POMPEIU PROBLEM FOR THE HEISENBERG BALL

Abstract

The standard Pompeiu results for complex balls in the setting of the Heisenberg group $\bf H^n$ are also shown to carry over to Heisenberg balls. This extension is important because it allows for integration over sets of the same dimension as the ambient space $\bf H^n$. Several different concepts of the Heisenberg ball are considered, using differing definitions for the metric on $\bf H^n$. For purposes of this work, none of these is more natural than the others. The results for each of the spaces $L^2$, $L^p$ for $1 \leq p \lt \infty$, and for $L^\infty$, are directly comparable to the Pompeiu results for complex balls in $\bf H^n$, as in [2, 3, 4]. The natural expression for the Pompeiu problem in $\bf H^n$ is integration over complex balls, and the extension to the Heisenberg ball builds upon the methods for this case. The extra dimension primarily leads to extra complexity in the integrals. At the level of $L^\infty$, where the results require balls of two radii satisfying appropriate conditions, the additional dimension adds to the complexity in the functions defining the conditions for the radii. The different concepts of the Heisenberg ball lead to different forms for these arithmetic conditions defining the radii. The differences between these balls can also be seen when they are rotated with $\bf H^n$, an issue to be further considered in a later work. The standard Pompeiu results have been extended to all of the cases considered here.