Rigidity and stability of the Leibniz and the chain rule

Abstract

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.