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).

Taylor formula for homogeneous groups and applications (Italian Title: La Formula di Taylor per i Gruppi Omogenei ed Applicazioni) / A. Bonfiglioli. - In: SEMINARI DI ANALISI MATEMATICA BRUNO PINI. - ISSN 2240-2837. - STAMPA. - Anno Accademico 2007/2008:(2008), pp. 43-69.

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

#### Abstract

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).
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2008
Taylor formula for homogeneous groups and applications (Italian Title: La Formula di Taylor per i Gruppi Omogenei ed Applicazioni) / A. Bonfiglioli. - In: SEMINARI DI ANALISI MATEMATICA BRUNO PINI. - ISSN 2240-2837. - STAMPA. - Anno Accademico 2007/2008:(2008), pp. 43-69.
A. Bonfiglioli
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11585/128949`
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