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

Hermann König*, Vitali Milman

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Let k∈N. Consider maps T:Ck(R)→C(R) and A1,A2:Ck-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).

Original languageEnglish
Pages (from-to)2546-2569
Number of pages24
JournalJournal of Functional Analysis
Issue number4
StatePublished - 15 Feb 2014


  • Chain rule
  • Operator equations
  • Schwarzian derivative


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