Smoothness of the Radon-Nikodym derivative of a convolution of orbital measures on compact symmetric spaces of rank one

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Abstract

Let G/K be a compact symmetric space of rank one. The aim of this paper is to give sufficient conditions for the Cv -smoothness of the Radon Nikodym derivative fa1,...,ap = d (μa1 * ... * μap of the convolution μa1 *...*μap of some orbital measures μai, with respect to the Haar measure μG of G. This generalizes some of the main results in [12], in the case of compact rank one symmetric spaces, where the absolute continuity of the measure μa1 * ... * μap with respect to dμG was considered. Our main result generalizes also the main results in [1] and [7], where the L2-regularity was considered. As a consequence of our main result, we give sufficient conditions for fa1,...,ap to be in Lq (G, dμG) for all q ge; 1 and for the Fourier series of fa1,...,ap to converge absolutely and uniformly to fa1,...,ap.

Original languageEnglish
Pages (from-to)211-222
Number of pages12
JournalAsian Journal of Mathematics
Volume22
Issue number2
DOIs
Publication statusPublished - Jan 1 2018

Fingerprint

Radon-Nikodym Derivative
Radon
Fourier series
Compact Space
Symmetric Spaces
Convolution
Smoothness
Derivatives
Haar Measure
Generalise
Absolute Continuity
Sufficient Conditions
Regularity
Converge

Keywords

  • Orbital measures
  • Radon-Nikodym derivative
  • Symmetric spaces

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

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title = "Smoothness of the Radon-Nikodym derivative of a convolution of orbital measures on compact symmetric spaces of rank one",
abstract = "Let G/K be a compact symmetric space of rank one. The aim of this paper is to give sufficient conditions for the Cv -smoothness of the Radon Nikodym derivative fa1,...,ap = d (μa1 * ... * μap of the convolution μa1 *...*μap of some orbital measures μai, with respect to the Haar measure μG of G. This generalizes some of the main results in [12], in the case of compact rank one symmetric spaces, where the absolute continuity of the measure μa1 * ... * μap with respect to dμG was considered. Our main result generalizes also the main results in [1] and [7], where the L2-regularity was considered. As a consequence of our main result, we give sufficient conditions for fa1,...,ap to be in Lq (G, dμG) for all q ge; 1 and for the Fourier series of fa1,...,ap to converge absolutely and uniformly to fa1,...,ap.",
keywords = "Orbital measures, Radon-Nikodym derivative, Symmetric spaces",
author = "Boudjemaa Anchouche and Sanjiv Gupta",
year = "2018",
month = "1",
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language = "English",
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journal = "Asian Journal of Mathematics",
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T1 - Smoothness of the Radon-Nikodym derivative of a convolution of orbital measures on compact symmetric spaces of rank one

AU - Anchouche, Boudjemaa

AU - Gupta, Sanjiv

PY - 2018/1/1

Y1 - 2018/1/1

N2 - Let G/K be a compact symmetric space of rank one. The aim of this paper is to give sufficient conditions for the Cv -smoothness of the Radon Nikodym derivative fa1,...,ap = d (μa1 * ... * μap of the convolution μa1 *...*μap of some orbital measures μai, with respect to the Haar measure μG of G. This generalizes some of the main results in [12], in the case of compact rank one symmetric spaces, where the absolute continuity of the measure μa1 * ... * μap with respect to dμG was considered. Our main result generalizes also the main results in [1] and [7], where the L2-regularity was considered. As a consequence of our main result, we give sufficient conditions for fa1,...,ap to be in Lq (G, dμG) for all q ge; 1 and for the Fourier series of fa1,...,ap to converge absolutely and uniformly to fa1,...,ap.

AB - Let G/K be a compact symmetric space of rank one. The aim of this paper is to give sufficient conditions for the Cv -smoothness of the Radon Nikodym derivative fa1,...,ap = d (μa1 * ... * μap of the convolution μa1 *...*μap of some orbital measures μai, with respect to the Haar measure μG of G. This generalizes some of the main results in [12], in the case of compact rank one symmetric spaces, where the absolute continuity of the measure μa1 * ... * μap with respect to dμG was considered. Our main result generalizes also the main results in [1] and [7], where the L2-regularity was considered. As a consequence of our main result, we give sufficient conditions for fa1,...,ap to be in Lq (G, dμG) for all q ge; 1 and for the Fourier series of fa1,...,ap to converge absolutely and uniformly to fa1,...,ap.

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KW - Radon-Nikodym derivative

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