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On the Limiting Weak-type Behaviors for Maximal Operators Associated with Power Weighted Measure

  • Xianming Hou (a1) and Huoxiong Wu (a1)
Abstract

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.

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Supported by the NNSF of China (Nos. 11771358, 11471041) and the NSF of Fujian Province of China (No. 2015J01025). Huoxiong Wu is the corresponding author.

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[1] Coifman, R. and Weiss, G., Analyse harmonique non-commutative sur certains espaces homogènes . Lecture Note in Mathematics, 242, Springer-Verlag, Berlin-New York, 1971.
[2] Ding, Y. and Lai, X., L 1-Dini conditions and limiting behavior of weak type estimates for singular integrals . Rev. Mat. Iberoam. 33(2017), no. 4, 12671284. https://doi.org/10.4171/RMI/971.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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