}$ is an eigenfunction of the Laplace–Beltrami operator with eigenvalue $-(\lambda ^2 + \rho ^2)$. For a function f on X, we define the Fourier transform of f by

$$\begin{aligned} \tilde{f}(\lambda , \xi ) := \int _X f(x) e^{(-i\lambda -\rho )B_{\xi }(x)} \mathrm{{d}}vol(x) \end{aligned}$$

for all $\lambda \in \mathbb {C}, \xi \in \partial X$ for which the integral converges. We prove a Fourier inversion formula

$$\begin{aligned} f(x) = C_0 \int _{0}^{\infty } \int _{\partial X} \tilde{f}(\lambda , \xi ) e^{(i\lambda - \rho )B_{\xi }(x)} \mathrm{{d}}\lambda _o(\xi ) |c(\lambda )|^{-2} \mathrm{{d}}\lambda \end{aligned}$$

for $f \in C^{\infty }_c(X)$, where c is a certain function on $\mathbb {R} - \{0\}$, $\lambda _o$ is the visibility measure on $\partial X$ with respect to the basepoint $o \in X$ and $C_0 > 0$ is a constant. We also prove a Plancherel theorem, and a version of the Kunze–Stein phenomenon.