Resolvent Estimates and Smoothing for Homogeneous Operators on Graded Lie Groups

TL;DR

This paper establishes uniform resolvent estimates and constructs globally smooth operators for a broad class of homogeneous operators on graded Lie groups, extending classical results to non-commutative settings.

Contribution

It introduces a general framework for resolvent estimates and smoothing operators for homogeneous operators on graded Lie groups, including non-commutative cases like Rockland operators.

Findings

Uniform resolvent estimates for homogeneous operators.

Construction of globally smooth operators without invariance assumptions.

Enhanced results for stratified groups using Hardy inequalities.

Abstract

By using commutator methods, we show uniform resolvent estimates and obtain globally smooth operators for self-adjoint injective homogeneous operators $H$ on graded groups, including Rockland operators, sublaplacians and many others. Left or right invariance is not required. Typically the globally smooth operator has the form $T=V|H|^{1/2}$, where $V$ only depends on the homogeneous structure of the group through Sobolev spaces, the homogeneous dimension and the minimal and maximal dilation weights. For stratified groups improvements are obtained, by using a Hardy-type inequality. Some of the results involve refined estimates in terms of real interpolation spaces and are valid in an abstract setting. Even for the commutative group $\R^N$ some new classes of operators are treated.