Lecture Event
Pompeiu's Problem on Damek-Ricci spaces
Dr. Norbert Peyerimhoff
7 Nov 2007, 16:30
In 1929, the Rumanian mathematician Dimitrie Pompeiu
asked the following question: given a continuous function f on R2 and
a compact set K. Assume that the integral of f over all images of K
under rigid motions vanishes. Does this imply that the function f
itself is zero? The answer is negative in the case that K is a disk.
But it can be shown that the conclusion holds in the case that f
vanishes on all disks of radius r_1 and of radius r_2, as long as
r_1 and r_2 avoid a certain countable set of radii. It is natural
to ask similar questions in more general geometries. In this talk
we discuss the same (two radius) problem in Damek-Ricci spaces. These
spaces became famous as counterexamples of the Lichnerowitch conjecture,
namely that every harmonic space should be a rank one symmetric
space.
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