### Abstract

Let L_{γ} =-1/4 (Σ^{n}_{j=1}](X ^{2}_{j} + Y^{2}_{j}) + i_{γ}T) where γ ∈ ℂ, and X_{j}, Y_{j} and T are the left-invariant vector fields of the Heisenberg group structure for ℝ^{n} × ℝ^{n} × ℝ. We explicitly compute the Fourier transform (in the spatial variables) of the fundamental solution of the heat equation ∂_{s}p =-L_{γ}p.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 ℂ^{n} with weight exp(-τP(z _{1},...,z_{n})), where P(z_{1},..., z_{n}) = 1/2(| Imz_{1}|^{2} +...+ I Imz_{n}|^{2})and τ ∈ℝ.

Original language | English (US) |
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Pages (from-to) | 937-944 |

Number of pages | 8 |

Journal | Proceedings of the American Mathematical Society |

Volume | 137 |

Issue number | 3 |

DOIs | |

State | Published - Mar 1 2009 |

Externally published | Yes |

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics