# Hokkaido Mathematical Journal

## Hokkaido Mathematical Journal, 34 (2005) pp.435-458

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### 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 \cite{Te} before. When we investigate ${\rm \R}^d$, we prove a new covering lemma of ${\rm \R}^d$. 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.

MSC(Primary) 42B25 maximal operator, covering lemma, non-homogeneous