Sharp estimates of the modified Hardy Littlewood maximal operator on the nonhomogeneous space via covering lemmas

Abstract

In this paper we consider the modified maximal operator on the separable metric space. Define $M_kf(x)= \sup_{r>0} \frac{1}{\mu(B(x,kr))}\int_{B(x,r)}|f(y)|d\mu(y)$ and $ M_{k,uc}f(x) = \sup_{x \in B(y,r)} \frac{1}{\mu(B(y,kr))}\int_{B(y,r)}|f(z)|d\mu(z)$ respectively. We investigate in what parameter $k$ the weak $(1,1)$-inequality holds for $M_k$ and $M_{k,uc}$ in general metric space and Euclidean space. The proofs are sharper than the method of Vitali's covering lemma. This attempt is partially done by Yutaka Terasawa [9] before. When we investigate ${\mathbf R}^d$, we prove a new covering lemma of ${\mathbf R}^d$. We also show that our condition on parameter $k$ is sharp. In connection with this we consider the dual inequality of Stein type and its applications.