Weighted weak type estimates for certain maximal operators in spaces of homogeneous type

Bull. Korean Math. Soc. 1999 Vol. 36, No. 1, 25-31

Yoon-Jae Yoo Kyungpook National University

Abstract : Let $\nu$ be a positive Borel measure on a space of homogeneous type $(X,d,\mu)$, satisfying the doubling property. A condition on a weight $w$ for which a maximal operator $ M_\nu f(x)$ defined by $$ M_\nu f(x)=\sup_{r>0} \frac {1 }{\nu(B(x,r))} \int_{B(x,r)} |f(y)|d\mu(y), $$ is of weak type $(p,p)$ with respect to $(\nu,w)$, is that there exists a constant $C$ such that $C\le w(y)$ for a.e. $y \in B(x,r)$ if $p=1$, and $\left (\frac{1}{\nu(B(x,r))} \int_{B(x,r)} w(y)^{-\frac{1}{p-1}}d\mu(y)\right)^{p-1} \le C, $ if $1

Keywords : maximal operator, spaces of homogeneous type, weights, weak type