Boundedness and Sobolev-Type Estimates for the Exponentially Damped Riesz Potential with Applications to the Regularity Theory of Elliptic PDEs
This paper investigates a new class of fractional integral operators, namely, the exponentially damped Riesz-type operators within the framework of variable exponent Lebesgue spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><...
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2025-07-01
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| Series: | Fractal and Fractional |
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| author | Waqar Afzal Mujahid Abbas Jorge E. Macías-Díaz Armando Gallegos Yahya Almalki |
| author_facet | Waqar Afzal Mujahid Abbas Jorge E. Macías-Díaz Armando Gallegos Yahya Almalki |
| author_sort | Waqar Afzal |
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| description | This paper investigates a new class of fractional integral operators, namely, the exponentially damped Riesz-type operators within the framework of variable exponent Lebesgue spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mrow><mi mathvariant="monospace">p</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></msup></semantics></math></inline-formula>. To the best of our knowledge, the boundedness of such operators has not been addressed in any existing functional setting. We establish their boundedness under appropriate log-Hölder continuity and growth conditions on the exponent function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="monospace">p</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></semantics></math></inline-formula>. To highlight the novelty and practical relevance of the proposed operator, we conduct a comparative analysis demonstrating its effectiveness in addressing convergence, regularity, and stability of solutions to partial differential equations. We also provide non-trivial examples that illustrate not only these properties but also show that, under this operator, a broader class of functions becomes locally integrable. The exponential decay factor notably broadens the domain of boundedness compared to classical Riesz and Bessel–Riesz potentials, making the operator more versatile and robust. Additionally, we generalize earlier results on Sobolev-type inequalities previously studied in constant exponent spaces by extending them to the variable exponent setting through our fractional operator, which reduces to the classical Riesz potential when the decay parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>λ</mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. Applications to elliptic PDEs are provided to illustrate the functional impact of our results. Furthermore, we develop several new structural properties tailored to variable exponent frameworks, reinforcing the strength and applicability of the proposed theory. |
| format | Article |
| id | doaj-art-cad20783c8c847b0b686eeaa52544b3c |
| institution | Kabale University |
| issn | 2504-3110 |
| language | English |
| publishDate | 2025-07-01 |
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| series | Fractal and Fractional |
| spelling | doaj-art-cad20783c8c847b0b686eeaa52544b3c2025-08-20T03:58:26ZengMDPI AGFractal and Fractional2504-31102025-07-019745810.3390/fractalfract9070458Boundedness and Sobolev-Type Estimates for the Exponentially Damped Riesz Potential with Applications to the Regularity Theory of Elliptic PDEsWaqar Afzal0Mujahid Abbas1Jorge E. Macías-Díaz2Armando Gallegos3Yahya Almalki4Abdus Salam School of Mathematical Sciences, Government College University, 68-B, New Muslim Town, Lahore 54600, PakistanDepartment of Mechanical Engineering Sciences, Faculty of Engineering and the Built Environment, Doornfontein Campus, University of Johannesburg, Johannesburg 2092, South AfricaDepartment of Mathematics and Didactics of Mathematics, Tallinn University, 10120 Tallinn, EstoniaDepartamento de Ciencias Exactas y Tecnología, Centro Universitario de los Lagos, Universidad de Guadalajara, Enrique Díaz de León 1144, Colonia Paseos de la Montaña, Lagos de Moreno 47460, Jalisco, MexicoDepartment of Mathematics, College of Sciences, King Khalid University, Abha 61413, Saudi ArabiaThis paper investigates a new class of fractional integral operators, namely, the exponentially damped Riesz-type operators within the framework of variable exponent Lebesgue spaces <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>L</mi><mrow><mi mathvariant="monospace">p</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></msup></semantics></math></inline-formula>. To the best of our knowledge, the boundedness of such operators has not been addressed in any existing functional setting. We establish their boundedness under appropriate log-Hölder continuity and growth conditions on the exponent function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="monospace">p</mi><mo>(</mo><mo>·</mo><mo>)</mo></mrow></semantics></math></inline-formula>. To highlight the novelty and practical relevance of the proposed operator, we conduct a comparative analysis demonstrating its effectiveness in addressing convergence, regularity, and stability of solutions to partial differential equations. We also provide non-trivial examples that illustrate not only these properties but also show that, under this operator, a broader class of functions becomes locally integrable. The exponential decay factor notably broadens the domain of boundedness compared to classical Riesz and Bessel–Riesz potentials, making the operator more versatile and robust. Additionally, we generalize earlier results on Sobolev-type inequalities previously studied in constant exponent spaces by extending them to the variable exponent setting through our fractional operator, which reduces to the classical Riesz potential when the decay parameter <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>λ</mo><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula>. Applications to elliptic PDEs are provided to illustrate the functional impact of our results. Furthermore, we develop several new structural properties tailored to variable exponent frameworks, reinforcing the strength and applicability of the proposed theory.https://www.mdpi.com/2504-3110/9/7/458Sobolev inequalityexponentially damped Riesz operatorHardy–Littlewood maximal operatorvariable Lebesgue spacesboundedness of fractional operatorsregularity of elliptic equations |
| spellingShingle | Waqar Afzal Mujahid Abbas Jorge E. Macías-Díaz Armando Gallegos Yahya Almalki Boundedness and Sobolev-Type Estimates for the Exponentially Damped Riesz Potential with Applications to the Regularity Theory of Elliptic PDEs Fractal and Fractional Sobolev inequality exponentially damped Riesz operator Hardy–Littlewood maximal operator variable Lebesgue spaces boundedness of fractional operators regularity of elliptic equations |
| title | Boundedness and Sobolev-Type Estimates for the Exponentially Damped Riesz Potential with Applications to the Regularity Theory of Elliptic PDEs |
| title_full | Boundedness and Sobolev-Type Estimates for the Exponentially Damped Riesz Potential with Applications to the Regularity Theory of Elliptic PDEs |
| title_fullStr | Boundedness and Sobolev-Type Estimates for the Exponentially Damped Riesz Potential with Applications to the Regularity Theory of Elliptic PDEs |
| title_full_unstemmed | Boundedness and Sobolev-Type Estimates for the Exponentially Damped Riesz Potential with Applications to the Regularity Theory of Elliptic PDEs |
| title_short | Boundedness and Sobolev-Type Estimates for the Exponentially Damped Riesz Potential with Applications to the Regularity Theory of Elliptic PDEs |
| title_sort | boundedness and sobolev type estimates for the exponentially damped riesz potential with applications to the regularity theory of elliptic pdes |
| topic | Sobolev inequality exponentially damped Riesz operator Hardy–Littlewood maximal operator variable Lebesgue spaces boundedness of fractional operators regularity of elliptic equations |
| url | https://www.mdpi.com/2504-3110/9/7/458 |
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