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Cauchy Difference Operator in Some Orlicz Spaces

  • Stanisław SiudutEmail author
Chapter

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 →, \(f:G \rightarrow \mathbb {R}\) and the Cauchy difference \({\mathcal {C}}{}f(x,y)=f(x\cdot y)-f(x)-f(y)\) of f belongs to \({\mathcal {L}}^{\varphi }_{\lambda \times \lambda }(G \times G,\mathbb {R})\), then there exists unique additive \(A:G \to \mathbb {R}\) such that \(f-A \in {\mathcal {L}}^{\varphi }_{\lambda }(G,\mathbb {R})\). Moreover, \(\Vert f-A \Vert _{\varphi } \leq K \Vert {\mathcal {C}}{}f \Vert _{\varphi },\) where K = 1 if λ(G) ≥ 1, K = 1 + (λ(G))−1 if λ(G) < 1. Similar result we also obtain without associativity of ⋅ but with \(f \in {\mathcal {L}}^{\varphi }_{\lambda }(G,\mathbb {R})\) and with measurability of \({\mathcal {C}}{}f\). In this case A = 0 and the Cauchy difference \({\mathcal {C}} : L^{\varphi }(G,\mathbb {R}) \rightarrow L^{\varphi }(G \times G,\mathbb {R})\) is linear continuous and continuously invertible on its image, where \(L ^{\varphi }(G,\mathbb {R})\) denotes the space of equivalence classes of functions in \({\mathcal {L}}^{\varphi }(G,\mathbb {R})\). Moreover, \({\mathcal {C}}\) is compact iff \(L ^{\varphi }(G,\mathbb {R})\) has a finite dimension.

Let (G, ⋅, λ) be a measurable group with a complete, left-invariant and σ-finite measure λ such that λ(G) = . If φ is a φ-function, \(f:G \rightarrow \mathbb {R}\) and \({\mathcal {C}}{}f\in {\mathcal {L}}^{\varphi }_{\lambda \times \lambda }(G\times G,\mathbb {R})\), then there exists a unique additive \(A:G\rightarrow \mathbb {R}\), which is equal to f λ a.e.

Keywords

Cauchy difference operator Orlicz spaces Ulam stability 

Mathematics Subject Classification (2010)

Primary 39B82; Secondary 46E30 

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Authors and Affiliations

  1. 1.Institute of MathematicsPedagogical UniversityKrakówPoland

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