Orlicz functions that do not satisfy the $\Delta _2$-condition and high order Gâteaux smoothness in $h_M(\Gamma )$
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- by Milen Ivanov, Stanimir Troyanski and Nadia Zlateva;
- Proc. Amer. Math. Soc. 152 (2024), 2007-2019
- DOI: https://doi.org/10.1090/proc/16664
- Published electronically: February 23, 2024
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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.
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Bibliographic Information
- Milen Ivanov
- Affiliation: Radiant Life Technologies Ltd., Nicosia, Cyprus
- MR Author ID: 624831
- Email: milen@radiant-life-technologies.com
- Stanimir Troyanski
- Affiliation: Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Bl. 8, Acad. G. Bonchev Str., 1113 Sofia, Bulgaria; and Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100 Murcia, Spain
- MR Author ID: 174580
- Email: stroya@um.es
- Nadia Zlateva
- Affiliation: Sofia University, Faculty of Mathematics and Informatics, 5 James Bourchier Blvd., 1164 Sofia, Bulgaria
- MR Author ID: 357306
- ORCID: 0000-0002-6223-9055
- Email: zlateva@fmi.uni-sofia.bg
- Received by editor(s): June 17, 2023
- Received by editor(s) in revised form: September 2, 2023, and September 7, 2023
- Published electronically: February 23, 2024
- Additional Notes: The study of the first and third authors was financed by the European Union-NextGenerationEU, through the National Recovery and Resilience Plan of the Republic of Bulgaria, project SUMMIT BG-RRP-2.004-0008-C01. The second author was supported by Grant PID2021-122126NB-C32 financed by MCIN/AEI /10.13039/501100011033 / FEDER, UE.
- Communicated by: Stephen Dilworth
- © Copyright 2024 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 2007-2019
- MSC (2020): Primary 46B45; Secondary 46E30, 49J50
- DOI: https://doi.org/10.1090/proc/16664
- MathSciNet review: 4728470
Dedicated: Dedicated to Rumen Maleev, in memoriam