Pompeiu problem for complex ellipsoids on the Heisenberg group
by Der-Chen Chang   Wayne Eby  

Vol. 2 No. 3 (2007) P.731~P.768

ABSTRACT

We extend results of the Pompeiu problem on the Heisenberg group $\mathbf{H}^n$ from spheres to complex ellipsoids. These results also tell us what happens for spheres and complex ellipsoids on the anisotropic Heisenberg group, $\mathbf{H}^n_\mathbf{a}$ . The results for $L^2$, $L^p$, and $L^\infty$ have the same character as previous results for spheres on $\mathbf{H}^n$. However, when moving to $L^\infty$ and including rotations, we maintain the result from Euclidean space that only one complex ellipsoid is needed.

KEYWORDS
Pompeiu problem, Heisenberg group, Laguerre functions, complex ellipsoids, Bessel functions, Gelfand transform, Tauberian theorem

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 32A10, 30E99

MILESTONES

Received: 2006-11-21
Revised :
Accepted:

Download Full Content