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On the images of Sobolev spaces under the heat kernel transform on the Heisenberg group
Author(s) -
Radha R.,
Thangavelu S.,
Naidu D. Venku
Publication year - 2013
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201100233
Subject(s) - mathematics , sobolev space , heisenberg group , pointwise , heat kernel , hilbert space , kernel (algebra) , order (exchange) , space (punctuation) , group (periodic table) , combinatorics , image (mathematics) , mathematical analysis , physics , quantum mechanics , linguistics , philosophy , finance , artificial intelligence , computer science , economics
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.