Extension of submultiplicativity and supermultiplicativity of Orlicz functions. (Q5932434)

The authors consider an Orlicz function \(\varphi \), submultiplicative (supermultiplicative) at infinity, and they prove that a necessary and sufficient condition for the existence of Orlicz functions \(\psi \), equivalent to \(\varphi \) at infinity and submultiplicative (supermultiplicative) on the whole of \(\mathbb R^n\), is the \(\Delta _2\)-condition for \(\varphi \) at infinity. They also prove an analogous theorem for an Orlicz function \(\varphi \) supermultiplicative at the origin, when the necessary and sufficient condition turns out to be \(\varphi ^{*}\in \Delta _2\) at the origin, where \(\varphi ^{*}\) is the conjugate Orlicz function to \(\varphi \). For Orlicz functions submultiplicative at infinity (supermultiplicative at the origin) the equivalent function \(\psi \) even satisfies \(\lim _{t\to 0}\psi (t)/t=0\) (\(\lim _{t\to \infty}\psi (t)/t=\infty \)).NEWLINENEWLINEThe sufficiency for functions submultiplicative at infinity or at zero was proved by \textit{H. Hudzik} and \textit{L. Maligranda} [Indag. Math., New Ser. 3, 313--321 (1992; Zbl 0802.46042)].