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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2024-10-01
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| author | Dong Ji Yunan Cui |
| author_facet | Dong Ji Yunan Cui |
| author_sort | Dong Ji |
| collection | DOAJ |
| description | 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. |
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| institution | OA Journals |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-10-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-79caffda89a046dfbc272bc4b43023332025-08-20T02:11:04ZengMDPI AGAxioms2075-16802024-10-01131069610.3390/axioms13100696Monotonicities of Quasi-Normed Orlicz SpacesDong Ji0Yunan Cui1Department of Mathematics, Harbin University of Science and Technology, Harbin 150080, ChinaDepartment of Mathematics, Harbin University of Science and Technology, Harbin 150080, ChinaIn 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.https://www.mdpi.com/2075-1680/13/10/696<i>b</i>-Orlicz functionOrlicz spacesquasi-normstrict monotonicitylower local uniform monotonicityupper local uniform monotonicity |
| spellingShingle | Dong Ji Yunan Cui Monotonicities of Quasi-Normed Orlicz Spaces Axioms <i>b</i>-Orlicz function Orlicz spaces quasi-norm strict monotonicity lower local uniform monotonicity upper local uniform monotonicity |
| title | Monotonicities of Quasi-Normed Orlicz Spaces |
| title_full | Monotonicities of Quasi-Normed Orlicz Spaces |
| title_fullStr | Monotonicities of Quasi-Normed Orlicz Spaces |
| title_full_unstemmed | Monotonicities of Quasi-Normed Orlicz Spaces |
| title_short | Monotonicities of Quasi-Normed Orlicz Spaces |
| title_sort | monotonicities of quasi normed orlicz spaces |
| topic | <i>b</i>-Orlicz function Orlicz spaces quasi-norm strict monotonicity lower local uniform monotonicity upper local uniform monotonicity |
| url | https://www.mdpi.com/2075-1680/13/10/696 |
| work_keys_str_mv | AT dongji monotonicitiesofquasinormedorliczspaces AT yunancui monotonicitiesofquasinormedorliczspaces |