A Simplified Calculation for the Fundamental Solution to the Heat Equation on the Heisenberg Group

TL;DR

This paper derives an explicit Fourier transform of the fundamental solution to the heat equation on the Heisenberg group, simplifying calculations for related heat equations and operators in complex analysis.

Contribution

It provides a simplified explicit computation of the Fourier transform of the fundamental solution for the heat equation on the Heisenberg group, including applications to the -heat equation and weighted -operator.

Findings

Explicit Fourier transform of the fundamental solution derived.

Simplified computation for -heat equation on the Heisenberg group.

Explicit kernel for weighted -operator heat equation in .

Abstract

Let $L = -1/4 (\sum_{j=1}^n(X_j^2+Y_j^2)+i\gamma T)$ where $\gamma$ is a complex number, $X_j$, $Y_j$, and $T$ are the left invariant vector fields of the Heisenberg group structure for $R^n \times R^n \times R$. We explicitly compute the Fourier transform (in the spatial variables) of the fundamental solution of the Heat Equation $\partial_s\rho = -L\rho$. As a consequence, we have a simplified computation of the Fourier transform of the fundamental solution of the $\Box_b$-heat equation on the Heisenberg group and an explicit kernel of the heat equation associated to the weighted dbar-operator in $C^n$ with weight $\exp(-\tau P(z_1,...,z_n))$ where $P(z_1,...,z_n) = 1/2(x_1^2 + >... x_n^2)$, $z_j=x_j+iy_j$, and $\tau\in R$.