WEIGHTED LOCAL MORREY SPACES

We discuss the boundedness of linear and sublinear operators in two types of weighted local Morrey spaces. One is defined by Natasha Samko in 2008. The other is defined by Yasuo Komori-Furuya and Satoru Shirai in 2009. We characterize the class of weights for which the Hardy-Littlewood maximal operator is bounded. Under a certain integral condition it turns out that the singular integral operators are bounded if and only if the Hardy-Littlewood maximal operator is bounded. As an application of the characterization, the power weight function vertical bar . vertical bar(alpha) is considered. The condition on a for which the Hardy-Littlewood maximal operator is bounded can be described completely.