The chain rule functional equation on RnRn

We study operators T   from C1(Rn,Rn)C1(Rn,Rn) to C(Rn,L(Rn,Rn))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⩾2n⩾2 the operator T is of the form(Tf)(x)=|detf′(x)p|H(f(x))f′(x)H(x)−1(Tf)(x)=|detf′(x)|pH(f(x))f′(x)H(x)−1 for a suitable p⩾0p⩾0 and H∈C(Rn,GL(n))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=1n=1. In this setting the non-commutativity of the linear operators L(Rn,Rn)L(Rn,Rn) from RnRn to RnRn creates additional difficulties but also clarifies and enriches the understanding of the problem.