On the images of Sobolev spaces under the heat kernel transform on the Heisenberg group (Q2852506)

The aim of this paper is to obtain certain characterizations for the image of a Sobolev space \(H^s(\mathbb{H}^n)=(I+\mathcal{L})^sL^2(\mathbb{H}^n)\) on the Heisenberg group \(\mathbb{H}^n\) under the heat kernel transform \(\mathcal{H}_{t}=e^{-t\triangle}\) where \(\mathcal{L}\) is the sublaplacian of \(\mathbb{H}^n\) and \(\triangle\) is the Laplace operator on \(\mathbb{H}^n\).NEWLINENEWLINEThe authors give three types of characterizations for the image of a Sobolev space \(H^m(\mathbb{H}^n)\) of positive order \(m \in \mathbb{N}\) under the heat kernel transform \(\mathcal{H}_{t}\), using a direct integral of Bergman spaces, a direct sum of two weighted Bergman spaces, and certain unitary representations of the Heisenberg group \(\mathbb{H}^n\) which can be realized on the Hilbert space of Hilbert-Schmidt operators on \(L^2(\mathbb{R}^n)\).NEWLINENEWLINEThey also show that the image of a Sobolev space \(H^{-s}(\mathbb{H}^n)\) of negative order \(-s<0\) under the heat kernel transform \(\mathcal{H}_{t}\) is a direct sum of two weighted Bergman spaces.NEWLINENEWLINEFinally, they try to obtain some pointwise estimates for the functions in the image of Schwartz class on the Heisenberg group \(\mathbb{H}^n\) under the heat kernel transform \(\mathcal{H}_{t}\).