Rigidity of the Chain Rule and Nearly Submultiplicative Functions

Abstract

Assume that $T: C^{1}(\mathbb{R}) \rightarrow C(\mathbb{R})$ nearly satisfies the chain rule in the sense that

$$\displaystyle{\vert T(\,f \circ g)(x) - (Tf)(g(x))(Tg)(x)\vert \leq S(x,(\,f \circ g)(x),g(x))}$$

holds for all $f,g \in C^{1}(\mathbb{R})$ and $x \in \mathbb{R}$, where $S: \mathbb{R}^{3} \rightarrow \mathbb{R}$ is a suitable fixed function. We show under a weak non-degeneracy and a weak continuity assumption on T that S may be chosen to be 0, i.e. that T satisfies the chain rule operator equation, the solutions of which are explicitly known. We also determine the solutions of one-sided chain rule inequalities like

$$\displaystyle{T(\,f \circ g)(x) \leq (Tf)(g(x))(Tg)(x) + S(x,(\,f \circ g)(x),g(x))}$$

under a further localization assumption. To prove the above results, we investigate the solutions of nearly submultiplicative inequalities on $\mathbb{R}$

$$\displaystyle{\phi (\alpha \beta ) \leq \phi (\alpha )\phi (\beta ) + d}$$

and characterize the nearly multiplicative functions on $\mathbb{R}$

$$\displaystyle{\vert \phi (\alpha \beta ) -\phi (\alpha )\phi (\beta )\vert \leq d\ }$$

under weak restrictions on ϕ.