Cauchy Difference Operator in Some Orlicz Spaces

Stanisław Siudut ()
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Stanisław Siudut: Pedagogical University, Institute of Mathematics

Chapter Chapter 17 in Ulam Type Stability, 2019, pp 365-381 from Springer

Abstract: Abstract Let (G, ⋅, λ) be a measurable group with a complete, left-invariant and finite measure λ. If φ is a convex φ-function satisfying conditions φ(u)∕u → 0 as u → 0, φ(u)∕u →∞ as u →∞, f : G → ℝ $$f:G \rightarrow \mathbb {R}$$ and the Cauchy difference C f ( x , y ) = f ( x ⋅ y ) − f ( x ) − f ( y ) $${\mathcal {C}}{}f(x,y)=f(x\cdot y)-f(x)-f(y)$$ of f belongs to L λ × λ φ ( G × G , ℝ ) $${\mathcal {L}}^{\varphi }_{\lambda \times \lambda }(G \times G,\mathbb {R})$$ , then there exists unique additive A : G → ℝ $$A:G \to \mathbb {R}$$ such that f − A ∈ L λ φ ( G , ℝ ) $$f-A \in {\mathcal {L}}^{\varphi }_{\lambda }(G,\mathbb {R})$$ . Moreover, ∥ f − A ∥ φ ≤ K ∥ C f ∥ φ , $$\Vert f-A \Vert _{\varphi } \leq K \Vert {\mathcal {C}}{}f \Vert _{\varphi },$$ where K = 1 if λ(G) ≥ 1, K = 1 + (λ(G))−1 if λ(G)

Keywords: Cauchy difference operator; Orlicz spaces; Ulam stability; Primary 39B82; Secondary 46E30 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/978-3-030-28972-0_17

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