Pre-Compactness of Sets and Compactness of Commutators for Riesz Potential in Global Morrey-Type Spaces

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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Bibliographic Details
Main Authors: Nurzhan Bokayev, Victor Burenkov, Dauren Matin, Aidos Adilkhanov
Format: Article
Language:English
Published: MDPI AG 2024-11-01
Series:Mathematics
Subjects:
commutator
Riesz potential
compactness
global Morrey space
<i>VMO</i>
Online Access:https://www.mdpi.com/2227-7390/12/22/3533
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Summary: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>.
ISSN:2227-7390

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