Local uncertainty inequalities for compact groups

Price, John F.; Sitaram, Alladi

doi:10.1090/S0002-9939-1988-0943063-1

Local uncertainty inequalities for compact groups
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by John F. Price and Alladi Sitaram PDF

Proc. Amer. Math. Soc. 103 (1988), 441-447 Request permission

Abstract:

Conditions are established on $\alpha ,\beta \in {\mathbf {R}}$ for there to exist a constant $K = K(\alpha ,\beta )$ such that \[ \sum \limits _{\gamma \in E} {d(\gamma )\operatorname {tr} (\hat f{{(\gamma )}^ * }\hat f(\gamma )) \leq K{{\left ( {\sum \limits _{\gamma \in E} {d{{(\gamma )}^2}} } \right )}^\alpha }{{\left \| {{w^\beta }f} \right \|}_2}} \] for all $f \in {L^1}(G)$ and $E \subseteq \hat G$ where $G$ is a compact metric group, $\hat G$ is its dual, $\hat f$ is the Fourier transform of $f$ and $w:G \to {{\mathbf {R}}^ + }$ is the function taking $x \in G$ to the area of the ball in $G$ with centre $e$ and $x$ on its boundary. This is followed by a partial analogy for compact riemannian manifolds.

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