Analysis and Mathematical Physics

, Volume 9, Issue 4, pp 2109–2132

# Necessary and sufficient conditions for the bounds of the commutator of a Littlewood-Paley operator with fractional differentiation

Article

## Abstract

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 $$0<\alpha < 1$$.

## Keywords

Commutator Littlewood-Paley operator Rough kernel BMO Sobolev spaces

42B20 42B25

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© Springer Nature Switzerland AG 2019

## Authors and Affiliations

• Xiongtao Wu
• 1
• Yanping Chen
• 2
Email author
• Liwei Wang
• 3
• Wenyu Tao
• 4
1. 1.Department of Mathematics, School of Mathematics and StatisticsHengyang Normal UniversityHengyangChina
2. 2.Department of Applied Mathematics, School of Mathematics and PhysicsUniversity of Science and Technology BeijingBeijingChina
3. 3.School of Mathematics and PhysicsAnhui Polytechnic UniversityWuhuChina
4. 4.School of Mathematics and PhysicsUniversity of Science and Technology BeijingBeijingChina