Finding Heat Kernels Using the Laguerre Calculus

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


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
$$K(x) = \frac{f(\theta )} {{r}^{2}},\qquad x = {x}_{1} + i{x}_{2} = r{e}^{i\theta },$$
$$f(\theta ) ={ \sum \nolimits }_{m\in \mathbb{Z},m\neq 0}{f}_{m}{e}^{im\theta }.$$
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.


Fundamental Solution Heat Kernel Heisenberg Group Inverse Fourier Transform Convolution Operator 
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Copyright information

© Springer Science+Business Media, LLC 2011

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

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