Hokkaido Mathematical Journal

A remark on modified Morrey spaces on metric measure spaces

Yoshihiro SAWANO, Tetsu SHIMOMURA, and Hitoshi TANAKA}

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Abstract

Morrey norms, which are originally endowed with two parameters, are considered on general metric measure spaces. Volberg, Nazarov and Treil showed that the modified Hardy-Littlewood maximal operator is bounded on Legesgue spaces. The modified Hardy-Littlewood maximal operator is known to be bounded on Morrey spaces on Euclidean spaces, if we introduce the third parameter instead of adopting a natural extension of Morrey spaces. When it comes to geometrically doubling, as long as an auxiliary parameter is introduced suitably, the Morrey norm does not depend on the third parameter and this norm extends naturally the original Morrey norm. If the underlying space has a rich geometric structure, there is still no need to introduce auxiliary parameters. However, an example shows that this is not the case in general metric measure spaces. In this paper, we present an example showing that Morrey spaces depend on the auxiliary parameters.

Article information

Source
Hokkaido Math. J., Volume 47, Number 1 (2018), 1-15.

Dates
First available in Project Euclid: 13 March 2018

Permanent link to this document
https://projecteuclid.org/euclid.hokmj/1520928055

Digital Object Identifier
doi:10.14492/hokmj/1520928055

Mathematical Reviews number (MathSciNet)
MR3773722

Zentralblatt MATH identifier
06853588

Subjects
Primary: 42B35: Function spaces arising in harmonic analysis
Secondary: 42B25: Maximal functions, Littlewood-Paley theory

Keywords
Morrey space non-doubling metric measure spaces

Citation

SAWANO, Yoshihiro; SHIMOMURA, Tetsu; TANAKA}, Hitoshi. A remark on modified Morrey spaces on metric measure spaces. Hokkaido Math. J. 47 (2018), no. 1, 1--15. doi:10.14492/hokmj/1520928055. https://projecteuclid.org/euclid.hokmj/1520928055


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