Submultiplicative functions and operator inequalities

Hermann König ¹   ; Vitali Milman ²

¹ Mathematisches Seminar Universität Kiel 24098 Kiel, Germany
² School of Mathematical Sciences Tel Aviv University Ramat Aviv, Tel Aviv 69978, Israel

Studia Mathematica, Tome 223 (2014) no. 3, pp. 217-231

Résumé

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 $0 A \le 1$.

Détail

DOI : 10.4064/sm223-3-3

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