Orlicz functions that do not satisfy the $\Delta _2$-condition and high order Gâteaux 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 $\limsup _{t\to 0}M(t)/t^p \!>0$ for some $p\ge 1$, there exists a positive sequence $T=(t_k)_{k=1}^\infty$ tending to zero and such that \begin{equation*} \sup _{k\in \mathbb {N}}\frac {M(ct_k)}{M(t_k)} <\infty ,\text { for all }c>1, \end{equation*} 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.