Submultiplicative functions and operator inequalities
Volume 223 / 2014
Abstract
Let $T:C^1(\mathbb{R}) \to C(\mathbb{R})$ be an operator satisfying the “chain rule inequality” \[ T(f \circ g) \le (Tf) \circ g \cdot Tg, \quad f,g \in C^1(\mathbb{R}). \] Imposing a weak continuity and a non-degeneracy condition on $T$, we determine the form of all maps $T$ satisfying this inequality together with $T(-\mathop{\rm Id}\nolimits)(0)< 0$. They have the form \[ Tf = \begin{cases}(H \circ f / H) f'^p, & f' \ge 0,\\ -A (H \circ f / H ) |f'|^p, & f' < 0, \end{cases} \] with $p>0$, $H \in C(\mathbb{R})$, $A \ge 1$. For $A=1$, these are just the solutions of the chain rule operator equation. To prove this, we characterize the submultiplicative, measurable functions $K$ on $\mathbb{R}$ which are continuous at $0$ and $1$ and satisfy $K(-1)< 0< K(1)$. Any such map $K$ has the form \[ K(\alpha) = \begin{cases} \alpha^p, &\alpha \ge 0,\\ -A |\alpha|^p,&\alpha < 0,\end{cases} \] with $A \ge 1$ and $p>0$. Corresponding statements hold in the supermultiplicative case with $0< A \le 1$.
