The chain rule functional equation on Rn

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.