Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 42, 2017, 367-391

GENERAL MAXIMAL OPERATORS AND THE REVERSE HÖLDER CLASSES

Hiroki Saito and Hitoshi Tanaka

Kogakuin University, Academic Support Center
2665–1, Nakanomachi, Hachioji-shi Tokyo, 192–0015, Japan; j1107703 'at' gmail.com

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 City, Ibaraki, 305-8521 Japan; htanaka 'at' k.tsukuba-tech.ac.jp

Abstract. By a basis in Rⁿ we mean a collection of open and bounded sets B. In this paper we show that, if the general maximal operator M_B is bounded on L^p(Rⁿ) for p > 1 and the weight w belongs to the reverse Hölder RH_∞,B class, then the weighted maximal operator M_B,w is bounded on L^p(Rⁿ,w) for p > 1. When the general basis B has dyadic substructure with the Stein property, we investigate the equivalence between the Muckenhoupt class A_∞,B and the reverse Hölder class RH_1,B. We also discuss equivalent ways of defining the reverse Hölder class RH_1,B.

2010 Mathematics Subject Classification: Primary 42B25; Secondary 42B35.

Key words: General maximal operator, Gehring lemma, Fujii–Wilson constant, Muckenhoupt weight class, reverse Hölder weight class, Stein property.

Reference to this article: H. Saito and H. Tanaka: General maximal operators and the reverse Hölder classes. Ann. Acad. Sci. Fenn. Math. 42 (2017), 367-391.

Full document as PDF file

https://doi.org/10.5186/aasfm.2017.4227