<i>L<sup>p</sup></i>-Mapping Properties of a Class of Spherical Integral Operators
In this paper, we study a class of spherical integral operators <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">I</mi><mi mathvariant="san...
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MDPI AG
2023-08-01
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| Online Access: | https://www.mdpi.com/2075-1680/12/9/802 |
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| author | Laith Hawawsheh Ahmad Qazza Rania Saadeh Amjed Zraiqat Iqbal M. Batiha |
| author_facet | Laith Hawawsheh Ahmad Qazza Rania Saadeh Amjed Zraiqat Iqbal M. Batiha |
| author_sort | Laith Hawawsheh |
| collection | DOAJ |
| description | In this paper, we study a class of spherical integral operators <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">I</mi><mi mathvariant="sans-serif">Ω</mi></msub><mi>f</mi></mrow></semantics></math></inline-formula>. We prove an inequality that relates this class of operators with some well-known Marcinkiewicz integral operators by using the classical Hardy inequality. We also attain the boundedness of the operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">I</mi><mi mathvariant="sans-serif">Ω</mi></msub><mi>f</mi></mrow></semantics></math></inline-formula> for some <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><mn>2</mn></mrow></semantics></math></inline-formula> whenever <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Ω</mi></semantics></math></inline-formula> belongs to a certain class of Lebesgue spaces. In addition, we introduce a new proof of the optimality condition on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Ω</mi></semantics></math></inline-formula> in order to obtain the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula>-boundedness of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">I</mi><mi mathvariant="sans-serif">Ω</mi></msub></semantics></math></inline-formula>. Generally, the purpose of this work is to set up new proofs and extend several known results connected with a class of spherical integral operators. |
| first_indexed | 2024-03-10T23:03:20Z |
| format | Article |
| id | doaj.art-38adf2fbfe83499bada433e75c1b37c6 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-10T23:03:20Z |
| publishDate | 2023-08-01 |
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| spelling | doaj.art-38adf2fbfe83499bada433e75c1b37c62023-11-19T09:31:56ZengMDPI AGAxioms2075-16802023-08-0112980210.3390/axioms12090802<i>L<sup>p</sup></i>-Mapping Properties of a Class of Spherical Integral OperatorsLaith Hawawsheh0Ahmad Qazza1Rania Saadeh2Amjed Zraiqat3Iqbal M. Batiha4School of Basic Sciences and Humanities, German Jordanian University, Amman 11180, JordanDepartment of Mathematics, Faculty of Science, Zarqa University, Zarga 13110, JordanDepartment of Mathematics, Faculty of Science, Zarqa University, Zarga 13110, JordanDepartment of Mathematics, Al Zaytoonah University, Amman 11733, JordanDepartment of Mathematics, Al Zaytoonah University, Amman 11733, JordanIn this paper, we study a class of spherical integral operators <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">I</mi><mi mathvariant="sans-serif">Ω</mi></msub><mi>f</mi></mrow></semantics></math></inline-formula>. We prove an inequality that relates this class of operators with some well-known Marcinkiewicz integral operators by using the classical Hardy inequality. We also attain the boundedness of the operator <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">I</mi><mi mathvariant="sans-serif">Ω</mi></msub><mi>f</mi></mrow></semantics></math></inline-formula> for some <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><mn>2</mn></mrow></semantics></math></inline-formula> whenever <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Ω</mi></semantics></math></inline-formula> belongs to a certain class of Lebesgue spaces. In addition, we introduce a new proof of the optimality condition on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="sans-serif">Ω</mi></semantics></math></inline-formula> in order to obtain the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mn>2</mn></msup></semantics></math></inline-formula>-boundedness of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="script">I</mi><mi mathvariant="sans-serif">Ω</mi></msub></semantics></math></inline-formula>. Generally, the purpose of this work is to set up new proofs and extend several known results connected with a class of spherical integral operators.https://www.mdpi.com/2075-1680/12/9/802singular integralssquare functionsmaximal functions |
| spellingShingle | Laith Hawawsheh Ahmad Qazza Rania Saadeh Amjed Zraiqat Iqbal M. Batiha <i>L<sup>p</sup></i>-Mapping Properties of a Class of Spherical Integral Operators Axioms singular integrals square functions maximal functions |
| title | <i>L<sup>p</sup></i>-Mapping Properties of a Class of Spherical Integral Operators |
| title_full | <i>L<sup>p</sup></i>-Mapping Properties of a Class of Spherical Integral Operators |
| title_fullStr | <i>L<sup>p</sup></i>-Mapping Properties of a Class of Spherical Integral Operators |
| title_full_unstemmed | <i>L<sup>p</sup></i>-Mapping Properties of a Class of Spherical Integral Operators |
| title_short | <i>L<sup>p</sup></i>-Mapping Properties of a Class of Spherical Integral Operators |
| title_sort | i l sup p sup i mapping properties of a class of spherical integral operators |
| topic | singular integrals square functions maximal functions |
| url | https://www.mdpi.com/2075-1680/12/9/802 |
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