THE SMOOTHNESS of ORBITAL MEASURES on NONCOMPACT SYMMETRIC SPACES

Sanjiv Kumar Gupta, Kathryn E. Hare

Research output: Contribution to journalArticlepeer-review

Abstract

Let be an irreducible symmetric space, where G is a noncompact, connected Lie group and K is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any r=r (G/K)continuous orbital measures has its density function in L2(G) and hence is an absolutely continuous measure with respect to the Haar measure. The number r is approximately the rank G/K of. For the special case of the orbital measures vai, supported on the double cosets KaiK, where ai belongs to the dense set of regular elements, we prove the sharp result that va1∗va2 ϵ L2 except for the symmetric space of Cartan class when the convolution of three orbital measures is needed (even though va1∗va2 is absolutely continuous).

Original languageEnglish
Article numberS1446788721000033
JournalJournal of the Australian Mathematical Society
DOIs
Publication statusAccepted/In press - 2021
Externally publishedYes

Keywords

  • orbital measure
  • spherical function
  • symmetric space

ASJC Scopus subject areas

  • Mathematics(all)

Cite this