Fractional integral operators on Hardy local Morrey spaces with variable exponents

Abstract

We establish the mapping properties of the fractional integral operators on the Hardy local Morrey spaces with variable exponents by using the extrapolation theory. The local Morrey spaces with variable exponents are generalizations and extensions of the local Morrey spaces, the Lebesgue spaces with variable exponents and the Morrey spaces with variable exponents. The Hardy local Morrey spaces with variable exponents are the Hardy spaces built on the local Morrey spaces with variable exponents. Our main result extends and generalizes the mapping properties of the fractional integral operators on the Hardy spaces, the local Morrey spaces, the Hardy spaces with variable exponents and the local Morrey spaces with variable exponents. We obtain our main result by extending the J.L. Rubio de Francia extrapolation theory to the local Morrey spaces with variable exponents. This method was originally developed by J.L. Rubio de Francia on the weighted Lebesgue spaces and recently it has been extended to a number of function spaces such as the Morrey spaces with variable exponents, the local Morrey spaces with variable exponents and the Morrey-Banach spaces. We further extend it to the local Morrey spaces with variable exponents. By using the mapping properties of the fractional integral operators on the weighted Hardy spaces, we establish the mapping properties of the fractional integral operators on the Hardy local Morrey spaces with variable exponents.