Submultiplicative functions and operator inequalities

Hermann König ; Vitali Milman

Studia Mathematica, Tome 223 (2014), p. 217-231 / Harvested from The Polish Digital Mathematics Library

Résumé

Let T: C¹(ℝ) → C(ℝ) be an operator satisfying the “chain rule inequality” T(f∘g) ≤ (Tf)∘g⋅Tg, f,g ∈ C¹(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(-Id)(0) < 0. They have the form Tf = ⎧ $(H∘f/H)f^{\prime p}$, f’ ≥ 0, ⎨ ⎩ $-A(H∘f/H)|f^{\prime }|^{p}$, f’ < 0, with p > 0, H ∈ C(ℝ), A ≥ 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 ℝ which are continuous at 0 and 1 and satisfy K(-1) < 0 < K(1). Any such map K has the form K(α) = ⎧ $α^{p}$, α ≥ 0, ⎨ ⎩ $-A|α|^{p}$, α < 0, with A ≥ 1 and p > 0. Corresponding statements hold in the supermultiplicative case with 0 < A ≤ 1.

Détail

Publié le : 2014-01-01

Zbl 1317.39034

EUDML-ID : urn:eudml:doc:285878