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.

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Acknowledgements

The first author would like to thank Swagato K. Ray and Rudra P. Sarkar for generously sharing their time and knowledge over the course of numerous educative and enjoyable discussions. The other two authors like to thank the MFO for hospitality during their stay in the “Research in Pairs” program in 2019 and the SFB/TR191 “Symplectic structures in geometry, algebra and dynamics”. This article generalizes an earlier version by the first author in the case of negatively curved harmonic manifolds.

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Correspondence to Kingshook Biswas.

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Biswas, K., Knieper, G. & Peyerimhoff, N. The Fourier Transform on Harmonic Manifolds of Purely Exponential Volume Growth. J Geom Anal 31, 126–163 (2021). https://doi.org/10.1007/s12220-019-00253-9

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Keywords

  • Harmonic manifolds
  • Fourier transform
  • Busemann functions

Mathematics Subject Classification

  • 53C25