Hardy's theorem for the Jacobi transform
Let \(a, b, \lambda \in \mathbb{C}\), \(\rho = a + b + 1\), and let \(t\) be a strictly positive real variable. The Jacobi function \(\varphi_\lambda^{a,b}\) (of order \((a,b)\)) is defined by \[ \varphi_\lambda^{a,b}(t) = {}_2F_1 (\tfrac{1}{2}(\rho + i\lambda),\tfrac{1}{2}(\rho - i\lambda),a + 1;-\sinh^2(t)), \] where \({}_2F_1\) is the Gauss hypergeometric function, and the Jacobi transform (of order \((a,b)\)) is given by \[ \widehat{f}^{a,b}(\lambda) = \int_{\mathbb{R}_+}\,f(t) \varphi_\lambda^{a,b}(t)(2\sinh(t))^{2a+1}(2\cosh(t))^{2b+1}\,dt \] for all even functions \(f\) on \(\mathbb{R}\) and all complex numbers \(\lambda\) for which the right hand side is well-defined.\newline The following generalization of Hardy's theorem [\textit{G. H. Hardy}, J. Lond. Math. Soc. 8, 227--231 (1933; Zbl 0007.30403)] is proven in this paper. Let \(a,b \in \mathbb{C}\), \(a \not\in -\mathbb{N}\), and \(\rho = a+ b + 1\) . Assume that \(f\) is an even measurable function on \(\mathbb{R}\) satisfying \[ | f(t)| \leq C(1 + | t| )^Me^{-\Re\rho| t| }e^{-\alpha| t| ^2} \qquad (t \in \mathbb{R}) \] and \[ | \widehat{f}^{a,b}(\lambda)| \leq Ce^{-\beta| \lambda| ^2} \qquad(\lambda \in \mathbb{R}) \] for nonnegative real constants \(C, M, \alpha, \beta\) with \(M \geq \Re a + \frac{1}{2}\) and \(\alpha\beta = \frac{1}{4}\). Then there exists a constant \(c\) such that \(\widehat{f}^{a,b}(\lambda) = ce^{-\beta\lambda^2}\). This theorem is part of the main result of the paper: an \(L^p\) version of Hardy's theorem for the Jacobi transform, which is then applied to study the Fourier transform on real hyperbolic spaces. The author shows that Hardy's theorem can be generalized to functions on the real hyperbolic spaces \(\text{SO}_0(1,n)/ \text{SO}_0(n)\) \((n \in \mathbb{N}^*)\), where the subscript ``\(_0\)'' denotes the identity component, but not to the spaces \(\text{SO}_0(m,n)/ \text{SO}_0(m-1,n)\) with \(m,n \in \mathbb{N}^*\) and \(m > 1\).
scientific article; zbMATH DE number 2068258