Hardy's theorem for the helgason Fourier transform on noncompact rank one symmetric spaces

Volume 94 / 2002

S. Thangavelu Colloquium Mathematicum 94 (2002), 263-280 MSC: Primary 22E30; Secondary 43A30. DOI: 10.4064/cm94-2-8


Let $G$ be a semisimple Lie group with Iwasawa decomposition $G=KAN$. Let $ X=G/K$ be the associated symmetric space and assume that $X$ is of rank one. Let $M$ be the centraliser of $A$ in $K$ and consider an orthonormal basis $ \{Y_{\delta,j}: \delta \in \widehat K_0,\, 1 \leq j \leq d_\delta \}$ of $L^2 (K/M)$ consisting of $K$-finite functions of type $\delta$ on $K/M$. For a function $f$ on $X$ let $\skew 6\widetilde {f} (\lambda,b)$, $\lambda \in \mathbb C$, be the Helgason Fourier transform. Let $h_t$ be the heat kernel associated to the Laplace–Beltrami operator and let $ Q_\delta (i \lambda + \varrho )$ be the Kostant polynomials. We establish the following version of Hardy's theorem for the Helgason Fourier transform: Let $f$ be a function on $G/K$ which satisfies $ |f(ka_{r})|\leq Ch_{t}(r).$ Further assume that for every $\delta $ and $j$ the functions $$ F_{\delta,j}(\lambda )=Q_{\delta }(i\lambda +\varrho )^{-1}\int_{K/M} \skew 6\widetilde{f}(\lambda ,b)Y_{\delta ,j}(b)\,db $$ satisfy the estimates $|F_{\delta,j}(\lambda )|\leq C_{\delta ,j}e^{-t\lambda ^{2}}$ for $\lambda \in \mathbb R$. Then $f$ is a constant multiple of the heat kernel~$h_{t}$.


  • S. ThangaveluStat-Math Division
    Indian Statistical Institute
    8th Mile, Mysore Road
    Bangalore-560 059, India

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