Calderón Operator on Local Morrey Spaces with Variable Exponents

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.

Parabolic local Campanato estimates for commutators of parabolic fractional maximal and integral operators with rough kernel

In this paper, the author introduces parabolic generalized local Morrey spaces and parabolic local Campanato spaces, respectively and also establishes parabolic local Campanato estimates for commutators of parabolic fractional maximal and integral operators with rough kernel on parabolic generalized local Morrey spaces.