Let
$\unicode[STIX]{x1D6FD}\geqslant 0$
, let
$e_{1}=(1,0,\ldots ,0)$
be a unit vector on
$\mathbb{R}^{n}$
, and let
$d\unicode[STIX]{x1D707}(x)=|x|^{\unicode[STIX]{x1D6FD}}dx$
be a power weighted measure on
$\mathbb{R}^{n}$
. For
$0\leqslant \unicode[STIX]{x1D6FC}<n$
, let
$M_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6FC}}$
be the centered Hardy-Littlewood maximal function and fractional maximal functions associated with measure
$\unicode[STIX]{x1D707}$
. This paper shows that for
$q=n/(n-\unicode[STIX]{x1D6FC})$
,
$f\in L^{1}(\mathbb{R}^{n},d\unicode[STIX]{x1D707})$
,
$$\begin{eqnarray}\displaystyle \lim _{\unicode[STIX]{x1D706}\rightarrow 0+}\unicode[STIX]{x1D706}^{q}\unicode[STIX]{x1D707}(\{x\in \mathbb{R}^{n}:M_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6FC}}f(x)>\unicode[STIX]{x1D706}\})=\frac{\unicode[STIX]{x1D714}_{n-1}}{(n+\unicode[STIX]{x1D6FD})\unicode[STIX]{x1D707}(B(e_{1},1))}\Vert f\Vert _{L^{1}(\mathbb{R}^{n},d\unicode[STIX]{x1D707})}^{q}, & & \displaystyle \nonumber\\ \displaystyle \lim _{\unicode[STIX]{x1D706}\rightarrow 0+}\unicode[STIX]{x1D706}^{q}\unicode[STIX]{x1D707}\left(\left\{x\in \mathbb{R}^{n}:\left|M_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6FC}}f(x)-\frac{\Vert f\Vert _{L^{1}(\mathbb{R}^{n},d\unicode[STIX]{x1D707})}}{\unicode[STIX]{x1D707}(B(x,|x|))^{1-\unicode[STIX]{x1D6FC}/n}}\right|>\unicode[STIX]{x1D706}\right\}\right)=0, & & \displaystyle \nonumber\end{eqnarray}$$
which is new and stronger than the previous result even if
$\unicode[STIX]{x1D6FD}=0$
. Meanwhile, the corresponding results for the un-centered maximal functions as well as the fractional integral operators with respect to measure
$\unicode[STIX]{x1D707}$
are also obtained.