Weyl asymptotic formulas in the nilpotent Lie group setting

TL;DR

This paper establishes spectral asymptotic formulas for negative fractional powers of hypoelliptic operators on graded Lie groups, extending classical results and connecting to Connes' integration formula.

Contribution

It generalizes known spectral asymptotics to hypoelliptic operators with anisotropic symbols on graded Lie groups, including variable coefficient cases.

Findings

Derived a spectral asymptotic formula for hypoelliptic operators on graded Lie groups.

Extended Birman and Solomyak's methods to nilpotent Lie group setting.

Connected spectral asymptotics to Connes' integration formula.

Abstract

The asymptotic properties of negative order pseudo-differential operators have been an important part of the spectral theory since H.Weyl's classical results. In this paper, we derive a spectral asymptotic formula for the negative fractional powers of hypoelliptic operators on graded Lie groups. Such operators have anisotropically homogeneous principal symbols; for these, our results generalize known results of Birman and Solomyak from 1977. Additionally, our work implies a version of Connes' integration formula for hypoelliptic operators on graded Lie groups. Our methods allow us to extend results from constant-coefficient operators to those with smoothly varying coefficients. The principal technique is to adapt the singular value perturbation arguments of Birman and Solomyak to the setting of nilpotent Lie groups. The decomposing of graded Lie groups is inspired by Folland and Stein in their development of harmonic analysis on homogeneous groups.