Laguerre Calculus and its applications in complex analysis and harmonic analysis.
he Heisenberg group is the simplest, non-commutative, nilpotent Lie group. It arises in two fundamental but different settings in analysis. On the one hand, it can be realized as the boundary of the unit ball in several complex variables. On the other hand, there is its genesis in the context of Quantum Mechanics. The Laguerre calculus is the symbolic tensor calculus induced by the Laguerre function on the Heisenberg group. In this talk, I will introduce the Heisenberg group and its Lie algebra from the setting of complex analysis. Then I will use the Laguerre calculus to solve the $\bar\partial$-Neumann problem in the non-isotropic Siegel Domain. If time permits, I will also talk about its application to the Heat kernel, power of sub-Laplacian and Riesz transform on the Heisenberg group.