An Operator Equation Generalizing the Leibniz Rule for the Second Derivative

Part of the Lecture Notes in Mathematics book series (LNM, volume 2050)


We determine all operators \(T : {C}^{2}(\mathbb{R}) \rightarrow C(\mathbb{R})\) and \(A : {C}^{1}(\mathbb{R}) \rightarrow C(\mathbb{R})\) which satisfy the equation
$$T(f \cdot g) = (Tf) \cdot g + f \cdot (Tg) + (Af) \cdot (Ag)\ ;\quad f,g \in {C}^{2}(\mathbb{R}).$$
This operator equation models the second order Leibniz rule for (fg) with \(Af = \sqrt{2}f'\). Under a mild regularity and non-degeneracy assumption on A, we show that the operators T and A have to be of a very restricted type. In addition to the operator solutions S of the Leibniz rule derivation equation corresponding to A = 0,
$$S(f \cdot g) = (Sf) \cdot g + f \cdot (Sg)\ ;\quad f,g \in {C}^{2}(\mathbb{R})\text{ or }{C}^{1}(\mathbb{R})\,$$
which are of the form
$$Sf = bf' + af\ln \vert f\vert,\quad a,b \in C(\mathbb{R}),$$
T and A may be of the following three types
$$\begin{array}{lll} Tf & = \frac{1} {2}{d}^{2}f'' &,\ Af = d\,f' \\ Tf & = \frac{1} {2}{d}^{2}f{(\ln \vert f\vert )}^{2} &,\ Af = d\,f\ln \vert f\vert \\ Tf & = {d}^{2}f(\epsilon \vert f{\vert }^{p} - 1)&,\ Af = d\,f(\epsilon \vert f{\vert }^{p} - 1) \end{array}$$
for suitable continuous functions d, c and p and where ε is either 1 or sgnf and p ≥ − 1. The last operator solution is degenerate in the sense that T is a multiple of A. We also determine all solutions of (1) if T and A operate only on positive \({C}^{2}(\mathbb{R})\)-functions or \({C}^{2}(\mathbb{R})\)-functions which are nowhere zero.


Functional Equation Additive Function Operator Equation Operator Solution Leibniz Rule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We would like to thank the referee for valuable suggestions and remarks, in particular for pointing out a gap in the original localization argument. This led us to the example in the following section.


  1. 1.
    J. Aczél, in Lectures on Functional Equations and Their Applications. Mathematics in Science and Engineering, vol. 19 (Academic, New York, 1966)Google Scholar
  2. 2.
    J. Aczél, J. Dhombres, in Functional Equations in Several Variables. Encyclopedia of Mathematics and Its Applications, vol. 31 (Cambridge University Press, Cambridge, 1989)Google Scholar
  3. 3.
    S. Artstein-Avidan, H. König, V. Milman, The chain rule as a functional equation. J. Funct. Anal. 259, 2999–3024 (2010)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    H. Goldmann, P. Šemrl, Multiplicative derivations on C(X). Monatsh. Math. 121, 189–197 (1996)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    H. König, V. Milman, A functional equation characerizing the second derivative. J. Funct. Anal. 261, 876–896 (2011)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    H. König, V. Milman, Characterizing the derivative and the entropy function by the Leibniz rule, with an appendix by D. Faifman. J. Funct. Anal. 261, 1325–1344 (2011)MATHCrossRefGoogle Scholar
  7. 7.
    H. König, V. Milman, An operator equation characterizing the Laplacian. Algebra and Analysis (to appear)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Mathematisches SeminarChristian-Albrechts-Universität zu KielKielGermany
  2. 2.Sackler Faculty of Exact Sciences, Department of MathematicsTel Aviv UniversityTel AvivIsrael

Personalised recommendations