Calderón Operator on Local Morrey Spaces with Variable Exponents

Abstract

In this paper, we establish the boundedness of the Calderón operator on local Morrey spaces with variable exponents. We obtain our result by extending the extrapolation theory of Rubio de Francia to the local Morrey spaces with variable exponents. The exponent functions of the local Morrey spaces with the exponent functions are only required to satisfy the log-Hölder continuity assumption at the origin and infinity only. As special cases of the main result, we have Hardy’s inequalities, the Hilbert inequalities and the boundedness of the Riemann–Liouville and Weyl averaging operators on local Morrey spaces with variable exponents.

Modular Hadamard, Riemann-Liouville and Weyl fractional integrals

“…Furthermore, Corollary 4.7 also yields the mapping properties for the one-sided maximal operators M − α and M + α on Orlicz spaces and Lorentz-Orlicz spaces. In addition, we can also obtain the mapping properties for R α and W α by using the extrapolation such as the results in [16]. For simplicity, we skip the details and leave it to the reader.…”