Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2023
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2306.09654 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913455510913024 |
|---|---|
| author | Ivanov, Milen Troyanski, Stanimir Zlateva, Nadia |
| author_facet | Ivanov, Milen Troyanski, Stanimir Zlateva, Nadia |
| contents | We study Orlicz functions that do not satisfy the $Δ_2$-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 $Δ_2$ condition with respect to $T$.
Consequently, we show that for each Orlicz function with lower Boyd index $α_M < \infty$ 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 $Δ_2$ 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 $h_M(Γ)$ for $Γ$ uncountable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_09654 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Orlicz functions that do not satisfy the $Δ_2$-condition and high order Gateaux smoothness in $ h_M(Γ) $ Ivanov, Milen Troyanski, Stanimir Zlateva, Nadia Functional Analysis 46B45, 46E30, 49J50 We study Orlicz functions that do not satisfy the $Δ_2$-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 $Δ_2$ condition with respect to $T$. Consequently, we show that for each Orlicz function with lower Boyd index $α_M < \infty$ 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 $Δ_2$ 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 $h_M(Γ)$ for $Γ$ uncountable. |
| title | Orlicz functions that do not satisfy the $Δ_2$-condition and high order Gateaux smoothness in $ h_M(Γ) $ |
| topic | Functional Analysis 46B45, 46E30, 49J50 |
| url | https://arxiv.org/abs/2306.09654 |