scholarly journals On the behaviors of rough multilinear fractional integral and multi-sublinear fractional maximal operators both on product Lp and weighted Lp spaces

2020 ◽  
Vol 21 (7-8) ◽  
pp. 715-726
Author(s):  
Ferit Gürbüz
Keyword(s):  
Fractional Integral ◽  
Weak Type ◽  
Fractional Integration ◽  
Maximal Operators ◽  
Product Spaces ◽  
Type Estimate ◽  
Fractional Integral Operators ◽  
Multilinear Fractional Integral ◽  
Weighted Estimates ◽  
Lp Spaces

AbstractThe aim of this paper is to get the product ${L}^{p}$-estimates, weighted estimates and two-weighted estimates for rough multilinear fractional integral operators and rough multi-sublinear fractional maximal operators, respectively. The author also studies two-weighted weak type estimate on product ${L}^{p}\left({\mathrm{\mathbb{R}}}^{n}\right)$ for rough multi-sublinear fractional maximal operators. In fact, this article is the rough kernel versions of [C. E. Kenig and E. M. Stein, “Multilinear estimates and fractional integration,” Math. Res. Lett., vol. 6, pp. 1–15, 1999, Y. Shi and X. X. Tao, “Weighted ${L}_{p}$ boundedness for multilinear fractional integral on product spaces,” Anal. Theory Appl., vol. 24, no. 3, pp. 280–291, 2008]'s results.

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

2019 ◽  
Vol 62 (02) ◽  
pp. 313-326 ◽  
Author(s):  
Xianming Hou ◽  
Huoxiong Wu
Keyword(s):  
Fractional Integral ◽  
Weak Type ◽  
Integral Operators ◽  
Unit Vector ◽  
Maximal Functions ◽  
Maximal Operators ◽  
Fractional Integral Operators ◽  
Littlewood Maximal Function ◽  
Image Position ◽  
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 $q=n/(n-\unicode[STIX]{x1D6FC})$ , $f\in L^{1}(\mathbb{R}^{n},d\unicode[STIX]{x1D707})$ , $$\begin{eqnarray}\displaystyle \lim _{\unicode[STIX]{x1D706}\rightarrow 0+}\unicode[STIX]{x1D706}^{q}\unicode[STIX]{x1D707}(\{x\in \mathbb{R}^{n}:M_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6FC}}f(x)>\unicode[STIX]{x1D706}\})=\frac{\unicode[STIX]{x1D714}_{n-1}}{(n+\unicode[STIX]{x1D6FD})\unicode[STIX]{x1D707}(B(e_{1},1))}\Vert f\Vert _{L^{1}(\mathbb{R}^{n},d\unicode[STIX]{x1D707})}^{q}, &amp; &amp; \displaystyle \nonumber\\ \displaystyle \lim _{\unicode[STIX]{x1D706}\rightarrow 0+}\unicode[STIX]{x1D706}^{q}\unicode[STIX]{x1D707}\left(\left\{x\in \mathbb{R}^{n}:\left|M_{\unicode[STIX]{x1D707}}^{\unicode[STIX]{x1D6FC}}f(x)-\frac{\Vert f\Vert _{L^{1}(\mathbb{R}^{n},d\unicode[STIX]{x1D707})}}{\unicode[STIX]{x1D707}(B(x,|x|))^{1-\unicode[STIX]{x1D6FC}/n}}\right|>\unicode[STIX]{x1D706}\right\}\right)=0, &amp; &amp; \displaystyle \nonumber\end{eqnarray}$$ 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.

scholarly journals Two-weight norm inequalities for fractional integral operators with $A_{\lambda,\infty}$ weights

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Junren Pan ◽  
Wenchang Sun
Keyword(s):  
Fractional Integral ◽  
Integral Operators ◽  
Type Estimate ◽  
Norm Inequalities ◽  
Fractional Integral Operators ◽  
New Class ◽  
Formula Type

Abstract In this paper, we introduce a new class of weights, the $A_{\lambda, \infty}$Aλ,∞ weights, which contains the classical $A_{\infty}$A∞ weights. We prove a mixed $A_{p,q}$Ap,q–$A_{\lambda,\infty}$Aλ,∞ type estimate for fractional integral operators.

scholarly journals Pathway fractional integral operators of generalized k-wright function and k4-function

2017 ◽  
Vol 35 (2) ◽  
pp. 235 ◽  
Author(s):  
Dinesh Kumar ◽  
Ram Kishore Saxena ◽  
Jitendra Daiya
Keyword(s):  
Fractional Integral ◽  
Fractional Integration ◽  
Integral Operators ◽  
Wright Function ◽  
Finite Product ◽  
Fractional Integral Operators ◽  
Integral Formulas ◽  
Fractional Integration Operator ◽  
Composition Formula ◽  
Special Cases

In the present work we introduce a composition formula of the pathway fractional integration operator with finite product of generalized K-Wright function and K4-function. The obtained results are in terms of generalized Wright function.Certain special cases of the main results given here are also considered to correspond with some known and new (presumably) pathway fractional integral formulas.

Block Decomposition and Weighted Hausdorff Content

2019 ◽  
Vol 63 (1) ◽  
pp. 141-156
Author(s):  
Hiroki Saito ◽  
Hitoshi Tanaka ◽  
Toshikazu Watanabe
Keyword(s):  
Fractional Integral ◽  
Integral Operators ◽  
Maximal Operators ◽  
Block Decomposition ◽  
Fractional Integral Operators

AbstractBlock decomposition of $L^{p}$ spaces with weighted Hausdorff content is established for $0<p\leqslant 1$ and the Fefferman–Stein type inequalities are shown for fractional integral operators and some variants of maximal operators.

scholarly journals Weak Type Estimates of Variable Kernel Fractional Integral and Their Commutators on Variable Exponent Morrey Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Xukui Shao ◽  
Shuangping Tao
Keyword(s):  
Lebesgue Space ◽  
Variable Exponent ◽  
Fractional Integral ◽  
Weak Type ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Variable Kernel ◽  
Fractional Integral Operators ◽  
Weak Type Estimates ◽  
Weak Morrey Spaces

In this paper, the authors obtain the boundedness of the fractional integral operators with variable kernels on the variable exponent weak Morrey spaces based on the results of Lebesgue space with variable exponent as the infimum of exponent function p(·) equals 1. The corresponding commutators generated by BMO and Lipschitz functions are considered, respectively.

scholarly journals Estimates of Fractional Integral Operators on Variable Exponent Lebesgue Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-7
Author(s):  
Canqin Tang ◽  
Qing Wu ◽  
Jingshi Xu
Keyword(s):  
Integral Operator ◽  
Maximal Operator ◽  
Variable Exponent ◽  
Fractional Integral ◽  
Weak Type ◽  
Type Inequality ◽  
Integral Operators ◽  
Lebesgue Spaces ◽  
Fractional Integral Operators ◽  
Variable Exponent Lebesgue Spaces

By some estimates for the variable fractional maximal operator, the authors prove that the fractional integral operator is bounded and satisfies the weak-type inequality on variable exponent Lebesgue spaces.

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