Hardy Inequalities and Interrelations of Fractional Triebel–Lizorkin Spaces in a Bounded Uniform Domain
The interrelations of Triebel–Lizorkin spaces on smooth domains of Euclidean space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n<...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/4/637 |
| _version_ | 1797478373695422464 |
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| author | Jun Cao Yongyang Jin Yuanyuan Li Qishun Zhang |
| author_facet | Jun Cao Yongyang Jin Yuanyuan Li Qishun Zhang |
| author_sort | Jun Cao |
| collection | DOAJ |
| description | The interrelations of Triebel–Lizorkin spaces on smooth domains of Euclidean space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula> are well-established, whereas only partial results are known for the non-smooth domains. In this paper, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> is a non-smooth domain of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula> that is bounded and uniform. Suppose <i>p</i>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>n</mi><msub><mrow><mo stretchy="false">(</mo><mfrac><mn>1</mn><mi>p</mi></mfrac><mo>−</mo><mfrac><mn>1</mn><mi>q</mi></mfrac><mo stretchy="false">)</mo></mrow><mo>+</mo></msub><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><msub><mrow><mo stretchy="false">(</mo><mfrac><mn>1</mn><mi>p</mi></mfrac><mo>−</mo><mfrac><mn>1</mn><mi>q</mi></mfrac><mo stretchy="false">)</mo></mrow><mo>+</mo></msub><mo>:</mo><mo>=</mo><mo movablelimits="true" form="prefix">max</mo><mrow><mo stretchy="false">{</mo><mi>n</mi><mrow><mo stretchy="false">(</mo><mfrac><mn>1</mn><mi>p</mi></mfrac><mo>−</mo><mfrac><mn>1</mn><mi>q</mi></mfrac><mo stretchy="false">)</mo></mrow><mo>,</mo><mn>0</mn><mo stretchy="false">}</mo></mrow></mrow></semantics></math></inline-formula>. The authors show that three typical types of fractional Triebel–Lizorkin spaces, on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula>: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>F</mi><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow><mi>s</mi></msubsup><mrow><mo stretchy="false">(</mo><mo>Ω</mo><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mover accent="true"><mi>F</mi><mo>˚</mo></mover><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow><mi>s</mi></msubsup><mrow><mo stretchy="false">(</mo><mo>Ω</mo><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><msubsup><mover accent="true"><mi>F</mi><mo>˜</mo></mover><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow><mi>s</mi></msubsup><mrow><mo stretchy="false">(</mo><mo>Ω</mo><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, defined via the restriction, completion and supporting conditions, respectively, are identical if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> is E-thick and supports some Hardy inequalities. Moreover, the authors show the condition that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> is E-thick can be removed when considering only the density property <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>F</mi><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow><mi>s</mi></msubsup><mrow><mo stretchy="false">(</mo><mo>Ω</mo><mo stretchy="false">)</mo></mrow><mo>=</mo><msubsup><mover accent="true"><mi>F</mi><mo>˚</mo></mover><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow><mi>s</mi></msubsup><mrow><mo stretchy="false">(</mo><mo>Ω</mo><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, and the condition that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> supports Hardy inequalities can be characterized by some Triebel–Lizorkin capacities in the special case of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mi>q</mi><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula>. |
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| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T21:30:59Z |
| publishDate | 2022-02-01 |
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| series | Mathematics |
| spelling | doaj.art-efaf66c9fd4f4c82989edea7bdbc50cd2023-11-23T20:57:51ZengMDPI AGMathematics2227-73902022-02-0110463710.3390/math10040637Hardy Inequalities and Interrelations of Fractional Triebel–Lizorkin Spaces in a Bounded Uniform DomainJun Cao0Yongyang Jin1Yuanyuan Li2Qishun Zhang3Department of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, ChinaDepartment of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, ChinaDepartment of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, ChinaDepartment of Applied Mathematics, Zhejiang University of Technology, Hangzhou 310023, ChinaThe interrelations of Triebel–Lizorkin spaces on smooth domains of Euclidean space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula> are well-established, whereas only partial results are known for the non-smooth domains. In this paper, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> is a non-smooth domain of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>n</mi></msup></semantics></math></inline-formula> that is bounded and uniform. Suppose <i>p</i>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><mo>[</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>s</mi><mo>∈</mo><mo stretchy="false">(</mo><mi>n</mi><msub><mrow><mo stretchy="false">(</mo><mfrac><mn>1</mn><mi>p</mi></mfrac><mo>−</mo><mfrac><mn>1</mn><mi>q</mi></mfrac><mo stretchy="false">)</mo></mrow><mo>+</mo></msub><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><msub><mrow><mo stretchy="false">(</mo><mfrac><mn>1</mn><mi>p</mi></mfrac><mo>−</mo><mfrac><mn>1</mn><mi>q</mi></mfrac><mo stretchy="false">)</mo></mrow><mo>+</mo></msub><mo>:</mo><mo>=</mo><mo movablelimits="true" form="prefix">max</mo><mrow><mo stretchy="false">{</mo><mi>n</mi><mrow><mo stretchy="false">(</mo><mfrac><mn>1</mn><mi>p</mi></mfrac><mo>−</mo><mfrac><mn>1</mn><mi>q</mi></mfrac><mo stretchy="false">)</mo></mrow><mo>,</mo><mn>0</mn><mo stretchy="false">}</mo></mrow></mrow></semantics></math></inline-formula>. The authors show that three typical types of fractional Triebel–Lizorkin spaces, on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula>: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>F</mi><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow><mi>s</mi></msubsup><mrow><mo stretchy="false">(</mo><mo>Ω</mo><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mover accent="true"><mi>F</mi><mo>˚</mo></mover><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow><mi>s</mi></msubsup><mrow><mo stretchy="false">(</mo><mo>Ω</mo><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><msubsup><mover accent="true"><mi>F</mi><mo>˜</mo></mover><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow><mi>s</mi></msubsup><mrow><mo stretchy="false">(</mo><mo>Ω</mo><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, defined via the restriction, completion and supporting conditions, respectively, are identical if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> is E-thick and supports some Hardy inequalities. Moreover, the authors show the condition that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> is E-thick can be removed when considering only the density property <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>F</mi><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow><mi>s</mi></msubsup><mrow><mo stretchy="false">(</mo><mo>Ω</mo><mo stretchy="false">)</mo></mrow><mo>=</mo><msubsup><mover accent="true"><mi>F</mi><mo>˚</mo></mover><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow><mi>s</mi></msubsup><mrow><mo stretchy="false">(</mo><mo>Ω</mo><mo stretchy="false">)</mo></mrow></mrow></semantics></math></inline-formula>, and the condition that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> supports Hardy inequalities can be characterized by some Triebel–Lizorkin capacities in the special case of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mi>q</mi><mo><</mo><mo>∞</mo></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/10/4/637Triebel–Lizorkin spaceHardy inequalityuniform domainfractional Laplacian |
| spellingShingle | Jun Cao Yongyang Jin Yuanyuan Li Qishun Zhang Hardy Inequalities and Interrelations of Fractional Triebel–Lizorkin Spaces in a Bounded Uniform Domain Mathematics Triebel–Lizorkin space Hardy inequality uniform domain fractional Laplacian |
| title | Hardy Inequalities and Interrelations of Fractional Triebel–Lizorkin Spaces in a Bounded Uniform Domain |
| title_full | Hardy Inequalities and Interrelations of Fractional Triebel–Lizorkin Spaces in a Bounded Uniform Domain |
| title_fullStr | Hardy Inequalities and Interrelations of Fractional Triebel–Lizorkin Spaces in a Bounded Uniform Domain |
| title_full_unstemmed | Hardy Inequalities and Interrelations of Fractional Triebel–Lizorkin Spaces in a Bounded Uniform Domain |
| title_short | Hardy Inequalities and Interrelations of Fractional Triebel–Lizorkin Spaces in a Bounded Uniform Domain |
| title_sort | hardy inequalities and interrelations of fractional triebel lizorkin spaces in a bounded uniform domain |
| topic | Triebel–Lizorkin space Hardy inequality uniform domain fractional Laplacian |
| url | https://www.mdpi.com/2227-7390/10/4/637 |
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