Operator equations and domain dependence, the case of the Schwarzian derivative

Let k∈N. Consider maps T:C^k(R)→C(R) and A₁,A₂:C^k-1(R)→C(R) satisfying the operator equation T(f{ring operator}g)=(Tf){ring operator}g{dot operator}A₁g+(A₂f){ring operator}g{dot operator}Tg for all f,g∈Ck(R). We determine the form of all solutions (T, A₁, A₂) of this equation and study their dependence on the domain of T. For k=2 the equation models the second derivative chain rule and the solutions T, A₁ and A₂ are known. T, A₁ and A₂ are closely related local operators. We consider the case k≥3 and show that variants of the Schwarzian derivative appear in T if T depends non-trivially on the third derivative: there are d≠0, p≥2 and H∈C(R) such that Tf=[d(f‴f'^p-1-3/2(f″)2f'p-2)+|f'|^p+2H{ring operator}f-|f'|^p]{sgnf'}, A₁f=(f')²A₂f,A₂f=|f'|p{sgnf'}. The term {sgnf^'} may be present or not. For k≥4, there are no solutions T depending non-trivially on f^(k). The natural domains for T turn out to be Cl(R) for l∈{1, 2, 3} and C1(R) for A₁ and A₂. If T is restricted to Ck(R)-functions with non-vanishing derivative, we may allow p≥0. For p=0, the main term in T is the Schwarzian derivative S, Sf=(f‴f'-3/2(f″/f')²).