A Restriction Theorem for the Heisenberg Group

Abstract

Since the discovery by E. M. Stein (see Fefferman [5]) that the Fourier transform of an LP-function on R' has a well-defined restriction to the unit sphere S`' which is square integrable on Sn-1, if p is close enough to 1, various new restriction theorems have been proved (see e.g. [10], [22], [27], [29]). The importance of such theorems in harmonic analysis (see e.g. [5], [6], [24], [28]) as well as in the theory of partial differential equations ([18], [19], [25]) has become evident. It has turned out [29] that the crucial step in proving the Stein-Tomas restriction theorem is the following: Let a denote the surface measure of S n 1, and let K denote the Fourier transform of a. Then the operator Af = f * K is bounded from LP to LP', if p is sufficiently close to 1. 9 can be interpreted in terms of the Laplacian A on Rn: Let -A = foX dEx be the spectral decomposition of A. Then, formally, 9 = const. f106(1 X) dEx, where 6 denotes Dirac's point measure. Therefore, it is natural to study analogues for 9, where A is replaced by a more general positive differential operator L. In the case of second order elliptic operators on compact manifolds, this has been done by Sogge [25].