Orlicz functions that do not satisfy the Δ₂-condition and high order Gâteaux smoothness in ℎ_{𝑀}(Γ)

Abstract

We study Orlicz functions that do not satisfy the Δ 2 \Delta _2 -condition at zero. We prove that for every Orlicz function M M such that lim sup t → 0 M ( t ) / t p > 0 \limsup _{t\to 0}M(t)/t^p \!>0 for some p ≥ 1 p\ge 1 , there exists a positive sequence T = ( t k ) k = 1 ∞ T=(t_k)_{k=1}^\infty tending to zero and such that sup k ∈ N M ( c t k ) M ( t k ) > ∞ , for all c > 1 , \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 M satisfies the Δ 2 \Delta _2 condition with respect to T T . Consequently, we show that for each Orlicz function with lower Boyd index α M > ∞ \alpha _M > \infty there exists an Orlicz function N N such that: (a) there exists a positive sequence T = ( t k ) k = 1 ∞ T=(t_k)_{k=1}^\infty tending to zero such that N N satisfies the Δ 2 \Delta _2 condition with respect to T T , and (b) the space h N h_N is isomorphic to a subspace of h M 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 ( Γ ) h_M(\Gamma ) for Γ \Gamma uncountable.