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Article overview
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The Fourier transform on harmonic manifolds of purely exponential volume growth | | Kingshook Biswas
; Gerhard Knieper
; Norbert Peyerimhoff
; | | Date: |
8 May 2019 | | 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 $
ho =
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
$lambda in C$ the function $e^{(ilambda -
ho)B_{xi}}$ is an
eigenfunction of the Laplace-Beltrami operator with eigenvalue $-(lambda^2 +
ho^2)$.
For a function $f$ on $X$, we define the Fourier transform of $f$ by
$$ ilde{f}(lambda, xi) := int_X f(x) e^{(-ilambda -
ho)B_{xi}(x)}
dvol(x)$$ for all $lambda in C, xi in partial X$ for which the integral
converges. We prove a Fourier inversion formula $$f(x) = C_0 int_{0}^{infty}
int_{partial X} ilde{f}(lambda, xi) e^{(ilambda -
ho)B_{xi}(x)}
dlambda_o(xi) |c(lambda)|^{-2} dlambda$$ 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. | | Source: | arXiv, 1905.4112 | | Services: | Forum | Review | PDF | Favorites | |
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