A complete analogue of Hardy’s theorem on SL2(ℝ) and characterization of the heat kernel |
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Authors: | Rudra P Sarkar |
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Institution: | (1) Stat-Math Unit, Indian Statistical Institute, 203 B. T. Road, 700 108 Kolkata, India |
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Abstract: | A theorem of Hardy characterizes the Gauss kernel (heat kernel of the Laplacian) on ℝ from estimates on the function and its
Fourier transform. In this article we establisha full group version of the theorem for SL2(ℝ) which can accommodate functions with arbitraryK-types. We also consider the ‘heat equation’ of the Casimir operator, which plays the role of the Laplacian for the group.
We show that despite the structural difference of the Casimir with the Laplacian on ℝn or the Laplace—Beltrami operator on the Riemannian symmetric spaces, it is possible to have a heat kernel. This heat kernel
for the full group can also be characterized by Hardy-like estimates. |
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Keywords: | Hardy’ s theorem uncertainty principle heat kernel Casimir |
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