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

Let k∈Nk∈N. Consider maps T:Ck(R)→C(R)T:Ck(R)→C(R) and A1,A2:Ck−1(R)→C(R)A1,A2:Ck−1(R)→C(R) satisfying the operator equationT(f∘g)=(Tf)∘g⋅A1g+(A2f)∘g⋅TgT(f∘g)=(Tf)∘g⋅A1g+(A2f)∘g⋅Tg for all f,g∈Ck(R)f,g∈Ck(R). We determine the form of all solutions (T,A1,A2)(T,A1,A2) of this equation and study their dependence on the domain of T  . For k=2k=2 the equation models the second derivative chain rule and the solutions T  , A1A1 and A2A2 are known. T  , A1A1 and A2A2 are closely related local operators. We consider the case k⩾3k⩾3 and show that variants of the Schwarzian derivative appear in T if T   depends non-trivially on the third derivative: there are d≠0d≠0, p⩾2p⩾2 and H∈C(R)H∈C(R) such thatTf=[d(f‴f′p−1−32(f″)2f′p−2)+|f′|p+2H∘f−|f′|p]{sgnf′},A1f=(f′)2A2f,A2f=|f′|p{sgnf′}. The term {sgnf′} may be present or not. For k⩾4k⩾4, there are no solutions T   depending non-trivially on f(k)f(k). The natural domains for T   turn out to be Cl(R)Cl(R) for l∈{1,2,3}l∈{1,2,3} and C1(R)C1(R) for A1A1 and A2A2. If T   is restricted to Ck(R)Ck(R)-functions with non-vanishing derivative, we may allow p⩾0p⩾0. For p=0p=0, the main term in T is the Schwarzian derivative S  , Sf=(f‴f′−32(f″f′)2).