On Solvability of the Sonin–Abel Equation in the Weighted Lebesgue Space
In this paper we present a method of studying a convolution operator under the Sonin conditions imposed on the kernel. The particular case of the Sonin kernel is a kernel of the fractional integral Riemman–Liouville operator, other various types of the Sonin kernels are a Bessel-type function, funct...
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-07-01
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| Series: | Fractal and Fractional |
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| Online Access: | https://www.mdpi.com/2504-3110/5/3/77 |
| _version_ | 1797519185575673856 |
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| author | Maksim V. Kukushkin |
| author_facet | Maksim V. Kukushkin |
| author_sort | Maksim V. Kukushkin |
| collection | DOAJ |
| description | In this paper we present a method of studying a convolution operator under the Sonin conditions imposed on the kernel. The particular case of the Sonin kernel is a kernel of the fractional integral Riemman–Liouville operator, other various types of the Sonin kernels are a Bessel-type function, functions with power-logarithmic singularities at the origin e.t.c. We pay special attention to study kernels close to power type functions. The main our aim is to study the Sonin–Abel equation in the weighted Lebesgue space, the used method allows us to formulate a criterion of existence and uniqueness of the solution and classify a solution, due to the asymptotics of the Jacobi series coefficients of the right-hand side. |
| first_indexed | 2024-03-10T07:39:25Z |
| format | Article |
| id | doaj.art-2f9f8282169b46a08da82acd8874f173 |
| institution | Directory Open Access Journal |
| issn | 2504-3110 |
| language | English |
| last_indexed | 2024-03-10T07:39:25Z |
| publishDate | 2021-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Fractal and Fractional |
| spelling | doaj.art-2f9f8282169b46a08da82acd8874f1732023-11-22T13:09:12ZengMDPI AGFractal and Fractional2504-31102021-07-01537710.3390/fractalfract5030077On Solvability of the Sonin–Abel Equation in the Weighted Lebesgue SpaceMaksim V. Kukushkin0Moscow State University of Civil Engineering, 129337 Moscow, RussiaIn this paper we present a method of studying a convolution operator under the Sonin conditions imposed on the kernel. The particular case of the Sonin kernel is a kernel of the fractional integral Riemman–Liouville operator, other various types of the Sonin kernels are a Bessel-type function, functions with power-logarithmic singularities at the origin e.t.c. We pay special attention to study kernels close to power type functions. The main our aim is to study the Sonin–Abel equation in the weighted Lebesgue space, the used method allows us to formulate a criterion of existence and uniqueness of the solution and classify a solution, due to the asymptotics of the Jacobi series coefficients of the right-hand side.https://www.mdpi.com/2504-3110/5/3/77Riemann–Liouville operatorAbel equationJacobi polinomialsweighted Lebesgue spacesconvolution operatorsSonin conditions |
| spellingShingle | Maksim V. Kukushkin On Solvability of the Sonin–Abel Equation in the Weighted Lebesgue Space Fractal and Fractional Riemann–Liouville operator Abel equation Jacobi polinomials weighted Lebesgue spaces convolution operators Sonin conditions |
| title | On Solvability of the Sonin–Abel Equation in the Weighted Lebesgue Space |
| title_full | On Solvability of the Sonin–Abel Equation in the Weighted Lebesgue Space |
| title_fullStr | On Solvability of the Sonin–Abel Equation in the Weighted Lebesgue Space |
| title_full_unstemmed | On Solvability of the Sonin–Abel Equation in the Weighted Lebesgue Space |
| title_short | On Solvability of the Sonin–Abel Equation in the Weighted Lebesgue Space |
| title_sort | on solvability of the sonin abel equation in the weighted lebesgue space |
| topic | Riemann–Liouville operator Abel equation Jacobi polinomials weighted Lebesgue spaces convolution operators Sonin conditions |
| url | https://www.mdpi.com/2504-3110/5/3/77 |
| work_keys_str_mv | AT maksimvkukushkin onsolvabilityofthesoninabelequationintheweightedlebesguespace |