摘要：
For $$b\in L_{\mathrm{loc}}({\mathbb {R}}^n)$$ and $$0<\alpha <1$$, we use fractional differentiation to define a new type of commutator of the Littlewood-Paley g-function operator, namely $$\begin{aligned} g_{\Omega ,\alpha ;b}(f )(x) =\bigg (\int _0^\infty \bigg |\frac{1}{t} \int _{|x-y|\le t}\frac{\Omega (x-y)}{|x-y|^{n+\alpha -1}}(b(x)-b(y))f(y)\,dy\bigg |^2\frac{dt}{t}\bigg )^{1/2}. \end{aligned}$$Here, we obtain the necessary and sufficient conditions for the function b to guarantee that $$g_{\Omega ,\alpha ;b}$$ is a bounded operator on $$L^2({\mathbb {R}}^n)$$. More precisely, if $$\Omega \in L(\log ^+ L)^{1/2}{(S^{n-1})}$$ and $$b\in I_{\alpha }(BMO)$$, then $$g_{\Omega ,\alpha ;b}$$ is bounded on $$L^2({\mathbb {R}}^n)$$. Conversely, if $$g_{\Omega ,\alpha ;b}$$ is bounded on $$L^2({\mathbb {R}}^n)$$, then $$b \in Lip_\alpha ({\mathbb {R}}^n)$$ for $$0<\alpha < 1$$.
期刊：
JOURNAL OF INEQUALITIES AND APPLICATIONS,2018年2018(1):1-13 ISSN：1029-242X
通讯作者：
Huang, Haiwu
作者机构：
[Wu, Xiongtao; Huang, Haiwu] Hengyang Normal Univ, Coll Math & Stat, Hengyang, Peoples R China.;[Huang, Haiwu] Hunan Prov Key Lab Intelligent Informat Proc & Ap, Hengyang, Peoples R China.;[Zhang, Qingxia] Southwest Petr Univ, Sch Sci, Chengdu, Sichuan, Peoples R China.
通讯机构：
[Huang, Haiwu] H;Hengyang Normal Univ, Coll Math & Stat, Hengyang, Peoples R China.;Hunan Prov Key Lab Intelligent Informat Proc & Ap, Hengyang, Peoples R China.
关键词：
ANA random variables;Complete convergence;Weighed sums;Equivalent conditions
摘要：
In this investigation, some sufficient and necessary conditions of the complete convergence for weighted sums of asymptotically negatively associated (ANA, in short) random variables are presented without the assumption of identical distribution. As an application of the main results, the Marcinkiewicz-Zygmund type strong law of large numbers based on weighted sums of ANA cases is obtained. The results of this paper extend and generalize some well-known corresponding ones.
作者：
Wu, Xiongtao;Tao, Wenyu;Chen, Yanping*;Zhu, Kai
期刊：
Journal of Function Spaces,2018年2018:1-10 ISSN：2314-8896
通讯作者：
Chen, Yanping
作者机构：
[Wu, Xiongtao] Hengyang Normal Univ, Sch Math & Stat, Dept Math, Hengyang 421008, Peoples R China.;[Tao, Wenyu] Univ Sci & Technol Beijing, Sch Math & Phys, Beijing 100083, Peoples R China.;[Chen, Yanping] Univ Sci & Technol Beijing, Sch Math & Phys, Dept Appl Math, Beijing 100083, Peoples R China.;[Zhu, Kai] Beijing Normal Univ, Sch Math Sci, Beijing 100875, Peoples R China.
通讯机构：
[Chen, Yanping] U;Univ Sci & Technol Beijing, Sch Math & Phys, Dept Appl Math, Beijing 100083, Peoples R China.
摘要：
<jats:p>Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>L</mml:mi><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:mi mathvariant="normal">d</mml:mi><mml:mi mathvariant="normal">i</mml:mi><mml:mi mathvariant="normal">v</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>A</mml:mi><mml:mo>∇</mml:mo><mml:mo stretchy="false">)</mml:mo></mml:math> be a second-order divergence form elliptic operator, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M2"><mml:mrow><mml:mi>A</mml:mi></mml:mrow></mml:math> is an accretive <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M3"><mml:mi>n</mml:mi><mml:mo>×</mml:mo><mml:mi>n</mml:mi></mml:math> matrix with bounded measurable complex coefficients in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M4"><mml:msup><mml:mrow><mml:mi mathvariant="double-struck">R</mml:mi></mml:mrow><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:msup><mml:mo>.</mml:mo></mml:math> In this paper, we mainly establish the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M5"><mml:mrow><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> boundedness for the commutators generated by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><mml:mi>b</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mrow><mml:mi>I</mml:mi></mml:mrow><mml:mrow><mml:mi>α</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi>B</mml:mi><mml:mi>M</mml:mi><mml:mi>O</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math> and the square function related to fractional differentiation for second-order elliptic operators.</jats:p>
