arXiv 1802.072362018-02-21math.DG

负曲调和流形上的傅里叶变换

The Fourier transform on negatively curved harmonic manifolds

Kingshook Biswas

备注：: 32 pages

摘要：

设$X$是一个完备的单连通调和流形，其截面曲率$K$满足$K \leq -1$，$\partial X$表示$X$在无穷远处的边界。令$h > 0$表示$X$中半球的平均曲率，令$ρ= h/2$。固定一个基点$o \in X$，对于$ξ\in \partial X$，令$B_ξ$表示在$ξ$处的Busemann函数，使得$B_ξ(o) = 0$，则对于$λ\in \mathbb{C}$，函数$e^{(iλ- ρ)B_ξ}$是具有特征值$-(λ^2 + ρ^2)$的Laplace-Beltrami算子的特征函数。对于$X$上的函数$f$，我们定义了$f$的傅里叶变换为$$\tilde{f}(λ, ξ) := \int_X f(x) e^{(-iλ- ρ)B_ξ(x)} dvol(x)$$，对于所有积分收敛的$λ\in \mathbb{C}, ξ\in \partial X$。证明了一个关于$f \in C^{\infty}_c(X)$的傅里叶反演公式$$f(x) = C_0 \int_{0}^{\infty} \int_{\partial X} \tilde{f}(λ, ξ) e^{(iλ- ρ)B_ξ(x)} dλ_o(ξ) |c(λ)|^{-2} dλ$$，其中$c$是$\mathbb{R} - \{0\}$上的一个确定函数，$λ_o$是$\partial X$上关于基点$o \in X$的可见性测度，$C_0 > 0$是一个常数.我们还证明了一个Plancherel定理。这推广了非紧型秩一对称空间和负曲调和$NA$群（或Damek-Ricci空间）的相应结果。

英文摘要：

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).

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