Orlicz functions that do not satisfy the ${\Delta }_2$-condition and high order Gateaux smoothness in $h_M(\Gamma )$

Abstract

We study Orlicz functions that do not satisfy the ${\Delta }_2$-condition at zero. We prove that for every Orlicz function $M$ such that ${\mathrm{limsup}}_{t\to 0}M(t)/t^p>0$ for some $p\ge 1$, there exists a positive sequence $T=(t_k{)}_{k=1}^{\infty }$ tending to zero and such that $$\underset{k\in \mathbb{N}}{\sup }\frac{M(c{t}_k)}{M(t_k)}<\infty ,\text{ for all }c>1,$$ that is, $M$ satisfies the ${\Delta }_2$ condition with respect to $T$. Consequently, we show that for each Orlicz function with lower Boyd index ${\alpha }_M<\infty$ there exists an Orlicz function $N$ such that: (a) there exists a positive sequence $T=(t_k{)}_{k=1}^{\infty }$ tending to zero such that $N$ satisfies the ${\Delta }_2$ condition with respect to $T$, and (b) the space $h_N$ is isomorphic to a subspace of $h_M$ generated by one vector. We apply this result to find the maximal possible order of Gâteaux differentiability of a continuous bump function on the Orlicz space $h_M(\Gamma )$ for $\Gamma$ uncountable.

References in corpus (1)

• A note on fragmentability and weak-G_delta sets