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 harmonic manifolds except for the flat spaces. Denote by $h > 0$ the mean curvature of horospheres in $X$, and set $ρ= h/2$. Fixing a basepoint $o \in X$, for $ξ\in \partial X$, denote by $B_ξ$ the Busemann function at $ξ$ such that $B_ξ(o) = 0$. then for $λ\in \C$ 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 $λ\in \C, ξ\in \partial X$ 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, and a version of the Kunze-Stein phenomenon.

arXiv admin note: substantial text overlap with arXiv:1802.07236