The Boundedness of the Hardy-Littlewood Maximal Function and the Strong Maximal Function on the Space BMO

Author

Wenhao ZhangFollow

Graduation Year

2018

Date of Submission

4-2018

Document Type

Open Access Senior Thesis

Degree Name

Bachelor of Arts

Department

Mathematics

Reader 1

Winston Ou

Reader 2

Chiu-Yen Kao

Terms of Use & License Information

Terms of Use for work posted in Scholarship@Claremont.

Abstract

In this thesis, we present the space BMO, the one-parameter Hardy-Littlewood maximal function, and the two-parameter strong maximal function. We use the John-Nirenberg inequality, the relation between Muckenhoupt weights and BMO, and the Coifman-Rochberg proposition on constructing A₁ weights with the Hardy- Littlewood maximal function to show the boundedness of the Hardy-Littlewood maximal function on BMO. The analogous statement for the strong maximal function is not yet understood. We begin our exploration of this problem by discussing an equivalence between the boundedness of the strong maximal function on rectangular BMO and the fact that the strong maximal function maps A_∞ weights into the A₁ class. We then extend a multiparameter counterexample to the Coifman-Rochberg proposition proposed by Soria (1987) and discuss the difficulties in modifying it into an A_∞ counterexample that would disapprove the boundedness of the strong maximal function.

Recommended Citation

Zhang, Wenhao, "The Boundedness of the Hardy-Littlewood Maximal Function and the Strong Maximal Function on the Space BMO" (2018). CMC Senior Theses. 1907.
https://scholarship.claremont.edu/cmc_theses/1907