Abstract: We study a certain Fourier multiplier which degenerate or have singularities on the unit sphere in \(\mathbb{R}^n\). Necessary and sufficient conditions are obtained for this multiplier to be in the H\"{o}rmander class \(M_p^q\).
For citation: Gurov M. N.,Karasev D. N., Nogin V. A. On a certain fourier multiplier. Vladikavkazskii matematicheskii zhurnal [Vladikavkaz Math. J.], vol. 17, no. 1, pp.14-20. DOI 10.23671/VNC.2015.1.7286
1. Hormander L. Estimates for Translation Invariant Operators in \(Lp \)
Spaces. Acta Mathematica, 1960, vol. 104, pp. 93-140.
2. Karapetjanc A. N., Karasev D. N., Nogin V. A. Ocenki dlja
nekotoryh operatorov tipa potenciala s oscillirujushhimi jadrami,
Izvestija NAN Armenii [Proceedings of the NAS Armenia], 2003, vol.
38, no. 2, pp.37-62 (Russian).
3. Betilgiriev M. A., Karasev D. N., Nogin V. A. \(Lp-Lq\) Estimates
for Some Potential Type Operators with Oscillating Kernels.
Fractional Calculus & Applied Analysis, 2004, vol. 7, no. 2, pp.
213-241.
4. Karasev D. N. \(Lp \to Lq\) Estimates for Some Potential Type
Operators with Oscillating Kernels. Differential Equations
[Differentsial'nye Uravneniya, 2003, vol. 39, no. 3, pp. 418-420],
2003, vol. 39, no. 3, pp. 453-456.
5. Karasev D. N. \(Lp\to Lq\)-Estimates for Some Potential-Type
Operators with Oscillating Kernels. Fractional Calculus & Applied
Analysis, 2002, vol. 5, no. 2, pp. 131-153.
6. Karasev D. N., Nogin V. A. Inversion of Some Potential-Type
Operators with Oscillating Kernels in the Elliptic and Non-Elliptic
Cases. Integral Transforms & Special Functions, 2002, vol. 13, pp.
529-545.
7. Borjeson L. Estimates for the Bochner-Riesz Operator with
Negative Index. Ind. Univ. Math. J., 1986, vol. 35, no. 2, p.
225-233.
8. Sogge C. D. Eigenfunction and Bochner-Riesz Estimates on
Manifolds with Boundary. Math. Res. Lett., 2002, vol. 9, pp.
205-216.
9. Nogin V. A., Rubin B. S. Ocenki dlja potencialov s
oscillirujushhimi jadrami, svjazannyh s uravneniem Gel'mgol'ca,
Differential Equations [Differentsial'nye Uravneniya, 1990, vol. 26,
no. 9, pp. 1608-1613], 1990, vol. 26, no.9, pp.
10. Miyachi A. On Some Estimates for the Wave Equation in \(Lp\) and \(Hp\).
J. Fac. Sci. Univ. Tokyo. Sec. IA, 1980, vol. 27, pp. 331-354.
11. Karasev D. N., Nogin V. A. On Boundedness of Some Potential-Type
Operators with Oscillating Kernels. Math. Nachr., 2005, vol. 274,
no. 5, pp. 554-574.
12. Samko S. G., Kosteckaya G. S. Absoljutnaja integriruemost'
integralov Fur'e, Vestnik RUDN (Matematika), 1994, no.1,
pp. 138-168 (Russian).
13. Samko S. G. Hypersingular Integrals and Their Applications,
London-N.Y., Taylor&Frances, 2002, 358+xvii p. (Ser. Anal. Methods
and Special Functions, vol. 5).
The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses.
Сайт использует файлы cookie, необходимые для корректной работы сайта, и сервисы Яндекс-метрики, используемые для анализа статистики посещаемости, которые не содержат сведений, на основании которых можно идентифицировать личность пользователя. Продолжение пользования сайтом является согласием на применение данных технологий.
