The Fourier transform on negatively curved harmonic manifolds

Abstract

Let $X$ be a complete, simply connected harmonic manifold with sectional curvatures $K$ satisfying $K\le -1$, and let $\partial X$ denote the boundary at infinity of $X$. Let $h>0$ denote the mean curvature of horospheres in $X$, and let $\rho =h/2$. Fixing a basepoint $o\in X$, for $\xi \in \partial X$, let $B_{\xi }$ denote 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 $$\tilde{f}(\lambda ,\xi ):={\int }_{X}f(x)e^{(-i\lambda -\rho )B_{\xi }(x)}dvol(x)$$ for all $\lambda \in \mathbb{C},\xi \in \partial X$ for which the integral converges. We prove a Fourier inversion formula $$f(x)=C_0{\int }_0^{\infty }{\int }_{\partial X}\tilde{f}(\lambda ,\xi )e^{(i\lambda -\rho )B_{\xi }(x)}d{\lambda }_o(\xi )|c(\lambda ){|}^{-2}d\lambda$$ for $f\in C_c^{\infty }(X)$, 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. This generalizes the corresponding results for rank one symmetric spaces of noncompact type and negatively curved harmonic $NA$ groups (or Damek-Ricci spaces).

32 pages