Pre-Compactness of Sets and Compactness of Commutators for Riesz Potential in Global Morrey-Type Spaces
In this paper, we establish sufficient conditions for the pre-compactness of sets in the global Morrey-type spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><msubsup><m...
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2024-11-01
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| author | Nurzhan Bokayev Victor Burenkov Dauren Matin Aidos Adilkhanov |
| author_facet | Nurzhan Bokayev Victor Burenkov Dauren Matin Aidos Adilkhanov |
| author_sort | Nurzhan Bokayev |
| collection | DOAJ |
| description | In this paper, we establish sufficient conditions for the pre-compactness of sets in the global Morrey-type spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><msubsup><mi>M</mi><mrow><mi>p</mi><mi>θ</mi></mrow><mrow><mi>w</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></msubsup></mrow></semantics></math></inline-formula>. Our main result is the compactness of the commutators of the Riesz potential <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="[" close="]"><mi>b</mi><mo>,</mo><msub><mi>I</mi><mi>α</mi></msub></mfenced></semantics></math></inline-formula> in global Morrey-type spaces from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><msubsup><mi>M</mi><mrow><msub><mi>p</mi><mn>1</mn></msub><msub><mi>θ</mi><mn>1</mn></msub></mrow><mrow><msub><mi>w</mi><mn>1</mn></msub><mrow><mo>(</mo><mo>·</mo><mo>)</mo></mrow></mrow></msubsup></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><msubsup><mi>M</mi><mrow><msub><mi>p</mi><mn>2</mn></msub><msub><mi>θ</mi><mn>2</mn></msub></mrow><mrow><msub><mi>w</mi><mn>2</mn></msub><mrow><mo>(</mo><mo>·</mo><mo>)</mo></mrow></mrow></msubsup></mrow></semantics></math></inline-formula>. We also present new sufficient conditions for the commutator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="[" close="]"><mi>b</mi><mo>,</mo><msub><mi>I</mi><mi>α</mi></msub></mfenced></semantics></math></inline-formula> to be bounded from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><msubsup><mi>M</mi><mrow><msub><mi>p</mi><mn>1</mn></msub><msub><mi>θ</mi><mn>1</mn></msub></mrow><mrow><msub><mi>w</mi><mn>1</mn></msub><mrow><mo>(</mo><mo>·</mo><mo>)</mo></mrow></mrow></msubsup></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><msubsup><mi>M</mi><mrow><msub><mi>p</mi><mn>2</mn></msub><msub><mi>θ</mi><mn>2</mn></msub></mrow><mrow><msub><mi>w</mi><mn>2</mn></msub><mrow><mo>(</mo><mo>·</mo><mo>)</mo></mrow></mrow></msubsup></mrow></semantics></math></inline-formula>. In the proof of the theorem regarding the compactness of the commutator for the Riesz potential, we primarily utilize the boundedness condition for the commutator for the Riesz potential <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="[" close="]"><mi>b</mi><mo>,</mo><msub><mi>I</mi><mi>α</mi></msub></mfenced></semantics></math></inline-formula> in global Morrey-type spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><msubsup><mi>M</mi><mrow><mi>p</mi><mi>θ</mi></mrow><mrow><mi>w</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></msubsup></mrow></semantics></math></inline-formula>, and the sufficient conditions derived from the theorem on pre-compactness of sets in global Morrey-type spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><msubsup><mi>M</mi><mrow><mi>p</mi><mi>θ</mi></mrow><mrow><mi>w</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></msubsup></mrow></semantics></math></inline-formula>. |
| format | Article |
| id | doaj-art-068c02693b4749518cec1d61b8fea79b |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-068c02693b4749518cec1d61b8fea79b2024-11-26T18:11:44ZengMDPI AGMathematics2227-73902024-11-011222353310.3390/math12223533Pre-Compactness of Sets and Compactness of Commutators for Riesz Potential in Global Morrey-Type SpacesNurzhan Bokayev0Victor Burenkov1Dauren Matin2Aidos Adilkhanov3Faculty of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University, Astana 010000, KazakhstanV.A. Steklov Mathematical Institute, Russian Academy of Sciences, 42 Gubkin St, 117966 Moscow, RussiaFaculty of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University, Astana 010000, KazakhstanFaculty of Mechanics and Mathematics, L.N. Gumilyov Eurasian National University, Astana 010000, KazakhstanIn this paper, we establish sufficient conditions for the pre-compactness of sets in the global Morrey-type spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><msubsup><mi>M</mi><mrow><mi>p</mi><mi>θ</mi></mrow><mrow><mi>w</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></msubsup></mrow></semantics></math></inline-formula>. Our main result is the compactness of the commutators of the Riesz potential <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="[" close="]"><mi>b</mi><mo>,</mo><msub><mi>I</mi><mi>α</mi></msub></mfenced></semantics></math></inline-formula> in global Morrey-type spaces from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><msubsup><mi>M</mi><mrow><msub><mi>p</mi><mn>1</mn></msub><msub><mi>θ</mi><mn>1</mn></msub></mrow><mrow><msub><mi>w</mi><mn>1</mn></msub><mrow><mo>(</mo><mo>·</mo><mo>)</mo></mrow></mrow></msubsup></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><msubsup><mi>M</mi><mrow><msub><mi>p</mi><mn>2</mn></msub><msub><mi>θ</mi><mn>2</mn></msub></mrow><mrow><msub><mi>w</mi><mn>2</mn></msub><mrow><mo>(</mo><mo>·</mo><mo>)</mo></mrow></mrow></msubsup></mrow></semantics></math></inline-formula>. We also present new sufficient conditions for the commutator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="[" close="]"><mi>b</mi><mo>,</mo><msub><mi>I</mi><mi>α</mi></msub></mfenced></semantics></math></inline-formula> to be bounded from <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><msubsup><mi>M</mi><mrow><msub><mi>p</mi><mn>1</mn></msub><msub><mi>θ</mi><mn>1</mn></msub></mrow><mrow><msub><mi>w</mi><mn>1</mn></msub><mrow><mo>(</mo><mo>·</mo><mo>)</mo></mrow></mrow></msubsup></mrow></semantics></math></inline-formula> to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><msubsup><mi>M</mi><mrow><msub><mi>p</mi><mn>2</mn></msub><msub><mi>θ</mi><mn>2</mn></msub></mrow><mrow><msub><mi>w</mi><mn>2</mn></msub><mrow><mo>(</mo><mo>·</mo><mo>)</mo></mrow></mrow></msubsup></mrow></semantics></math></inline-formula>. In the proof of the theorem regarding the compactness of the commutator for the Riesz potential, we primarily utilize the boundedness condition for the commutator for the Riesz potential <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfenced separators="" open="[" close="]"><mi>b</mi><mo>,</mo><msub><mi>I</mi><mi>α</mi></msub></mfenced></semantics></math></inline-formula> in global Morrey-type spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><msubsup><mi>M</mi><mrow><mi>p</mi><mi>θ</mi></mrow><mrow><mi>w</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></msubsup></mrow></semantics></math></inline-formula>, and the sufficient conditions derived from the theorem on pre-compactness of sets in global Morrey-type spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><msubsup><mi>M</mi><mrow><mi>p</mi><mi>θ</mi></mrow><mrow><mi>w</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></msubsup></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2227-7390/12/22/3533commutatorRiesz potentialcompactnessglobal Morrey space<i>VMO</i> |
| spellingShingle | Nurzhan Bokayev Victor Burenkov Dauren Matin Aidos Adilkhanov Pre-Compactness of Sets and Compactness of Commutators for Riesz Potential in Global Morrey-Type Spaces Mathematics commutator Riesz potential compactness global Morrey space <i>VMO</i> |
| title | Pre-Compactness of Sets and Compactness of Commutators for Riesz Potential in Global Morrey-Type Spaces |
| title_full | Pre-Compactness of Sets and Compactness of Commutators for Riesz Potential in Global Morrey-Type Spaces |
| title_fullStr | Pre-Compactness of Sets and Compactness of Commutators for Riesz Potential in Global Morrey-Type Spaces |
| title_full_unstemmed | Pre-Compactness of Sets and Compactness of Commutators for Riesz Potential in Global Morrey-Type Spaces |
| title_short | Pre-Compactness of Sets and Compactness of Commutators for Riesz Potential in Global Morrey-Type Spaces |
| title_sort | pre compactness of sets and compactness of commutators for riesz potential in global morrey type spaces |
| topic | commutator Riesz potential compactness global Morrey space <i>VMO</i> |
| url | https://www.mdpi.com/2227-7390/12/22/3533 |
| work_keys_str_mv | AT nurzhanbokayev precompactnessofsetsandcompactnessofcommutatorsforrieszpotentialinglobalmorreytypespaces AT victorburenkov precompactnessofsetsandcompactnessofcommutatorsforrieszpotentialinglobalmorreytypespaces AT daurenmatin precompactnessofsetsandcompactnessofcommutatorsforrieszpotentialinglobalmorreytypespaces AT aidosadilkhanov precompactnessofsetsandcompactnessofcommutatorsforrieszpotentialinglobalmorreytypespaces |