The chain rule functional equation on Rn

Hermann König*, Vitali Milman

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


We study operators T from C1(Rn,Rn) to C(Rn,L(Rn,Rn)) satisfying the "chain rule". T(f·g)(x)=((Tf)·g)(x)(Tg)(x);f,g∈C1(Rn,Rn),x∈Rn. Assuming a local surjectivity and non-degeneracy condition, we show that for n≥2 the operator T is of the form. (Tf)(x)=|detf'(x)|pH(f(x))f'(x)H(x)-1 for a suitable p≥0 and H∈C(Rn,GL(n)). For even n there might be an additional factor sgn(detf'(x)). This is the multidimensional extension of our results (Artstein-Avidan et al., 2010 [3]) for n=1. In this setting the non-commutativity of the linear operators L(Rn,Rn) from Rn to Rn creates additional difficulties but also clarifies and enriches the understanding of the problem.

Original languageEnglish
Pages (from-to)861-875
Number of pages15
JournalJournal of Functional Analysis
Issue number4
StatePublished - 15 Aug 2011


FundersFunder number
Fields Institute
Alexander von Humboldt-Stiftung
United States-Israel Binational Science Foundation200 6079
Israel Science Foundation387/09


    • Automorphisms of GL(n)
    • Chain rule in R
    • Functional equation


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