# Finding Heat Kernels Using the Laguerre Calculus

• Ovidiu Calin
• Der-Chen Chang
• Kenro Furutani
• Chisato Iwasaki
Chapter
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

## Abstract

In this chapter, we are going to use a harmonic analysis method to construct the heat kernels and fundamental solutions of the sub-Laplacian on the Heisenberg group. This method relies on Laguerre calculus. We shall start with a beautiful idea of Mikhlin from his 1936 study of convolution operators on 2. Let K be a principal value (P.V.) convolution operator on 2:
$$\mathbf{K}(f)(x) {=\lim }_{\epsilon \rightarrow 0}{ \int \nolimits \nolimits }_{\vert y\vert >\epsilon }K(y)f(x - y)dy,$$
where fC0 (2) and KC (2 ∖ { (0, 0)}) is homogeneous of degree − 2 with vanishing mean value; i.e.,
$${\int \nolimits \nolimits }_{\vert y\vert =1}K(y)dy = 0.$$
Thus we can write
$$K(x) = \frac{f(\theta )} {{r}^{2}},\qquad x = {x}_{1} + i{x}_{2} = r{e}^{i\theta },$$
where
$$f(\theta ) ={ \sum \nolimits }_{m\in \mathbb{Z},m\neq 0}{f}_{m}{e}^{im\theta }.$$
Suppose that g is another smooth function on [0, 2π] with
$$g(\theta ) ={ \sum \nolimits }_{m\in \mathbb{Z},m\neq 0}{g}_{m}{e}^{im\theta }.$$
Then g induces a principal value convolution operator G on 2 with kernel $$\frac{g(\theta )} {{r}^{2}}$$. In [91], we found the following identity.

## Keywords

Fundamental Solution Heat Kernel Heisenberg Group Inverse Fourier Transform Convolution Operator
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Authors and Affiliations

• Ovidiu Calin
• 1
• Der-Chen Chang
• 2
• Kenro Furutani
• 3
• Chisato Iwasaki
• 4
1. 1.Department of MathematicsEastern Michigan UniversityYpsilantiUSA
2. 2.Department of Mathematics and StatisticsGeorgetown UniversityWashingtonUSA
3. 3.Department of Mathematics ScienceUniversity of TokyoNodaJapan
4. 4.Department of Mathematical ScienceUniversity of HyogoHimejiJapan