Abstract :
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)(T g)(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) =
det f
(x)
p
H
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(det f
(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.
© 2011 Elsevier Inc. All rights reserved.
