Laguerre calculus and Paneitz operator on the Heisenberg group
Laguerre calculus and Paneitz operator on the Heisenberg group
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DOI：
10.1007/s11425-009-0056-0
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发表时间：
2009-07
期刊：
影响因子：
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通讯作者：
D. Chang;Shun-Cheng Chang;Jingzhi Tie
中科院分区：
文献类型：
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作者：
D. Chang;Shun-Cheng Chang;Jingzhi Tie
拉盖尔微积分是对海森堡群进行调和分析的有力工具。许多次椭圆偏微分算子可以用Laguerre演算求逆。在这篇文章中，我们使用Laguerre演算来寻找Paneitz算子及其热方程基本解的显式核。在CR几何中起重要作用的Paneitz算子可以写成：$${\Mathcal{P}_\Alpha}={\Mathcal{L}_\Alpha}\BAR{\Mathcal{L}_\Alpha}=\FRAC{1}{4}\Left[{\sum\Limits_{j=1]^n{\Left({Z_j\bar Z_j+\bar Z_j_j}\Right)}}\Right]^2+\Alpha^2T^2$$这里“zj”j=1是复切向丛Tℂ(Hn)的子丛T(1，0)的一个正交基，T是“缺失方向”。算符是海森堡群上的次拉普拉斯算子，如果α不属于例外集Λα，则次拉普拉斯算子是次椭圆的。我们还构造了算子Whileα∈Λα的投影算子和相应的基本解。
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 “Zj”j=1nis an orthonormal basis for the subbundleT(1,0)of the complex tangent bundleTℂ(Hn) andTis the “missing direction”. The operatoris the sub-Laplacian on the Heisenberg group which is sub-elliptic ifαdoes not belong to an exceptional set Λα. We also construct projection operators and relative fundamental solution for the operatorwhileα∈ Λα.