Szeg\"o limit theorem and Heisenberg Laplacian

The aim of this paper is to obtain a version of the classical Szeg"o limit theorem, where instead of the operator of second derivative on a circle we consider the Heisenberg-H"ormander Laplacian in $L^2({\mathbb{R}}^3)$. Besides, we derive a sharp inequality for convex functions that we call Szeg"o-type inequality.