Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 67-93

WEIGHTED LOCAL MORREY SPACES

Shohei Nakamura, Yoshihiro Sawano and Hitoshi Tanaka

Tokyo Metropolitan University, Department of Mathematics and Information Sciences
Minami-Ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan; nakamura-shouhei 'at' ed.tmu.ac.jp

Tokyo Metropolitan University, Department of Mathematics and Information Sciences
Minami-Ohsawa 1-1, Hachioji-shi, Tokyo 192-0397, Japan; ysawano 'at' tmu.ac.jp
and Department of Mathematical Analysis and Theory of Functions
Peoples Friendship University of Russia (RUDN University)
6 Miklukho-Maklay St, Moscow, 117198 Russia

National University Corporation Tsukuba University of Technology
Research and Support Center on Higher Education for the Hearing and Visually Impaired
Kasuga 4-12-7, Tsukuba 305-8521, Japan; htanaka 'at' k.tsukuba-tech.ac.jp

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.

2010 Mathematics Subject Classification: Primary 42B25, 42B35, 26A33.

Key words: Local Morrey spaces of Samko type, local Morrey spaces of Komori–Shirai type, weights.

Reference to this article: S. Nakamura, Y. Sawano and H. Tanaka: Weighted local Morrey spaces. Ann. Acad. Sci. Fenn. Math. 45 (2020), 67-93.

Full document as PDF file

https://doi.org/10.5186/aasfm.2020.4504