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Fractional integrals and derivatives: mapping properties

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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 19 Issue 3

Abstract

This survey is aimed at the audience of readers interested in the information on mapping properties of various forms of fractional integration operators, including multidimensional ones, in a large scale of various known function spaces.

As is well known, the fractional integrals defined in this or other forms improve in some sense the properties of the functions, at least locally, while fractional derivatives to the contrary worsen them. With the development of functional analysis this simple fact led to a number of important results on the mapping properties of fractional integrals in various function spaces.

In the one-dimensional case we consider both Riemann-Liouville and Liouville forms of fractional integrals and derivatives. In the multidimensional case we consider in particular mixed Liouville fractional integrals, Riesz fractional integrals of elliptic and hyperbolic type and hypersingular integrals. Among the function spaces considered in this survey, the reader can find Hölder spaces, Lebesgue spaces, Morrey spaces, Grand spaces and also weighted and/or variable exponent versions.

MSC 2010: Primary 26A33; Secondary 46E30
Key Words and Phrases: mapping properties; fractional integral; Riesz potential; hypersingular integrals; fractional derivatives; Lebesgue spaces; variable exponent Lebesgue spaces; Hölder spaces; variable exponent Hölder spaces; Grand spaces, Morrey spaces

Acknowledgements

H. Rafeiro was partially supported by Pontificia Universidad Javeriana under the research project “Study of mapping properties of fractional integrals and derivatives”, ID PROY: 7446.

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Received: 2015-11-15
Published Online: 2016-6-28
Published in Print: 2016-6-1

© 2016 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA-volume 19-3-2016)
  4. Survey Paper
  5. Fractional integrals and derivatives: mapping properties
  6. Research Paper
  7. Riesz fractional integrals in grand lebesgue spaces on ℝn
  8. Survey Paper
  9. United lattice fractional integro-differentiation
  10. Research Paper
  11. Integral equations of fractional order in Lebesgue spaces
  12. Research Paper
  13. General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems
  14. Research Paper
  15. Multilinear integral operators in weighted grand Lebesgue spaces
  16. Research Paper
  17. Fractional integration operator on some radial rays and intertwining for the Dunkl operator
  18. Research Paper
  19. Pseudo almost automorphy of semilinear fractional differential equations in Banach Spaces
  20. Research Paper
  21. Existence and uniqueness of global solutions of caputo-type fractional differential equations
  22. Research Paper
  23. Perfect nonlinear observers of fractional descriptor continuous-time nonlinear systems
https://doi.org/10.1515/fca-2016-0032
Keywords for this article
mapping properties; fractional integral; Riesz potential; hypersingular integrals; fractional derivatives; Lebesgue spaces; variable exponent Lebesgue spaces; Hölder spaces; variable exponent Hölder spaces; Grand spaces, Morrey spaces

Articles in the same Issue

  1. Frontmatter
  2. Editorial Note
  3. FCAA related news, events and books (FCAA-volume 19-3-2016)
  4. Survey Paper
  5. Fractional integrals and derivatives: mapping properties
  6. Research Paper
  7. Riesz fractional integrals in grand lebesgue spaces on ℝn
  8. Survey Paper
  9. United lattice fractional integro-differentiation
  10. Research Paper
  11. Integral equations of fractional order in Lebesgue spaces
  12. Research Paper
  13. General time-fractional diffusion equation: some uniqueness and existence results for the initial-boundary-value problems
  14. Research Paper
  15. Multilinear integral operators in weighted grand Lebesgue spaces
  16. Research Paper
  17. Fractional integration operator on some radial rays and intertwining for the Dunkl operator
  18. Research Paper
  19. Pseudo almost automorphy of semilinear fractional differential equations in Banach Spaces
  20. Research Paper
  21. Existence and uniqueness of global solutions of caputo-type fractional differential equations
  22. Research Paper
  23. Perfect nonlinear observers of fractional descriptor continuous-time nonlinear systems
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