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Russian Mathematics

, Volume 63, Issue 6, pp 1–7 | Cite as

On Invertibility of Convolution Type Operators in Morrey Spaces

  • O. G. AvsyankinEmail author
Article
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Abstract

We consider integral convolution operators in the Morrey spaces. For these operators we obtain the necessary and sufficient conditions of their invertibility. Moreover, we study the Banach algebra generated by all convolution operators with summable kernels and identity operator. For this algebra we construct the symbolic calculus, in terms of which we obtain the invertibility criterion of convolution operators.

Key words

Morrey space convolution operator symbol invertibility Banach algebra 

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Notes

Acknowledgments

Author expresses deep gratitude to professor V.I. Burenkov for useful discussion of work’s results.

Funding

The work is supported by Russian Found of Fundamental Researches, grants nos. 18-01-00094-A and 18-51-06005-Az_a.

References

  1. 1.
    Burenkov, V.I. "Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces. I", Eurasian Math. J. 3(3), 11–32 (2012).MathSciNetzbMATHGoogle Scholar
  2. 2.
    Burenkov, V.I. "Recent progress in studying the boundedness of classical operators of real analysis in general Morrey-type spaces. II", Eurasian Math. J. 4(1), 21–45 (2013).MathSciNetzbMATHGoogle Scholar
  3. 3.
    Morrey, C.B. "On the solutions of quasi-linear elliptic partial differential equations", Trans. Amer. Math. Soc. 43, 126–166 (1938).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Burenkov, V.I., Tararykova, T.V. "Analogue of Young’s inequality for convolutions of functions for general Morrey-type spaces", Funktzionalny. prostranstva, teorija pribl., smezhnye vopr. analiza for 110-birthday of academician S. M.Nikolskij. Tr. MIAN 293, 113–132 (2016) [In Russian].MathSciNetzbMATHGoogle Scholar
  5. 5.
    Burenkov, V.I., Tararykova, T.V. “Young’s inequality for convolutions in Morrey-type spaces”, Eurasian Math. J. 7(2), 92–99 (2016).MathSciNetzbMATHGoogle Scholar
  6. 6.
    Avsyankin, O.G. "On compactness of convolution-type operators in Morrey spaces", Matem. zametki 102(4), 483–489 (2017) [in Russian].MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Avsyankin, O.G. "On integral Volterra-type operators with homogeneous kernels in weight L p-spaces", Izv. Vuz. Matem., 11, 3–12 (2017) [in Russian].Google Scholar
  8. 8.
    Simonenko, I.B. "Convolution type operators in cones", Matem. sb. 74(2), 298–314 (1967) [in Russian].MathSciNetzbMATHGoogle Scholar
  9. 9.
    Gohberg, I.Tz., Feldman, I.A. Equations in convolution and projection methods of their solutions (Nauka. Moscow, 1971) [in Russian].Google Scholar
  10. 10.
    Gelfand, I.M., Rajkov, D.A., Shilov, G.E. Commutative normed rings (Fizmatgiz. Moscow, 1960) [in Russian].Google Scholar
  11. 11.
    Shilov, G.E. Mathematical analysis (functions of one variable). P. 3 (Nauka. Moscow, 1970) [in Russian].Google Scholar

Copyright information

© Allerton Press, Inc. 2019

Authors and Affiliations

  1. 1.Southern Federal UniversityRostov-on-DonRussia

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