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Heat Kernel in the Framework of Zero Order Mehler–Fock Transform

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Abstract

The main aim of this paper is to study the notion of the heat kernel associated with the zero order Mehler–Fock transform. The upper bound of the heat kernel is obtained. We discuss its some properties and fundamental solution of a generalized diffusion equation. Weierstrass type integral transform with the heat kernel is established. Further boundedness of the Weierstrass integral transform on Sobolev space is discussed and obtained its inversion formula also. Moreover we obtain Heisenberg type inequality associated with the Mehler–Fock transform and heat kernel.

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Acknowledgements

Authors are very thankful to the anonymous reviewer for his valuable and constructive comments. The first author of this work is supported by Science and Engineering Research Board, Gov. of India under Grant No. EMR/2016/005141.

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Authors and Affiliations

  1. Department of Applied Mathematics, Indian Institute of Technology (Indian School of Mines), Dhanbad, 826004, India

    Akhilesh Prasad & Sandeep Kumar Verma

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  1. Akhilesh Prasad
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  2. Sandeep Kumar Verma
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Correspondence to Akhilesh Prasad.

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Communicated by Jussi Behrndt, Fabrizio Colombo, Sergey Naboko.

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Prasad, A., Verma, S.K. Heat Kernel in the Framework of Zero Order Mehler–Fock Transform. Complex Anal. Oper. Theory 13, 3235–3249 (2019). https://doi.org/10.1007/s11785-019-00921-2

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  • DOI: https://doi.org/10.1007/s11785-019-00921-2

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