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

ABSTRACT

In this paper we prove the boundedness of certain sublinear operators , , generated by fractional integral operators with rough kernels , , from one generalized local Morrey space to another , , , and from the space to the weak space , , . In the case b belongs to the local Campanato space and is a linear operator, we find the sufficient conditions on the pair which ensures the boundedness of the commutator operators from to , , , , . In all cases the conditions for the boundedness of are given in terms of Zygmund-type integral inequalities on , which do not assume any assumption on monotonicity of in r.

KEYWORDS:

• Sublinear operator
• fractional integral operator
• generalized local Morrey spaces

AMS Subject Classifications:

• 42B20
• 42B25
• 42B35

Acknowledgements

We are indebted to the Referees for very careful reading and many valuable suggestions and comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of E. A. Gadjieva was partially supported by the grant of Presidium Azerbaijan National Academy of Science 2015.