# Cauchy Difference Operator in Some Orlicz Spaces

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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