Finding Heat Kernels Using the Laguerre Calculus
Ovidiu L. Calin, Der‐Chen Chang, Kenrô Furutani, Chisato Iwasaki · Applied and numerical harmonic analysis · 2010
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 ℝ 2. Let K be a principal value (P.V.) convolution operator on ℝ 2: $$\mathbf{K}(f)(x) {=\lim }_{\epsilon \rightarrow 0}{ \int olimits olimits }_{\vert y\vert >\epsilon }K(y)f(x - y)dy,$$ where f ∈ C 0 ∞ (ℝ 2) and K ∈ C ∞ (ℝ 2 ∖ { (0, 0)}) is homogeneous of degree − 2 with vanishing mean value; i.e., $${\int olimits olimits }_{\vert y\vert =1}K(y)dy = 0.$$ Thus we can write $$K(x) = \frac{f(\theta )} {{r}^{2}},\qquad x = {x}_{1} + i{x}_{2} = r{e}^{i\theta },$$ where $$f(\theta ) ={ \sum olimits }_{m\in \mathbb{Z},m eq 0}{f}_{m}{e}^{im\theta }.$$ Suppose that g is another smooth function on [0, 2π] with $$g(\theta ) ={ \sum olimits }_{m\in \mathbb{Z},m eq 0}{g}_{m}{e}^{im\theta }.$$ Then g induces a principal value convolution operator G on ℝ 2 with kernel $$\frac{g(\theta )} {{r}^{2}}$$ . In [91], we found the following identity.