## Abstract

Let k∈N. Consider maps T:C^{k}(R)→C(R) and A_{1},A_{2}:C^{k-1}(R)→C(R) satisfying the operator equation T(f{ring operator}g)=(Tf){ring operator}g{dot operator}A_{1}g+(A_{2}f){ring operator}g{dot operator}Tg for all f,g∈Ck(R). We determine the form of all solutions (T, A_{1}, A_{2}) 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, A_{1} and A_{2} are known. T, A_{1} and A_{2} 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+2}H{ring operator}f-|f'|^{p}]{sgnf'}, A_{1}f=(f')^{2}A_{2}f,A_{2}f=|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 A_{1} and A_{2}. 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}).

Original language | English |
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Pages (from-to) | 2546-2569 |

Number of pages | 24 |

Journal | Journal of Functional Analysis |

Volume | 266 |

Issue number | 4 |

DOIs | |

State | Published - 15 Feb 2014 |

## Keywords

- Chain rule
- Operator equations
- Schwarzian derivative