Boundedness of some sublinear operators and their commutators on generalized local Morrey spaces

Balakishiyev A., Gadjieva E., Gürbüz F., ŞERBETÇİ A.

Complex Variables and Elliptic Equations, vol.63, no.11, pp.1620-1641, 2018 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 63 Issue: 11
  • Publication Date: 2018
  • Doi Number: 10.1080/17476933.2017.1403423
  • Journal Name: Complex Variables and Elliptic Equations
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Page Numbers: pp.1620-1641
  • Keywords: fractional integral operator, generalized local Morrey spaces, Sublinear operator
  • Hakkari University Affiliated: No

Abstract

In this paper we prove the boundedness of certain sublinear operators TΩα, TΩα, α ∈ (0,n), generated by fractional integral operators with rough kernels Ω ∈ Ls(Sn−1), s > 1, from one generalized local Morrey space (Formula presented.) to another (Formula presented.), (Formula presented.), (Formula presented.), and from the space (Formula presented.) to the weak space (Formula presented.), (Formula presented.), (Formula presented.). In the case b belongs to the local Campanato space (Formula presented.) and TΩbα is a linear operator, we find the sufficient conditions on the pair (ϕ1ϕ2) which ensures the boundedness of the commutator operators TΩbα from (Formula presented.) to (Formula presented.), (Formula presented.), (Formula presented.), (Formula presented.), (Formula presented.). In all cases the conditions for the boundedness of (Formula presented.) are given in terms of Zygmund-type integral inequalities on (ϕ1ϕ2), which do not assume any assumption on monotonicity of (ϕ1ϕ2) in r.