NDLI: Orbital Measures on SU(2) / SO(2)

Content Provider	Springer Nature Link
Author	Gupta, Sanjiv Kumar Anchouche, Boudjemâa Plagne, Alain
Copyright Year	2015
Abstract	We let $$U=SU(2)$$ , $$K=SO(2)$$ and denote by $$N_{U} (K)$$ the normalizer of K in U. For a an element of $$U\backslash N_{U} (K)$$ , we let $$\mu _{a}$$ be the normalized singular measure supported in KaK. For p a positive integer, it was proved by Gupta and Hare (Monatsh Math 159:27–59, 2010) that $$\mu _{a}^{(p)},$$ the convolution of p copies of $$\mu _{a}$$ , is absolutely continuous with respect to the Haar measure of the group U as soon as $$p \ge 2$$ . The aim of this paper is to go a step further by proving the following two results : (i) for every a in $$U\backslash N_{U}(K)$$ and every integer $$p \ge 3$$ , the Radon–Nikodym derivative of $$\mu _{a}^{(p)}$$ with respect to the Haar measure $$m_{U}$$ on U, namely $$\hbox {d}\mu _{a}^{(p)} / \hbox {d}m_{U}$$ , is in $$L^{2}(U)$$ , and (ii) there exist a in $$U\backslash N_{U}( K)$$ for which $$\hbox {d}\mu _{a}^{(2)}/\hbox {d}m_{U}$$ is not in $$L^{2}(U),$$ hence a counter example to the dichotomy conjecture stated by Gupta and Hare (Bull Aust Math Soc 79:513–522, 2009). Since $$L^{2}(U) \subseteq L^{1} (U)$$ , our result gives in particular a new proof of the main result of Gupta and Hare (Monatsh Math 159:27–59, 2010) when $$p>2$$ .
Ending Page	520
Page Count	28
Starting Page	493
File Format	PDF
ISSN	00269255
e-ISSN	14365081
Journal	Monatshefte für Mathematik
Issue Number	4
Volume Number	178
Language	English
Publisher	Springer Vienna
Publisher Date	2015-09-03
Publisher Place	Vienna
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Symmetric spaces Analysis on other specific Lie groups Symmetric space Integration on manifolds; measures on manifolds Harmonic analysis Absolutely continuous measure Dichotomy conjecture Mathematics Exponential sums Analytic combinatorics Bi-invariant measure
Content Type	Text
Resource Type	Article
Subject	Mathematics

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