Geodesic
Rigidity and stability of the Leibniz and the chain rule
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Orthogonal series, approximation theory, and related problems, Tome 280 (2013), pp. 198-214.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study rigidity and stability properties of the Leibniz and chain rule operator equations. We describe which non-degenerate operators $V,T_1,T_2,A\colon C^k(\mathbb R)\to C(\mathbb R)$ satisfy equations of the generalized Leibniz and chain rule type for $f,g\in C^k(\mathbb R)$, namely, $V(f\cdot g)=(T_1f)\cdot g+f\cdot(T_2g)$ for $k=1$, $V(f\cdot g)=(T_1f)\cdot g+f\cdot(T_2g)+(Af)\cdot(Ag)$ for $k=2$, and $V(f\circ g)=(T_1f)\circ g\cdot(T_2g)$ for $k=1$. Moreover, for multiplicative maps $A$, we consider a more general version of the first equation, $V(f\cdot g)=(T_1f)\cdot(Ag)+(Af)\cdot(T_2g)$ for $k=1$. In all these cases, we completely determine all solutions. It turns out that, in any of the equations, the operators $V$, $T_1$ and $T_2$ must be essentially equal. We also consider perturbations of the chain and the Leibniz rule, $T(f\circ g)=Tf\circ g\cdot Tg+B(f\circ g,g)$ and $T(f\cdot g)=Tf\cdot g+f\cdot Tg+B(f,g)$, and show under suitable conditions on $B$ in the first case that $B=0$ and in the second case that the solution is a perturbation of the solution of the standard Leibniz rule equation. 
@article{TM_2013_280_a12,
     author = {Hermann K\"onig and Vitali Milman},
     title = {Rigidity and stability of the {Leibniz} and the chain rule},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {198--214},
     publisher = {mathdoc},
     volume = {280},
     year = {2013},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2013_280_a12/}
}
TY  - JOUR
AU  - Hermann König
AU  - Vitali Milman
TI  - Rigidity and stability of the Leibniz and the chain rule
JO  - Trudy Matematicheskogo Instituta imeni V.A. Steklova
PY  - 2013
SP  - 198
EP  - 214
VL  - 280
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/TM_2013_280_a12/
LA  - en
ID  - TM_2013_280_a12
ER  - 
%0 Journal Article
%A Hermann König
%A Vitali Milman
%T Rigidity and stability of the Leibniz and the chain rule
%J Trudy Matematicheskogo Instituta imeni V.A. Steklova
%D 2013
%P 198-214
%V 280
%I mathdoc
%U http://geodesic.mathdoc.fr/item/TM_2013_280_a12/
%G en
%F TM_2013_280_a12
Hermann König; Vitali Milman. Rigidity and stability of the Leibniz and the chain rule. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Orthogonal series, approximation theory, and related problems, Tome 280 (2013), pp. 198-214. http://geodesic.mathdoc.fr/item/TM_2013_280_a12/

[1] Aczél J., Lectures on functional equations and their applications, Acad. Press, New York, 1966 | MR | Zbl

[2] Artstein-Avidan S., Faifman D., Milman V., “On multiplicative maps of continuous and smooth functions”, Geometric aspects of functional analysis, Israel seminar 2006–2010, Lect. Notes Math., 2050, Springer, Berlin, 2012, 35–59 | DOI | MR | Zbl

[3] Artstein-Avidan S., König H., Milman V., “The chain rule as a functional equation”, J. Funct. Anal., 259 (2010), 2999–3024 | DOI | MR | Zbl

[4] Kestelman H., “On the functional equation $f(x+y) = f(x) + f(y)$”, Fundam. math., 34 (1947), 144–147 | MR | Zbl

[5] König H., Milman V., “Characterizing the derivative and the entropy function by the Leibniz rule”, J. Funct. Anal., 261 (2011), 1325–1344 | DOI | MR | Zbl

[6] König H., Milman V., “An operator equation generalizing the Leibniz rule for the second derivative”, Geometric aspects of functional analysis, Israel seminar 2006–2010, Lect. Notes Math., 2050, Springer, Berlin, 2012, 279–299 | DOI | MR | Zbl

[7] König H., Milman V., “An operator equation characterizing the Laplacian”, Algebra i analiz, 24:4 (2012), 137–155