Weighted local Morrey spaces

Abstract

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 | · | α is considered. The condition on α for which the Hardy-Littlewood maximal operator is bounded can be described completely.

Singular and fractional integral operators on weighted local Morrey spaces

“…Generalized weighted (global and local) Morrey spaces have been considered by several authors. For general weighted local Morrey spaces we refer for instance to [16].…”

“…We recall that B denotes the smallest ball centered at the origin containing B. The main idea for this reduction comes from [16]. We adapt here to local Morrey spaces the proof given in [3, Proposition 2.1] for global Morrey spaces.…”

“…The result for n/p − n/q = α and β 1 = β 2 is in [16]. When β 1 = β 2 = 0 this is known as Spanne's result (usually stated for global Morrey spaces).…”

“…The characterization of the weights w for which the Hardy-Littlewood maximal operator is bounded on global Morrey spaces is an open problem even for the most familiar cases of ϕ mentioned in the preceding paragraph. Nevertheless, a remarkable result of S. Nakamura, Y. Sawano, and H. Tanaka in [16] gives necessary and sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator M on weighted local Morrey spaces. In [4] the authors of this paper obtained a simplified characterization of the same result in the form of a Muckenhoupt-type condition adapted to the Morrey setting.…”

“…They are given by expressions which are similar to (1.2) and (1.3), respectively. In the case of M α a necessary and sufficient condition was already obtained in [16]. As we did for the Hardy-Littlewood maximal operator, we simplify the characterization and give it by means of a single condition, namely, (1.4) sup…”