Taylor formula for homogeneous groups and applications

In this paper, we provide a Taylor formula with integral remainder in the setting of homogeneous groups in the sense of Folland and Stein (Hardy spaces on homogeneous groups. Mathematical notes, vol 28. Princeton University Press, Princeton, 1982). 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).