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Abstrakty
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.
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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
217-231
Opis fizyczny
Daty
wydano
2014
Twórcy
autor
- Mathematisches Seminar, Universität Kiel, 24098 Kiel, Germany
autor
- School of Mathematical Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv 69978, Israel
Bibliografia
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_4064-sm223-3-3