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Submultiplicative functions and operator inequalities

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König H., Milman V.

Studia Mathematica

2014

tom 223

nr 3

217-231

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 = ⎧ $(H∘f/H)f'^{p}$, f' ≥ 0, ⎨ ⎩ $-A(H∘f/H)|f'|^{p}$, 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.

Ograniczanie wyników

1 Studia Mathematica

1 König H.

1 Milman V.

1 2014

Wyniki wyszukiwania

Submultiplicative functions and operator inequalities