A generalized Beurling theorem for some Lie groups (Q2191935)

The Beurling theorem [\textit{L. Hörmander}, Ark. Mat. 29, No.~2, 237--240 (1991; Zbl 0755.42009)] mentioned in the title is a version of the Uncertainty Principle (UP) that extends Hardy's UP [\textit{G.~H. Hardy}, J. Lond. Math. Soc. 8, 227--231 (1933; Zbl 0007.30403)]. This UP states that a function \(f\) and its Fourier transform \(\widehat{f}\) -- here normalized through \(\widehat{f}(\xi)=\int_{\mathbb{R}^d}f(x)e^{-i\langle x,\xi\rangle}\,\mbox{d}x\) -- cannot both decrease faster than some Gaussian. A sharper version due to [\textit{A.~Bonami} et al., Rev. Mat. Iberoam. 19, No.~1, 23--55 (2003; Zbl 1037.42010), Corollary~3.1] serves as starting point of this paper: {Theorem.} If \(f\in L^2(\mathbb{R}^d)\) is such that \[ \int_{\mathbb{R}^d}\int_{\mathbb{R}^d}\frac{|f(x)||\widehat{f}(\xi)|}{(1+\|x\|+\|\xi\|)^N}e^{\|x\|\|\xi\|} \,\mbox{d}x\,\mbox{d}\xi<+\infty, \tag{1} \] then \(f=0\) if \(N\leq d\), while for \(N>d\) the function \(f\) is a Hermite function \(f(x)=P(x)e^{-a\|x\|^2}\), \(a>0\), \(P\) a polynomial of degree \({<(N-d)/2}\). The first result of the paper is to extend this result by showing that the conclusion still holds if one replaces \(|f(x)||\widehat{f}(\xi)|\) with \(|f(x)|^p|\widehat{f}(\xi)|^q\), \(p,q\geq1\). In some sense this interpolates between the result of Hardy (\(p=q=+\infty\)) and [\textit{A. Bonami} et al., loc. cit.] (\(p=q=1\)). The proof mainly uses Hölder's inequality so as to show that the \(p,q\) condition compares to the \(p=q=1\) case and then apply [\textit{A. Bonami} et al., loc. cit.]. The result is then extended to two families of Lie groups: the Heisenberg group and semi-direct products of \(\mathbb{R}^d\) with compact sub-groups of \(\mathrm{Aut}(\mathbb{R}^d)\) and Heisenberg groups. In both cases the Fourier transform is replaced by the Fourier-Helgason transform and a characterisation of functions satisfying the right analogue of (1) is provided (with the critical index). The key in both cases is to relate the Fourier-Helgason transform to the Euclidean Fourier transform via an integral transform of \(f\) and then to apply the euclidean result to this transform. The results in this paper nicely complement previous results of the second author and collaborators (see references in the paper) as well as those of Thangavellu (see [\textit{G. B. Folland} and \textit{A.~Sitaram}, J. Fourier Anal. Appl. 3, No.~3, 207--238 (1997; Zbl 0885.42006); \textit{S.~Thangavelu}, An introduction to the uncertainty principle. Hardy's theorem on Lie groups. With a foreword by Gerald B.~Folland. Boston, MA: Birkhäuser (2004; Zbl 1188.43010)] for more references on this subject).