TY - JOUR

T1 - Rigidity and stability of the Leibniz and the chain rule

AU - König, Hermann

AU - Milman, Vitali

N1 - Funding Information:
We would like to thank D. Faifman for some useful discussions, in particular concerning Theorem 9. This work was supported in part by the Alexander von Humboldt Foundation, by ISF grant 387/09 and by BSF grant 200 6079.

PY - 2013

Y1 - 2013

N2 - We study rigidity and stability properties of the Leibniz and chain rule operator equations. We describe which non-degenerate operators V, T1, T2,A: Ck(ℝ) → C(ℝ) satisfy equations of the generalized Leibniz and chain rule type for f, g ∈ Ck(ℝ), namely, V (f · g) = (T1f) · g + f · (T2g) for k = 1, V (f · g) = (T1f) · g + f · (T2g) + (Af) · (Ag) for k = 2, and V (f ○ g) = (T1f) ○ g · (T2g) for k = 1. Moreover, for multiplicative maps A, we consider a more general version of the first equation, V (f · g) = (T1f) · (Ag) + (Af) · (T2g) for k = 1. In all these cases, we completely determine all solutions. It turns out that, in any of the equations, the operators V, T1 and T2 must be essentially equal. We also consider perturbations of the chain and the Leibniz rule, T (f ○ g) = Tf ○ g · Tg + B(f ○ g, g) and T (f · g) = Tf · g + f · 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.

AB - We study rigidity and stability properties of the Leibniz and chain rule operator equations. We describe which non-degenerate operators V, T1, T2,A: Ck(ℝ) → C(ℝ) satisfy equations of the generalized Leibniz and chain rule type for f, g ∈ Ck(ℝ), namely, V (f · g) = (T1f) · g + f · (T2g) for k = 1, V (f · g) = (T1f) · g + f · (T2g) + (Af) · (Ag) for k = 2, and V (f ○ g) = (T1f) ○ g · (T2g) for k = 1. Moreover, for multiplicative maps A, we consider a more general version of the first equation, V (f · g) = (T1f) · (Ag) + (Af) · (T2g) for k = 1. In all these cases, we completely determine all solutions. It turns out that, in any of the equations, the operators V, T1 and T2 must be essentially equal. We also consider perturbations of the chain and the Leibniz rule, T (f ○ g) = Tf ○ g · Tg + B(f ○ g, g) and T (f · g) = Tf · g + f · 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.

UR - http://www.scopus.com/inward/record.url?scp=84876447890&partnerID=8YFLogxK

U2 - 10.1134/S0081543813010136

DO - 10.1134/S0081543813010136

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:84876447890

SN - 0081-5438

VL - 280

SP - 191

EP - 207

JO - Proceedings of the Steklov Institute of Mathematics

JF - Proceedings of the Steklov Institute of Mathematics

IS - 1

ER -