Local Uncertainty Inequalities for Locally Compact Groups

Abstract

Let $G$ be a locally compact unimodular group equipped with Haar measure $m$, $\hat G$ its unitary dual and $\mu$ the Plancherel measure (or something closely akin to it) on $\hat G$. When $G$ is a euclidean motion group, a non-compact semisimple Lie group or one of the Heisenberg groups we prove local uncertainty inequalities of the following type: given $\theta \in \left [ {0,\tfrac {1} {2}} \right .)$ there exists a constant ${K_\theta }$ such that for all $f$ in a certain class of functions on $G$ and all measurable $E \subseteq \hat G$, \[ {\left ( {\int _E {\operatorname {Tr} (\pi {{(f)}^{\ast }}\pi (f)) d\mu (\pi )} } \right )^{1/2}} \leqslant {K_\theta }\mu {(E)^\theta }||{\phi _\theta }f|{|_2}\] where ${\phi _\theta }$ is a certain weight function on $G$ (for which an explicit formula is given). When $G = {{\mathbf {R}}^k}$ the inequality has been established with ${\phi _\theta }(x) = |x{|^{k\theta }}$.