TY - JOUR
T1 - The chain rule functional equation on Rn
AU - König, Hermann
AU - Milman, Vitali
N1 - Funding Information:
* Corresponding author. E-mail address: [email protected] (H. König). 1 Supported in part by the Fields Institute. 2 Supported in part by the Alexander von Humboldt Foundation, by the Fields Institute, by ISF grant 387/09 and by BSF grant 200 6079.
PY - 2011/8/15
Y1 - 2011/8/15
N2 - 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.
AB - 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.
KW - Automorphisms of GL(n)
KW - Chain rule in R
KW - Functional equation
UR - http://www.scopus.com/inward/record.url?scp=79956012771&partnerID=8YFLogxK
U2 - 10.1016/j.jfa.2011.01.006
DO - 10.1016/j.jfa.2011.01.006
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AN - SCOPUS:79956012771
SN - 0022-1236
VL - 261
SP - 861
EP - 875
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 4
ER -