Paley-Wiener estimates for the Heisenberg group

The Paley-Wiener space PW(G) on a stratified Lie group G is defined via the spectral decomposition of the associated sub-Laplacian. In this paper, we show that functions in PW(H), where H denotes the Heisenberg group, extend to an entire function on the complexification H-C, satisfying a growth estimate of exponential order two. We also show that a converse, characterizing elements of PW(H) only in terms of pointwise growth behaviour of the entire extension, is not available. (C) 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim