The sharp constant for truncated Hardy-Littlewood maximal inequality

Abstract

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