Complex powers for a class of infinite order hypoelliptic operators

TL;DR

This paper demonstrates that complex powers of certain infinite order hypoelliptic pseudodifferential operators can be represented as hypoelliptic operators with ultrasmoothing remainders, and applies this to analyze associated heat kernels.

Contribution

It establishes a representation of complex powers of infinite order hypoelliptic operators as hypoelliptic pseudodifferential operators with ultrasmoothing remainders, advancing spectral analysis techniques.

Findings

Complex powers can be represented as hypoelliptic pseudodifferential operators.

Derived precise heat kernel estimates for these operators.

Applied results to study semigroups generated by square roots of hypoelliptic operators.

Abstract

We prove the complex powers of a class of infinite order hypoelliptic pseudodifferential operators can always be represented as hypoelliptic pseudodifferential operators modulo ultrasmoothing operators. We apply this result to the study of semigroups generated by square roots of non-negative hypoelliptic infinite order operators. For this purpose, we derive precise estimates of the corresponding heat kernel.