Laguerre calculus and Paneitz operator on the Heisenberg group

Abstract

Laguerre calculus is a powerful tool for harmonic analysis on the Heisenberg group. Many sub-elliptic partial differential operators can be inverted by Laguerre calculus. In this article, we use Laguerre calculus to find explicit kernels of the fundamental solution for the Paneitz operator and its heat equation. The Paneitz operator which plays an important role in CR geometry can be written as follows:

$$ {\mathcal{P}_\alpha} = {\mathcal{L}_\alpha} \bar {\mathcal{L}_\alpha} = \frac{1} {4}\left[ {\sum\limits_{j = 1}^n {\left( {Z_j \bar Z_j + \bar Z_j Z_j } \right)} } \right]^2 + \alpha ^2 T^2 $$

Here “Z _j ” ⁿ _j=1 is an orthonormal basis for the subbundle T ^(1,0) of the complex tangent bundle T _ℂ(H _n ) and T is the “missing direction”. The operator $\mathcal{L}_\alpha$ is the sub-Laplacian on the Heisenberg group which is sub-elliptic if α does not belong to an exceptional set Λ _α . We also construct projection operators and relative fundamental solution for the operator $\mathcal{L}_\alpha$ while α ∈ Λ _α .