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 $\limsup_{t\to0}M(t)/t^p >0$ for some $p\ge1$, there exists a positive sequence $T=(t_k)_{k=1}^\infty$ tending to zero and such that $$ \sup_{k\in\mathbb{N}}\frac{M(ct_k)}{M(t_k)} <\infty,\text{ for all }c>1, $$ 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 G\^ateaux differentiability of a continuous bump function on the Orlicz space $h_M(\Gamma)$ for $\Gamma$ uncountable.