Fractional integrals and derivatives: Mapping properties

Humberto Rafeiro, Stefan Samko

Research output: Contribution to journalReview articlepeer-review

16 Citations (Scopus)

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.

Original languageEnglish
Pages (from-to)580-607
Number of pages28
JournalFractional Calculus and Applied Analysis
Volume19
Issue number3
DOIs
Publication statusPublished - Jun 1 2016
Externally publishedYes

Keywords

  • Grand spaces, Morrey spaces
  • Hölder spaces
  • Lebesgue spaces
  • Riesz potential
  • fractional derivatives
  • fractional integral
  • hypersingular integrals
  • mapping properties
  • variable exponent Hölder spaces
  • variable exponent Lebesgue spaces

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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