Monotonicities of Quasi-Normed Orlicz Spaces

Monotonicities of Quasi-Normed Orlicz Spaces

In this paper, we introduce a new Orlicz function, namely a <i>b</i>-Orlicz function, which is not necessarily convex. The Orlicz spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi&g...

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Bibliographic Details
Main Authors: Dong Ji, Yunan Cui
Format: Article
Language:English
Published: MDPI AG 2024-10-01
Series:Axioms
Subjects:
<i>b</i>-Orlicz function
Orlicz spaces
quasi-norm
strict monotonicity
lower local uniform monotonicity
upper local uniform monotonicity
Online Access:https://www.mdpi.com/2075-1680/13/10/696
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Summary:In this paper, we introduce a new Orlicz function, namely a <i>b</i>-Orlicz function, which is not necessarily convex. The Orlicz spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mo>Φ</mo></msup></semantics></math></inline-formula> generated by the <i>b</i>-Orlicz function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Φ</mo></semantics></math></inline-formula> equipped with a Luxemburg quasi-norm contain both classical spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mi>p</mi></msup><mrow><mo stretchy="false">(</mo><mi>p</mi><mo>≥</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>L</mi><mi>p</mi></msup><mrow><mo stretchy="false">(</mo><mn>0</mn><mo><</mo><mi>p</mi><mo><</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>. The Orlicz spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mo>Φ</mo></msup></semantics></math></inline-formula> are quasi-Banach spaces. Some basic properties in quasi-normed Orlicz spaces are discussed, and the criteria that a quasi-normed Orlicz space is strictly monotonic and lower (upper) locally uniformly monotonic are given.
ISSN:2075-1680

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