Qualitative uncertainty principle on certain Lie groups (Q6538793)

The qualitative uncertainty principle (QUP) on the Euclidean space \(\mathbb{R}^n\) states that for \(f\in L^1(\mathbb{R}^n)\), \(A=\{x\in \mathbb{R}^n : f(x)\neq 0\}\) and \(B=\{\xi \in \mathbb{R}^n : \widehat{f}(\xi)\neq 0\}\), if \(0<m(A)\ m(B)<\infty\), then \(f=0\) almost everywhere, where \(m\) denotes the Lebesgue measure on \(\mathbb{R}^n\). \N\textit{D.~Arnal} and \textit{J.~Ludwig} [Proc. Am. Math. Soc. 125, No.~4, 1071--1080 (1997; Zbl 0866.43002)]\Nextended the notion of QUP to unimodular groups. Let \(G\) be a unimodular group and \(\widehat{G}\) be the unitary dual of \(G\). Let \(\nu\) and \(\widehat{\nu}\) denote the Haar measure and Plancherel measure on \(G\) and \(\widehat{G}\), respectively. For \(f\in L^1(G)\), if \(\nu\{x \in G:f(x)\neq 0\}<\nu(G)\) and \(\int_{\widehat{G}}{\text{rank}(\pi(f))}\, d\widehat{\nu} < \infty\), then \(f=0\) almost everywhere.\N\NIn this paper, the authors discuss a version of Benedicks' theorem for the Weyl transform on certain step-two nilpotent Lie groups introduced by Moore and Wolf. Let \(G\) be a connected, simply connected, step-two nilpotent Lie group whose Lie algebra \(\mathfrak{g}\) has the orthogonal decomposition \(\mathfrak{g}=\mathfrak{b}\oplus \mathfrak{z}\) with \([\mathfrak{g},\mathfrak{b}]\subset \mathfrak{z}\) and \([\mathfrak{g}, [\mathfrak{g},\mathfrak{g}]]=\{0\}\), where \(\mathfrak{z}\) is the center of \(\mathfrak{g}\). If the Weyl transform on \(G\) is of finite rank, then the function has to be zero almost everywhere as long as the non-vanishing set for the function has finite measure.\N\NFurther, it is proved that if the Weyl transform of each Fourier-Wigner piece of a suitable function on the Heisenberg motion group is of finite rank, then the function has to be zero almost everywhere whenever the non-vanishing set for each Fourier-Wigner piece has finite measure.