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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Local uncertainty inequalities for locally compact groups


Authors: John F. Price and Alladi Sitaram
Journal: Trans. Amer. Math. Soc. 308 (1988), 105-114
MSC: Primary 22E30; Secondary 43A15
DOI: https://doi.org/10.1090/S0002-9947-1988-0946433-5
MathSciNet review: 946433
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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$,

$\displaystyle {\left( {\int_E {\operatorname{Tr} (\pi {{(f)}^{\ast}}\pi (f))\,d... ...\leqslant {K_\theta }\mu {(E)^\theta }\vert\vert{\phi _\theta }f\vert{\vert _2}$

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) = \vert x{\vert^{k\theta }}$.

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DOI: https://doi.org/10.1090/S0002-9947-1988-0946433-5
Article copyright: © Copyright 1988 American Mathematical Society