A simplified calculation for the fundamental solution to the heat equation on the Heisenberg group

Albert Boggess, Andrew Raich

Research output: Contribution to journalArticle

7 Scopus citations

Abstract

Let Lγ =-1/4 (Σnj=1](X 2j + Y2j) + iγT) where γ ∈ ℂ, and Xj, Yj and T are the left-invariant vector fields of the Heisenberg group structure for ℝn × ℝn × ℝ. We explicitly compute the Fourier transform (in the spatial variables) of the fundamental solution of the heat equation ∂sp =-Lγp.As a consequence, we have a simplified computation of the Fourier transform of the fundamental solution of the □b-heat equation on the Heisenberg group and an explicit kernel of the heat equation associated to the weighted ∂̄-operator in ℂn with weight exp(-τP(z 1,...,zn)), where P(z1,..., zn) = 1/2(| Imz1|2 +...+ I Imzn|2)and τ ∈ℝ.

Original languageEnglish (US)
Pages (from-to)937-944
Number of pages8
JournalProceedings of the American Mathematical Society
Volume137
Issue number3
DOIs
StatePublished - Mar 1 2009
Externally publishedYes

Keywords

  • Fundamental solution
  • Heat equation
  • Heat kernel
  • Heisenberg group
  • Kohn laplacian

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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