<i>L<sup>p</sup></i> Bounds for Rough Maximal Operators along Surfaces of Revolution on Product Domains
In this paper, we study the boundedness of rough Maximal integral operators along surfaces of revolution on product domains. For several classes of surfaces, we establish appropriate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/2/193 |
| _version_ | 1797343088380739584 |
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| author | Mohammed Ali Hussain Al-Qassem |
| author_facet | Mohammed Ali Hussain Al-Qassem |
| author_sort | Mohammed Ali |
| collection | DOAJ |
| description | In this paper, we study the boundedness of rough Maximal integral operators along surfaces of revolution on product domains. For several classes of surfaces, we establish appropriate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula> bounds of these Maximal operators under the assumption <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Ω</mo><mo>∈</mo><msup><mi>L</mi><mi>q</mi></msup><mrow><mo>(</mo><msup><mi mathvariant="double-struck">S</mi><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>×</mo><msup><mi mathvariant="double-struck">S</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for some <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>, and then we employ these bounds along with Yano’s extrapolation argument to obtain the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula> boundedness of the aforementioned integral operators under a weaker condition in which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> belongs to either the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>B</mi><mi>q</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mfrac><mn>2</mn><msup><mi>τ</mi><mo>′</mo></msup></mfrac><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msubsup><mrow><mo>(</mo><msup><mi mathvariant="double-struck">S</mi><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>×</mo><msup><mi mathvariant="double-struck">S</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> or to the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><msup><mrow><mo>(</mo><mi>l</mi><mi>o</mi><mi>g</mi><mi>L</mi><mo>)</mo></mrow><mrow><mn>2</mn><mo>/</mo><msup><mi>τ</mi><mo>′</mo></msup></mrow></msup><mrow><mo>(</mo><msup><mi mathvariant="double-struck">S</mi><mrow><mi>m</mi><mo>−</mo></mrow></msup><mo>×</mo><msup><mi mathvariant="double-struck">S</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Our results extend and improve many previously known results. |
| first_indexed | 2024-03-08T10:42:37Z |
| format | Article |
| id | doaj.art-c4d996f72a1e4b47880db4639ffd25f1 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-08T10:42:37Z |
| publishDate | 2024-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-c4d996f72a1e4b47880db4639ffd25f12024-01-26T17:31:03ZengMDPI AGMathematics2227-73902024-01-0112219310.3390/math12020193<i>L<sup>p</sup></i> Bounds for Rough Maximal Operators along Surfaces of Revolution on Product DomainsMohammed Ali0Hussain Al-Qassem1Department of Mathematics and Statistics, Faculty of Science and Arts, Jordan University of Science and Technology, P.O. Box 3030, Irbid 22110, JordanMathematics Program, Department of Mathematics, Statistics and Physics, College of Arts and Sciences, Qatar University, Doha 2713, QatarIn this paper, we study the boundedness of rough Maximal integral operators along surfaces of revolution on product domains. For several classes of surfaces, we establish appropriate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula> bounds of these Maximal operators under the assumption <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Ω</mo><mo>∈</mo><msup><mi>L</mi><mi>q</mi></msup><mrow><mo>(</mo><msup><mi mathvariant="double-struck">S</mi><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>×</mo><msup><mi mathvariant="double-struck">S</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for some <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>q</mi><mo>></mo><mn>1</mn></mrow></semantics></math></inline-formula>, and then we employ these bounds along with Yano’s extrapolation argument to obtain the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mi>p</mi></msup></semantics></math></inline-formula> boundedness of the aforementioned integral operators under a weaker condition in which <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo>Ω</mo></semantics></math></inline-formula> belongs to either the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msubsup><mi>B</mi><mi>q</mi><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mfrac><mn>2</mn><msup><mi>τ</mi><mo>′</mo></msup></mfrac><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msubsup><mrow><mo>(</mo><msup><mi mathvariant="double-struck">S</mi><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>×</mo><msup><mi mathvariant="double-struck">S</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula> or to the space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>L</mi><msup><mrow><mo>(</mo><mi>l</mi><mi>o</mi><mi>g</mi><mi>L</mi><mo>)</mo></mrow><mrow><mn>2</mn><mo>/</mo><msup><mi>τ</mi><mo>′</mo></msup></mrow></msup><mrow><mo>(</mo><msup><mi mathvariant="double-struck">S</mi><mrow><mi>m</mi><mo>−</mo></mrow></msup><mo>×</mo><msup><mi mathvariant="double-struck">S</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Our results extend and improve many previously known results.https://www.mdpi.com/2227-7390/12/2/193product spacessurfaces of revolutionrough kernelsMaximal integrals |
| spellingShingle | Mohammed Ali Hussain Al-Qassem <i>L<sup>p</sup></i> Bounds for Rough Maximal Operators along Surfaces of Revolution on Product Domains Mathematics product spaces surfaces of revolution rough kernels Maximal integrals |
| title | <i>L<sup>p</sup></i> Bounds for Rough Maximal Operators along Surfaces of Revolution on Product Domains |
| title_full | <i>L<sup>p</sup></i> Bounds for Rough Maximal Operators along Surfaces of Revolution on Product Domains |
| title_fullStr | <i>L<sup>p</sup></i> Bounds for Rough Maximal Operators along Surfaces of Revolution on Product Domains |
| title_full_unstemmed | <i>L<sup>p</sup></i> Bounds for Rough Maximal Operators along Surfaces of Revolution on Product Domains |
| title_short | <i>L<sup>p</sup></i> Bounds for Rough Maximal Operators along Surfaces of Revolution on Product Domains |
| title_sort | i l sup p sup i bounds for rough maximal operators along surfaces of revolution on product domains |
| topic | product spaces surfaces of revolution rough kernels Maximal integrals |
| url | https://www.mdpi.com/2227-7390/12/2/193 |
| work_keys_str_mv | AT mohammedali ilsuppsupiboundsforroughmaximaloperatorsalongsurfacesofrevolutiononproductdomains AT hussainalqassem ilsuppsupiboundsforroughmaximaloperatorsalongsurfacesofrevolutiononproductdomains |