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

S. Thangavelu¹

¹ Stat-Math Division Indian Statistical Institute 8th Mile, Mysore Road Bangalore-560 059, India

Colloquium Mathematicum, Tome 94 (2002) no. 2, pp. 263-280

Résumé

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~$h_{t}$.

Détail

DOI : 10.4064/cm94-2-8

Keywords: semisimple lie group iwasawa decomposition kan associated symmetric space assume rank centraliser consider orthonormal basis delta delta widehat leq leq delta consisting k finite functions type delta function skew widetilde lambda lambda mathbb helgason fourier transform heat kernel associated laplace beltrami operator delta lambda varrho kostant polynomials establish following version hardys theorem helgason fourier transform function which satisfies leq further assume every delta functions delta lambda delta lambda varrho int skew widetilde lambda delta satisfy estimates delta lambda leq delta t lambda lambda mathbb constant multiple heat kernel