Abstract
Consider the upper half plane S=ℝ×ℝ+ with the hyperbolic metric and the corresponding measure dxdy/y 2. We introduce a Weyl transform mapping functions on the space S×ℝ2 (viewed as the cotangent bundle of S) to operators on L 2-space on S, by using the Plancherel formula. It is proved that the Weyl transform with symbol in L p (p∈[1,2]) is not only bounded but also compact, while when 2<p<+∞, the Weyl transform is not a bounded operator.
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Mathematical Center of Chinese Ministry of Education, by NNSF of China No. 10990012 and No. 10826106, and RFDP of China No. 20060001010 and No. 200800010009.
The Project-sponsored by SRF for ROCS, SEM, China, and by NNSF of China No. 10871048 and No. 10931001.
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Peng, L., Zhao, J. Weyl transforms on the upper half plane. Rev Mat Complut 23, 77–95 (2010). https://doi.org/10.1007/s13163-009-0013-z
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DOI: https://doi.org/10.1007/s13163-009-0013-z