Intrinsic Taylor formula for non-homogeneous Kolmogorov-type Lie groups

TL;DR

This paper develops an intrinsic Taylor formula for a class of Lie groups associated with Kolmogorov operators, extending previous work by characterizing intrinsic H"older spaces without the homogeneity assumption.

Contribution

It introduces an intrinsic Taylor formula for non-homogeneous Lie groups related to Kolmogorov operators, generalizing prior homogeneous cases.

Findings

Established an intrinsic Taylor formula with remainder estimates.

Extended the characterization of intrinsic H"older spaces to non-homogeneous groups.

Demonstrated the Taylor polynomial's representation matches the homogeneous case.

Abstract

We prove an intrinsic Taylor-like formula for a class of Lie groups arising in the study of some sub-elliptic differential operators, namely the Kolmogorov operators. The estimate of the remainder is in terms of the intrinsic norm induced by such operators. These results extend the recent developments in a work by Pascucci and the present authors, where a full characterization of the intrinsic H\"older spaces and their Taylor polynomials were given under the additional assumption that the Lie group is homogeneous in the sense of Folland & Stein. Remarkably, the intrinsic Taylor polynomial admits the same representation as in the homogeneous case.