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

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. © 2008 American Mathematical Society.