Paley‐Wiener estimates for the Heisenberg group

Abstract

The Paley-Wiener space P W (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 P W (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 P W (H) only in terms of pointwise growth behaviour of the entire extension, is not available.

Paley–Wiener theorems for the $${\text {U}(n)}$$ U ( n ) -spherical transform on the Heisenberg group

“…Our characterizations are summarized in Therorem 8.4, Theorem 8.7 and Theorem 8.9. These results can be interpreted as a "real" spectral Paley-Wiener theorems for the spectral measure of the sublaplacian, a point of view which coincides with that of [16].…”

“…It does not look plausible to have a simple "complex variable" description of the entire functions which are in the range of the spherical, or inverse spherical, transform of the space of C ∞ -functions, or of distributions, with compact support, see also the comments in Fuhr [16], in a context that is closely related to ours.…”

“…Fourier transform and its inverse, but there are also "spectral" Paley-Wiener theorems, as in the already mentioned paper [16], where the condition of compact support on the transform of a given function is replaced by the condition that the function itself belongs to the image of the spectral measure of a compact set in R + associated to the sublaplacian (see also Strichartz [20], Bray [10], and Dann and Ólafsson [12] in other contexts).…”

Identification of Besov spaces via Littlewood Paley-Stein $g$ functions

“…A unified characterizations of Besov spaces in terms of atomic decomposition using a group representation theoretic approach was given by Feichtinger and Gröchenig ( [9]). New results in this direction in the context of Lie groups and homogeneous manifolds were recently published in [4,5,10,11], and [14]- [17]. For the classification of Besov spaces on compact Riemannian manifolds using continuous and time-frequency localized wavelets with higher vanishing moments we invite the reader to see [15,16].…”

A unified approach for Littlewood-Paley decomposition of abstract Besov spaces

“…In the classical level, this kind of decomposition using the spectral theoretic approach was proved in [8]. New results in this direction in the context of Lie groups and homogeneous manifolds were recently published in [2]- [3], [6,7], and [9]- [13]. In [10,11], the authors constructed continuous and time-frequency localized wavelets and applied them to the classification of Besov spaces on the compact Riemannian manifolds.…”