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

The aim of this paper is to obtain certain characterizations for the image of a Sobolev space on the Heisenberg group under the heat kernel transform. We give three types of characterizations for the image of a Sobolev space of positive order \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H^m(\mathbb {H}^n), m\in \mathbb {N}^n,$\end{document} under the heat kernel transform on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {H}^n,$\end{document} using direct sum and direct integral of Bergmann spaces and certain unitary representations of \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {H}^n$\end{document} which can be realized on the Hilbert space of Hilbert‐Schmidt operators on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L^2(\mathbb {R}^n).$\end{document} We also show that the image of Sobolev space of negative order \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$H^{-s}(\mathbb {H}^n), s(>0) \in \mathbb {R}$\end{document} is a direct sum of two weighted Bergman spaces. Finally, we try to obtain some pointwise estimates for the functions in the image of Schwartz class on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {H}^n$\end{document} under the heat kernel transform.