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Mat. Zametki, 2010, Volume 87, Issue 5, Pages 704–720 (Mi mz8717)  

This article is cited in 6 scientific papers (total in 6 papers)

Multidimensional Integral Operators with Homogeneous Kernels of Compact Type and Multiplicatively Weakly Oscillating Coefficients

V. M. Deundyak

Southern Federal University

Abstract: In the space $L_p(\mathbb R^n)$, $1<p<+\infty$, we consider a new class of integral operators with kernels homogeneous of degree $-n$, which includes the class of operators with homogeneous $SO(n)$-invariant kernels; we study the Banach algebra generated by such operators with multiplicatively weakly oscillating coefficients. For operators from this algebra, we define a symbol in terms of which we formulate a Fredholm property criterion and derive a formula for calculating the index. An important stage in obtaining these results is the establishment of the relationship between the operators of the class under study and the operators of one-dimensional convolution with weakly oscillating compact coefficients.

Keywords: multidimensional integral operator, operators with multiplicatively weakly oscillating coefficients, homogeneous kernel, convolution operator, the space $L_p(\mathbb R^n)$

DOI: https://doi.org/10.4213/mzm8717

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English version:
Mathematical Notes, 2010, 87:5, 672–686

Bibliographic databases:

Document Type: Article
UDC: 517.9
Received: 15.05.2009

Citation: V. M. Deundyak, “Multidimensional Integral Operators with Homogeneous Kernels of Compact Type and Multiplicatively Weakly Oscillating Coefficients”, Mat. Zametki, 87:5 (2010), 704–720; Math. Notes, 87:5 (2010), 672–686

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Deundyak V.M., Miroshnikova E.I., “Vychislenie indeksa mnogomernykh integralnykh operatorov s anizotropno odnomernymi yadrami kompaktnogo tipa”, Matematika i ee prilozheniya. zhurnal ivanovskogo matematicheskogo obschestva, 2011, no. 1, 39–48  elib
    2. V. M. Deundyak, E. I. Miroshnikova, “The boundedness and the Fredholm property of integral operators with anisotropically homogeneous kernels of compact type and variable coefficients”, Russian Math. (Iz. VUZ), 56:7 (2012), 1–14  mathnet  crossref  mathscinet
    3. Miroshnikova E.I., “Ogranichennost i obratimost integralnykh operatorov s odnorodnymi yadrami kompaktnogo tipa v nekotorykh vesovykh $l_p$-prostranstvakh”, Izv. vuzov. Severo-Kavkazskii region. Seriya: Estestvennye nauki, 2012, no. 2, 22–26  elib
    4. V. M. Deundyak, “Topological methods in solvability theory of multidimensional pair integral operators with homogeneous kernels of compact type”, Proc. Steklov Inst. Math., 278 (2012), 51–59  mathnet  crossref  mathscinet  isi
    5. Deundyak V.M., Lukin A.V., “Priblizhennyi metod resheniya operatornykh uravnenii svertki na gruppe $R^n$ s kompaktnymi koeffitsientami i prilozheniya”, Izvestiya vysshikh uchebnykh zavedenii. Severo-Kavkazskii region. Seriya: Estestvennye nauki, 2013, no. 6(178), 5–8  elib
    6. Elena M., “Boundedness and Invertibility of Multidimensional Integral Operators With Anisotropically Homogeneous Kernels in Weighted l-P-Spaces”, 10th International Conference on Mathematical Problems in Engineering, Aerospace and Sciences, AIP Conference Proceedings, 1637, ed. Sivasundaram S., Amer Inst Physics, 2014, 663–672  crossref  isi  scopus
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