Bessel–Riesz Operator in Variable Lebesgue Spaces <i>L<sup>p</sup></i><sup>(·)</sup>(<inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">R</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>)
This paper investigates the Bessel–Riesz operator within the framework of variable Lebesgue spaces. We extend existing results by establishing boundedness under more general conditions. The analysis is based on the Hardy–Littlewood maximal function, Hölder’s inequality, and dyadic decomposition tech...
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2025-05-01
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| author | Muhammad Nasir Fehaid Salem Alshammari Ali Raza |
| author_facet | Muhammad Nasir Fehaid Salem Alshammari Ali Raza |
| author_sort | Muhammad Nasir |
| collection | DOAJ |
| description | This paper investigates the Bessel–Riesz operator within the framework of variable Lebesgue spaces. We extend existing results by establishing boundedness under more general conditions. The analysis is based on the Hardy–Littlewood maximal function, Hölder’s inequality, and dyadic decomposition techniques. For a given domain space, we construct a suitable range space such that the operator remains bounded. Conversely, for a prescribed range space, we identify a corresponding domain space that guarantees boundedness. Illustrative examples are included to demonstrate the construction of such spaces. The main results hold when the essential infimum of the exponent function exceeds one, and we also establish weak-type estimates in the limiting case. |
| format | Article |
| id | doaj-art-514d7eb78e7f4a088bd23799b9225f8c |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-514d7eb78e7f4a088bd23799b9225f8c2025-08-20T03:32:27ZengMDPI AGAxioms2075-16802025-05-0114642910.3390/axioms14060429Bessel–Riesz Operator in Variable Lebesgue Spaces <i>L<sup>p</sup></i><sup>(·)</sup>(<inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">R</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>)Muhammad Nasir0Fehaid Salem Alshammari1Ali Raza2Abdus Salam School of Mathematical Sciences, Government College University, 68-B, New Muslim Town, Lahore 54600, PakistanDepartment of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11432, Saudi ArabiaAbdus Salam School of Mathematical Sciences, Government College University, 68-B, New Muslim Town, Lahore 54600, PakistanThis paper investigates the Bessel–Riesz operator within the framework of variable Lebesgue spaces. We extend existing results by establishing boundedness under more general conditions. The analysis is based on the Hardy–Littlewood maximal function, Hölder’s inequality, and dyadic decomposition techniques. For a given domain space, we construct a suitable range space such that the operator remains bounded. Conversely, for a prescribed range space, we identify a corresponding domain space that guarantees boundedness. Illustrative examples are included to demonstrate the construction of such spaces. The main results hold when the essential infimum of the exponent function exceeds one, and we also establish weak-type estimates in the limiting case.https://www.mdpi.com/2075-1680/14/6/429Bessel–Riesz operatorHardy–Littlewood maximal operatorvariable Lebesgue spaces |
| spellingShingle | Muhammad Nasir Fehaid Salem Alshammari Ali Raza Bessel–Riesz Operator in Variable Lebesgue Spaces <i>L<sup>p</sup></i><sup>(·)</sup>(<inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">R</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>) Axioms Bessel–Riesz operator Hardy–Littlewood maximal operator variable Lebesgue spaces |
| title | Bessel–Riesz Operator in Variable Lebesgue Spaces <i>L<sup>p</sup></i><sup>(·)</sup>(<inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">R</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>) |
| title_full | Bessel–Riesz Operator in Variable Lebesgue Spaces <i>L<sup>p</sup></i><sup>(·)</sup>(<inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">R</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>) |
| title_fullStr | Bessel–Riesz Operator in Variable Lebesgue Spaces <i>L<sup>p</sup></i><sup>(·)</sup>(<inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">R</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>) |
| title_full_unstemmed | Bessel–Riesz Operator in Variable Lebesgue Spaces <i>L<sup>p</sup></i><sup>(·)</sup>(<inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">R</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>) |
| title_short | Bessel–Riesz Operator in Variable Lebesgue Spaces <i>L<sup>p</sup></i><sup>(·)</sup>(<inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">R</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>) |
| title_sort | bessel riesz operator in variable lebesgue spaces i l sup p sup i sup · sup inline formula math display inline semantics mrow msub mi mathvariant double struck r mi mo mo msub mrow semantics math inline formula |
| topic | Bessel–Riesz operator Hardy–Littlewood maximal operator variable Lebesgue spaces |
| url | https://www.mdpi.com/2075-1680/14/6/429 |
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