Orbital Measures on SU(2)?/?SO(2)

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• 作者：Boudjemaa Anchouche ; Sanjiv Kumar Gupta ; Alain Plagne
• 关键词：Harmonic analysis ; Symmetric space ; Bi ; invariant measure ; Absolutely continuous measure ; Dichotomy conjecture ; Analytic combinatorics ; Exponential sums ; 43A80 ; 58C35 ; 53C35
• 刊名：Monatshefte f篓鹿r Mathematik
• 出版年：2015
• 出版时间：December 2015
• 年：2015
• 卷：178
• 期：4
• 页码：493-520
• 全文大小：497 KB
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• 作者单位：Boudjemaa Anchouche (1)
  Sanjiv Kumar Gupta (1)
  Alain Plagne (2)
  
  1. Department of Mathematics and Statistics, Sultan Qaboos University, PO Box 36, Al-Khoud 123, Muscat, Sultanate of Oman
  2. Centre de Mathématiques Laurent Schwartz, école polytechnique, 91128, Palaiseau Cedex, France
  
• 刊物主题：Mathematics, general;
• 出版者：Springer Vienna
• ISSN：1436-5081

文摘

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 159:27-9, 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 79:513-22, 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 159:27-9, 2010) when \(p>2\). Keywords Harmonic analysis Symmetric space Bi-invariant measure Absolutely continuous measure Dichotomy conjecture Analytic combinatorics Exponential sums