\(L_p-L_q\)-estimates for Generalized Riss Potentials with Oscillating Kernels

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.23671/VNC.2017.2.6503 \(L_p-L_q\)-estimates for Generalized Riss Potentials with Oscillating Kernels Gurov M. N. , Nogin V. A. Vladikavkaz Mathematical Journal 2017. Vol. 19. Issue 2. Abstract: We consider a class of multidimensional potential-type operators whose kernels are oscillating at infinity. The characteristics of these operators are infinitely differentiable homogeneous functions. We describe convex sets in the \((1/p;1/q)\)-plane for which these operators are bounded from \(L_p\) into \(L_q\) and indicate the domains where they are not bounded. In some cases we describe their \(\cal{L}\)-characteristics. To obtain these results we use a new method based on special representation of the symbols of multidimensional potential-type operators. To these representations of the symbols we apply the technique of Fourier-multipliers, which degenerate or have singularities on the unit sphere in \(\mathbb{R}^n\). Keywords: potential-type operators, oscillating kernel, method of Fourier multipliers, \(L_p-L_q\)-estimates, \(\cal L\)-characteristic. Language: Russian Download the full text For citation: Gurov M. N., Nogin V. A. \(L_p-L_q\)-estimates for generalized Riss potentials with oscillating kernels. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 19, no. 1, pp. 3-10. DOI 10.23671/VNC.2017.2.6503 + References 1. Betilgiriev M. A., Karasev D. N., Nogin V. A. \(L_p-L_q\)-estimates for some potential type operators with oscillating kernels. Fractional Calculus and Applied Analysis, 2004, vol. 7, no 2, pp. 213-241. 2. Borjeson L. Estimates for the Bochner-Riesz operator with negative index. Indiana University Mathematics Journal, 1986, vol. 35, no. 2, pp. 225-233. 3. Karapetyants A. N., Karasev D. N., Nogin V. A. \(L_p-L_q\)-estimates for some fractional acoustic potentials and some related operators. Fractional Calculus and Applied Analysis, 2005, vol. 7, no. 1, pp. 155-172. 4. Karasev D. N., Nogin V. A. On the boundness of some potential-type operators with oscillating kernels. Math. Nachr, 2005, vol. 278, no. 5, pp. 554-574. 5. Samko S. G. Hypersingular Integrals and their Applications [Internat. Series "Analytical Methods and Special Functions", Vol. 5]. London: Taylor and Frances, 2002, 376 p. 6. Stein E. M. Harmonic Analysis: Real-variable Method, Orthogonality, and Oscillatory Integrals. Princeton, NJ: Princeton Univ. Press, 1993, 355 p. 7. Gurov M. N. Fourier Transform of the One Oscillating Function. Izvestija vuzov. Severo-Kavkazskij region. Estestvennye nauki [Izv. Vyssh. Uchebn. Zaved. Sev.-Kavk. Reg. Estestv. Nauki], 2014, no. 5, pp. 11-14 (in Russian). 8. Gurov M. N., Karasev D. N., Nogin V. A. On a Certain Fourier Multiplier. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], 2015, vol. 17, no. 1, pp. 14-20 (in Russian). 9. Karapetyants A. N., Karasev D. N., Nogin V. A. Estimates for some potential-type operators with oscillating kernels. Proceedings of the NAS Armenia: Mathematics, 2003, vol. 38, no. 2, pp. 37-62. 10. Karasev D. N. \(L_p-L_q\)-estimates for some potential type operators with oscillating kernels. Differential Equations, 2003, vol. 39, no. 3, pp. 418-420. 11. Kostetskaya G. S., Samko S. G. Absolute integrability of fourier integrals. Vestnik RUDN (Math), 1994, no. 1, pp. 138-168. 12. Sobolev S. L. Some Applications of Functional Analysis in Mathematical Physics [Translations of mathematical monographs]. American Math. Society, 1991, 286 p. 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. ← Contents of issue


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