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.
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Research supported by N.B.H.M. Research Grant, Govt. of India.
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Sajith, G., Ratnakumar, P.K. Gelfand pairs,K-spherical means and injectivity on the Heisenberg group. J. Anal. Math. 78, 245–262 (1999). https://doi.org/10.1007/BF02791136
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DOI: https://doi.org/10.1007/BF02791136