A functional equation characterizing the second derivative

Consider an operator T:C2(R)→C(R) and isotropic maps A1,A2:C1(R)→C(R) such that the functional equationT(f∘g)=(Tf)∘g⋅A1g+(A2f)∘g⋅Tg;f,g∈C2(R) is satisfied on C2(R). The equation models the chain rule for the second derivative, in which case A1g=g′2 and A2f=f′. We show under mild non-degeneracy conditions - which imply that A1 and A2 are very different from T - that A1 and A2 must be of the very restricted form A1f=f′⋅A2f, A2f=|f′|p or sgn(f′)|f′|p, with p⩾1, and that any solution operator T has the formTf(x)=cA2(f(x))f′(x)f″(x)+(H(f(x))f′(x)−H(x))A2(f(x)),x∈R for some constant c∈R and some continuous function H. Conversely, any such map T satisfies the functional equation. Under some natural normalization condition, the only solution of the functional equation is Tf=f″ which means that the composition rule with some normalization condition characterizes the second derivative. If c=0, T does not depend on the second derivative. In this case, there are further solutions of the functional equation which we determine, too.