Gelfand pairs,K-spherical means and injectivity on the Heisenberg group |
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Authors: | G Sajith P K Ratnakumar |
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Institution: | (1) Ramanujan Institute for Advanced Study in Mathematics, University of Madras, 600 005 Chennai, India;(2) Stat-Math Unit Indian Statistical Institute, R.V. College Post, 560 059 Bangalore, India |
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Abstract: | We study the injectivity properties of the spherical mean value operators associated to the Gelfand pairs (H
n,K), whereK is a compact subgroup ofU(n). We show that these spherical mean value operators are injective onL
p Hn) for 1≤p<∞. Forp=∞, these operators are not injective. Nevertheless, if the spherical meansf*μ
i
overK-orbits of sufficiently many points (z
i,t
i) ∈H
n vanish, we identify a necessary and sufficient condition on the points (z
i,t
i) which guaranteesf=0. ForK=U(n), this is equivalent to the condition for the two-radius theorem.
Research supported by N.B.H.M. Research Grant, Govt. of India. |
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Keywords: | |
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