
On the limiting weaktype behaviors for maximal operators associated with power weighted measure
Let $\beta\ge 0$ and $e_1=(1,0,\ldots,0)$ is a unit vector on
$\mathbb{R}^{n}$, $d\mu(x)=x^\beta dx$ is a power weighted
measure on $\mathbb{R}^n$. For $0\le \alpha\lt n$, let $M_\mu^\alpha$
be the centered HardyLittlewood maximal function and fractional
maximal functions associated to measure $\mu$. This paper shows
that for $q=n/(n\alpha)$, $f\in L^1(\mathbb{R}^n,d\mu)$,
$$\lim\limits_{\lambda\to 0+}\lambda^q \mu(\{x\in\mathbb{R}^n:M_\mu^\alpha
f(x)\gt \lambda\})=\frac{\omega_{n1}}{(n+\beta)\mu(B(e_1,1))}\f\_{L^1(\mathbb{R}^n,
d\mu)}^q,$$
and
$$\lim_{\lambda\to 0+}\lambda^q \mu\Big(\Big\{x\in\mathbb{R}^n:\BigM_\mu^\alpha
f(x)\frac{\f\_{L^1(\mathbb{R}^n, d\mu)}}{\mu(B(x,x))^{1\alpha/n}}\Big\gt \lambda\Big\}\Big)=0,$$
which is new and stronger than the previous result even if $\beta=0$.
Meanwhile, the corresponding results for the uncentered maximal
functions as well as the fractional integral operators with respect
to measure $\mu$ are also obtained.
Keywords:limiting weak type behavior, power weight, HardyLittlewood maximal operator, fractional maximal operator, fractional integral Categories:42B20, 42B25 