Taylor formula for homogeneous groups and applications (Italian Title: La Formula di Taylor per i Gruppi Omogenei ed Applicazioni)

In this paper, we provide a Taylor formula with integral remainder in the setting of homogeneous groups in the sense of Folland and Stein. This formula allows us to give a simplified proof of the so-called `Taylor inequality'. As a by-product, we furnish an explicit expression for the relevant Taylor polynomials. Applications are provided. Among others, it is given a sufficient condition for the real-analiticity of a function whose higher order derivatives (in the sense of the Lie algebra) satisfy a suitable growth condition. Moreover, we prove the `L-harmonicity' of the Taylor polynomials related to a `L-harmonic' function, when L is a general homogenous left-invariant differential operator on a homogeneous group. (This result is one of the ingredients for obtaining Schauder estimates related to L).