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
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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. 
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     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},
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     year = {2013},
     volume = {280},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TM_2013_280_a12/}
}
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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/

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