On the images of Sobolev spaces under the heat kernel transform on the Heisenberg group |
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Authors: | R. Radha S. Thangavelu D. Venku Naidu |
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Affiliation: | 1. Department of Mathematics, Indian Institute of Technology, Chennai‐600 036, India.;2. Department of Mathematics, Indian Institute of Science, Bangalore‐560012, India. Phone: Phone: +91 80 22932711. Phone: +91 40 23017091 |
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Abstract: | 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 $H^m(mathbb {H}^n), min mathbb {N}^n,$ under the heat kernel transform on $mathbb {H}^n,$ using direct sum and direct integral of Bergmann spaces and certain unitary representations of $mathbb {H}^n$ which can be realized on the Hilbert space of Hilbert‐Schmidt operators on $L^2(mathbb {R}^n).$ We also show that the image of Sobolev space of negative order $H^{-s}(mathbb {H}^n), s(>0) in mathbb {R}$ 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 $mathbb {H}^n$ under the heat kernel transform. |
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Keywords: | Heisenberg group Sobolev space sublaplacian Hermite functions semigroup MSC (2010) Primary: 42B35 Secondary: 46E20 43A80 |
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