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

Let L_γ =-1/4 (Σⁿ_j=1](X ²_j + Y²_j) + i_γT) where γ ∈ ℂ, and X_j, Y_j 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 =-L_γp.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,...,z_n)), where P(z₁,..., z_n) = 1/2(| Imz₁|² +...+ I Imz_n|²)and τ ∈ℝ.