scholarly journals Norm Comparison Inequalities for the Composite Operator

2009 ◽  
Vol 2009 (1) ◽  
pp. 212915 ◽  
Author(s):  
Yuming Xing ◽  
Shusen Ding
2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Xuexin Li ◽  
Yong Wang ◽  
Yuming Xing

This paper obtains the Lipschitz and BMO norm estimates for the composite operator𝕄s∘Papplied to differential forms. Here,𝕄sis the Hardy-Littlewood maximal operator, andPis the potential operator. As applications, we obtain the norm estimates for the Jacobian subdeterminant and the generalized solution of the quasilinear elliptic equation.


2019 ◽  
Vol 109 (3) ◽  
pp. 320-339 ◽  
Author(s):  
GUOEN HU

AbstractLet $T_{1}$, $T_{2}$ be two Calderón–Zygmund operators and $T_{1,b}$ be the commutator of $T_{1}$ with symbol $b\in \text{BMO}(\mathbb{R}^{n})$. In this paper, by establishing new bilinear sparse dominations and a new weighted estimate for bilinear sparse operators, we prove that the composite operator $T_{1}T_{2}$ satisfies the following estimate: for $\unicode[STIX]{x1D706}>0$ and weight $w\in A_{1}(\mathbb{R}^{n})$, $$\begin{eqnarray}\displaystyle & & \displaystyle w(\{x\in \mathbb{R}^{n}:\,|T_{1}T_{2}f(x)|>\unicode[STIX]{x1D706}\})\nonumber\\ \displaystyle & & \displaystyle \qquad \lesssim [w]_{A_{1}}[w]_{A_{\infty }}\log (\text{e}+[w]_{A_{\infty }})\int _{\mathbb{R}^{n}}\frac{|f(x)|}{\unicode[STIX]{x1D706}}\log \bigg(\text{e}+\frac{|f(x)|}{\unicode[STIX]{x1D706}}\bigg)w(x)\,dx,\nonumber\end{eqnarray}$$ while the composite operator $T_{1,b}T_{2}$ satisfies $$\begin{eqnarray}\displaystyle & & \displaystyle w(\{x\in \mathbb{R}^{n}:\,|T_{1,b}T_{2}f(x)|>\unicode[STIX]{x1D706}\})\nonumber\\ \displaystyle & & \displaystyle \qquad \lesssim [w]_{A_{1}}[w]_{A_{\infty }}^{2}\log (\text{e}+[w]_{A_{\infty }})\int _{\mathbb{R}^{n}}\frac{|f(x)|}{\unicode[STIX]{x1D706}}\log ^{2}\bigg(\text{e}+\frac{|f(x)|}{\unicode[STIX]{x1D706}}\bigg)w(x)\,dx.\nonumber\end{eqnarray}$$


2018 ◽  
Vol 98 (5) ◽  
Author(s):  
Matthias Warschinke ◽  
Ryutaro Matsudo ◽  
Shogo Nishino ◽  
Toru Shinohara ◽  
Kei-Ichi Kondo

2018 ◽  
Vol 97 (3) ◽  
Author(s):  
Matthias Warschinke ◽  
Ryutaro Matsudo ◽  
Shogo Nishino ◽  
Toru Shinohara ◽  
Kei-Ichi Kondo

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Guannan Shi ◽  
Yuming Xing ◽  
Baiqing Sun

We first establish the local Poincaré inequality withL𝒜-averaging domains for the composition of the sharp maximal operator and potential operator, applied to the nonhomogenousA-harmonic equation. Then, according to the definition ofL𝒜-averaging domains and relative properties, we demonstrate the global Poincaré inequality withL𝒜-averaging domains. Finally, we give some illustrations for these theorems.


2001 ◽  
Vol 516 (3-4) ◽  
pp. 307-313 ◽  
Author(s):  
H. Verschelde ◽  
K. Knecht ◽  
K. Van Acoleyen ◽  
M. Vanderkelen
Keyword(s):  

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