The sharp constant for truncated Hardy-Littlewood maximal inequality

TL;DR

This paper determines the exact operator norms of truncated Hardy-Littlewood maximal operators, providing insights into the sharp constants involved in classical maximal inequalities.

Contribution

It explicitly calculates the $L^1$-norms of both the truncated and strong truncated Hardy-Littlewood maximal operators, advancing understanding of their sharp constants.

Findings

Computed the $L^1$-norm of the truncated Hardy-Littlewood maximal operator.

Derived the $L^1$-norm of the strong truncated Hardy-Littlewood maximal operator.

Provided potential implications for the sharp constant in classical Hardy-Littlewood maximal inequality.

Abstract

This paper focuses on the operator norm of the truncated Hardy-Littlewood maximal operator $M^b_a$ and the strong truncated Hardy-Littlewood maximal operator $\tilde{M}^{\boldsymbol{b}}_{\boldsymbol{a}}$, respectively. We first present the $L^1$-norm of $M^b_a$, and then the $L^1$-norm of $\tilde{M}^{\boldsymbol{b}}_{\boldsymbol{a}}$ is given. Our study may have some enlightening significance for the research on sharp constant for the classical Hardy-Littlewood maximal inequality.