The Fourier transform on negatively curved harmonic manifolds

Abstract

Let $X$ be a complete, simply connected harmonic manifold with sectional curvatures $K$ satisfying $K \leq -1$, and let $\partial X$ denote the boundary at infinity of $X$. Let $h > 0$ denote the mean curvature of horospheres in $X$, and let $ρ= h/2$. Fixing a basepoint $o \in X$, for $ξ\in \partial X$, let $B_ξ$ denote the Busemann function at $ξ$ such that $B_ξ(o) = 0$, then for $λ\in \mathbb{C}$ the function $e^{(iλ- ρ)B_ξ}$ is an eigenfunction of the Laplace-Beltrami operator with eigenvalue $-(λ^2 + ρ^2)$. For a function $f$ on $X$, we define the Fourier transform of $f$ by $$\tilde{f}(λ, ξ) := \int_X f(x) e^{(-iλ- ρ)B_ξ(x)} dvol(x)$$ for all $λ\in \mathbb{C}, ξ\in \partial X$ for which the integral converges. We prove a Fourier inversion formula $$f(x) = C_0 \int_{0}^{\infty} \int_{\partial X} \tilde{f}(λ, ξ) e^{(iλ- ρ)B_ξ(x)} dλ_o(ξ) |c(λ)|^{-2} dλ$$ for $f \in C^{\infty}_c(X)$, where $c$ is a certain function on $\mathbb{R} - \{0\}$, $λ_o$ 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).