Explicit fundamental solution for the operator \(L+\alpha |T|\) on the Gelfand pair \((\mathbb {H}_{n},U(n))\).

Abstract

By means of the spherical functions associated to the Gelfand pair \((\mathbb {H}_{n},U(n))\) we define the operator \(L+\alpha |T|\), where L denotes the Heisenberg sublaplacian and T denotes the central element of the Heisenberg Lie algebra, we establish a notion of fundamental solution and explicitly compute in terms of the Gauss hypergeometric function. For \(\alpha <n\) we use the Integral Representation Theorem to obtain a more detailed expression. Finally, we remark that when \(\alpha =0\) we recover the fundamental solution for the Heisenberg sublaplacian given by Folland.

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Notes

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    See also the book of Rainville on Special Functions [14] and the comprehensive work of [11].

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Correspondence to R. E. Vidal.

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Cardoso, I.E., Subils, M. & Vidal, R.E. Explicit fundamental solution for the operator \(L+\alpha |T|\) on the Gelfand pair \((\mathbb {H}_{n},U(n))\).. J. Pseudo-Differ. Oper. Appl. 12, 6 (2021). https://doi.org/10.1007/s11868-021-00375-1

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Keywords

  • Heisenberg group
  • Fundamental solution of pseudodifferential operator
  • Gelfand pair
  • Spherical functions

Mathematics Subject Classification

  • 47G30
  • 58J40