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Russia Rostov-on-Don
Section Mathematics
Title Projection method for solving equations for multidimensional operators with anisotropically homogeneous kernels of compact type
Author(-s) Deundyak V.M.ab, Lukin A.V.a
Affiliations Southern Federal Universitya, State Scientific Organization: Research Institute “Spetsvuzavtomatika”b
Abstract We consider the Banach algebra $\mathfrak{V}_{\mathbf{n}; p}$ of operators with anisotropically homogeneous kernels of compact type in $L_p$-space on the $\mathbb{R}^n$-group. Interest in the operators from $\mathfrak{V}_{\mathbf{n}; p}$ is motivated by their natural connection with the Mellin convolution operators and multidimensional multiplicative convolution operators on the $\mathbb{R}^n$-group, as well as by their applicability to the solution of problems with complex singularities. We describe the relationship of this algebra with the algebra of multidimensional convolution operators with compact coefficients using the similarity isomorphism. For the operators from the $\mathfrak{V}_{\mathbf{n}; p}$-algebra we obtain the criterion of applicability of the projection method for solving operator equations in terms of invertibility of some set of operators in cones. We prove the criterion of applicability using the reduction of the original equation to an equation for convolution operators with compact coefficients. The proof of the applicability of the projection method is sufficiently based on the new operator version of the local principle by A.V. Kozak in the theory of projection methods, which is a modification of the well-known local principle by I.B. Simonenko in the theory of the Fredholm property. In this paper, we give illustrative examples of the equations for operators with anisotropically homogeneous kernels of compact type, where we calculate the symbol and apply the developed projection method for these operators.
Keywords integral operator, homogeneous kernels, convolution operator, projection method, compact coefficients
UDC 517.988.8
MSC 47G10, 45L05, 45P05
DOI 10.20537/vm190202
Received 17 February 2019
Language Russian
Citation Deundyak V.M., Lukin A.V. Projection method for solving equations for multidimensional operators with anisotropically homogeneous kernels of compact type, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 2, pp. 153-165.
  1. Karapetiants N., Samko S. Equations with involutive operators, Boston: Birkhäuser, 2001.
  2. Avsyankin O.G. Multidimensional integral operators with bihomogeneous kernels: A projection method and pseudospectra, Siberian Mathematical Journal, 2006, vol. 47, issue 3, pp. 410-421.
  3. Avsyankin O.G. The projection method for matrix multidimensional dual integral operators with homogeneous kernels, Vladikavkazskii Matematicheskii Zhurnal, 2006, vol. 8, no. 1, pp. 3-10 (in Russian).
  4. Avsyankin O.G. Projection method for integral operators with homogeneous kernels perturbed by one-sided multiplicative shifts, Russian Mathematics, 2015, vol. 59, issue 2, pp. 7-13.
  5. Deundyak V.M. Multidimensional integral operators with homogeneous kernels of compact type and multiplicatively weakly oscillating coefficients, Mathematical Notes, 2010, vol. 87, issue 5-6, pp. 672-686.
  6. Deundyak V.M., Miroshnikova E.I. The boundedness and the Fredholm property of integral operators with anisotropically homogeneous kernels of compact type and variable coefficients, Russian Mathematics, 2012, vol. 56, issue 7, pp. 1-14.
  7. Rabinovich V., Roch S., Silbermann B. Limit operators and their applications in operator theory, Basel: Birkhäuser, 2004.
  8. Simonenko I.B. Operators of convolution type in cones, Mathematics of the USSR-Sbornik, 1967, vol. 3, no. 2, pp. 279-293.
  9. Deundyak V.M., Miroshnikova E.I. Multidimensional multiplicative convolutions and their applications to the theory of operators with homogeneous kernels, Trudy nauchnoi shkoly I.B. Simonenko: sbornik statei (Works of the scientific school of I.B. Simonenko: Transactions), Rostov-on-Don: Southern Federal University, 2010, pp. 67-78 (in Russian).
  10. Rabinovich V., Schulze B.-W., Tarkhanov N. $C$$*$ -algebras of singular integral operators in domains with oscillating conical singularities, Manuscripta Mathematica, 2002, vol. 108, issue 1, pp. 69-90.
  11. Kozak A.V. A local principle in the theory of projection methods, Soviet Mathematics. Doklady, 1973, vol. 14, pp. 1580-1583.
  12. Kozak A.V. A local principle in the projection methods theory, Integral'nye i differentsial'nye uravneniya i ikh prilozheniya: sbornik nauchnykh trydov (Integral and differential equations and their applications: Transactions), Elista, 1983, pp. 58-73 (in Russian).
  13. Simonenko I.B. Lokal'nyi metod v teorii invariantnykh otnositel'no sdviga operatorov i ih ogibayushchikh (The local method in the theory of operators invariant with respect to shifts and their envelopes), Rostov-on-Don: CVVR, 2007.
  14. Lukin A.V. Application of Simonenko–Kozak's local principe in the section method theory of solving convolution equations with operator coefficients, Vladikavkazskii Matematicheskii Zhurnal, 2016, vol. 18, no. 2, pp. 55-66 (in Russian).
  15. Deundyak V.M., Lukin A.V. Approximate method of solution of the convolution equations on a group $\mathbb{R}$$n$ with compact coefficients and applications, Izvestiya Vysshikh Uchebnykh Zavedenii. Severo-Kavkazskii Region. Estestvennye Nauki, 2013, vol. 6, pp. 5-8 (in Russian).
  16. Gohberg I.C., Fel'dman I.A. Convolution equations and projection methods for their solution, Providence, R.I.: Amer. Math. Soc., 2005. Original Russian text published in Gokhberg I.C., Fel'dman I.A. Uravneniya v svertkakh i proektsionnye metody ikh resheniya, Moscow: Nauka, 1971, 352 p.
  17. Deundyak V.M., Lukin A.V. An approximate method for solving the equations for multidimensional operators with anisotropically homogeneous kernels of compact type, Matematika i Ee Prilozheniya. Zhurnal Ivanovskogo Matematicheskogo Obshchestva, 2013, issue 1 (10), pp. 3-12 (in Russian).
  18. Grafakos L. Classical Fourier analysis, New York: Springer, 2008.
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