On the Limiting Weak-type Behaviors for Maximal Operators Associated with Power Weighted Measure
AbstractLet $\unicode[STIX]{x1D6FD}\geqslant 0$ , 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.