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Search: All articles in the CMB digital archive with keyword fractional maximal operator

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1. CMB Online first

Hou, Xianming; Wu, Huoxiong
On the limiting weak-type 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 Hardy-Littlewood maximal function and fractional maximal functions associated to measure $\mu$. This paper shows that for $q=n/(n-\alpha)$, $f\in L^1(\mathbb{R}^n,d\mu)$, $$\lim\limits_{\lambda\to 0+}\lambda^q \mu(\{x\in\mathbb{R}^n:M_\mu^\alpha f(x)\gt \lambda\})=\frac{\omega_{n-1}}{(n+\beta)\mu(B(e_1,1))}\|f\|_{L^1(\mathbb{R}^n, d\mu)}^q,$$ and $$\lim_{\lambda\to 0+}\lambda^q \mu\Big(\Big\{x\in\mathbb{R}^n:\Big|M_\mu^\alpha f(x)-\frac{\|f\|_{L^1(\mathbb{R}^n, d\mu)}}{\mu(B(x,|x|))^{1-\alpha/n}}\Big|\gt \lambda\Big\}\Big)=0,$$ which is new and stronger than the previous result even if $\beta=0$. Meanwhile, the corresponding results for the un-centered maximal functions as well as the fractional integral operators with respect to measure $\mu$ are also obtained.

Keywords:limiting weak type behavior, power weight, Hardy-Littlewood maximal operator, fractional maximal operator, fractional integral
Categories:42B20, 42B25

2. CMB 2016 (vol 60 pp. 131)

Gürbüz, Ferit
Some Estimates for Generalized Commutators of Rough Fractional Maximal and Integral Operators on Generalized Weighted Morrey Spaces
In this paper, we establish $BMO$ estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, respectively.

Keywords:fractional integral operator, fractional maximal operator, rough kernel, generalized commutator, $A(p,q)$ weight, generalized weighted Morrey space
Categories:42B20, 42B25

3. CMB 2016 (vol 60 pp. 586)

Liu, Feng; Wu, Huoxiong
Endpoint Regularity of Multisublinear Fractional Maximal Functions
In this paper we investigate the endpoint regularity properties of the multisublinear fractional maximal operators, which include the multisublinear Hardy-Littlewood maximal operator. We obtain some new bounds for the derivative of the one-dimensional multisublinear fractional maximal operators acting on vector-valued function $\vec{f}=(f_1,\dots,f_m)$ with all $f_j$ being $BV$-functions.

Keywords:multisublinear fractional maximal operators, Sobolev spaces, bounded variation
Categories:42B25, 46E35

4. CMB 2009 (vol 53 pp. 263)

Feuto, Justin; Fofana, Ibrahim; Koua, Konin
Weighted Norm Inequalities for a Maximal Operator in Some Subspace of Amalgams
We give weighted norm inequalities for the maximal fractional operator $ \mathcal M_{q,\beta }$ of Hardy–Littlewood and the fractional integral $I_{\gamma}$. These inequalities are established between $(L^{q},L^{p}) ^{\alpha }(X,d,\mu )$ spaces (which are superspaces of Lebesgue spaces $L^{\alpha}(X,d,\mu)$ and subspaces of amalgams $(L^{q},L^{p})(X,d,\mu)$) and in the setting of space of homogeneous type $(X,d,\mu)$. The conditions on the weights are stated in terms of Orlicz norm.

Keywords:fractional maximal operator, fractional integral, space of homogeneous type
Categories:42B35, 42B20, 42B25

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