$L^p$ compactness for Calderón type commutators

Ting Mei¹ ; Yong Ding²

¹School of Ethnic Minority Education Beijing University of Posts and Telecommunications 100876 Beijing, P.R. China and School of Mathematical Sciences Beijing Normal University 100875 Beijing, P.R. China
²School of Mathematical Sciences Beijing Normal University 100875 Beijing, P.R. China

Studia Mathematica, Tome 237 (2017) no. 1, pp. 1-23

Résumé

We discuss the $L^p$ compactness of Calderón type commutators $T_A$ defined by \begin{equation*} T_Af(x)=\text{p.v. }\int_{\mathbb R^n} \frac{\varOmega(x-y)}{|x-y|^{n+1}} R(A;x,y)f(y)\,dy, \end{equation*} where $R(A;x,y)=A(x)-A(y)-\nabla A(y)\cdot(x-y)$ with $D^\beta A\in \mathrm{BMO}(\mathbb R^n)$ for all $n\ge 2$ and $|\beta|=1$. Moreover, $\varOmega$ is homogeneous of degree zero and has a vanishing moment of order one on $\mathbb{S}^{n-1}$. We prove that both $T_A$ and its maximal operator $T_{A,*}$ are compact operators on $L^p(\mathbb R^n)$ for all $1 \lt p \lt \infty$ with $A$ satisfying some conditions. Moreover, the compactness of the fractional operators $I_{\alpha,A,m}$ and $M_{\alpha,A,m}$ is proved.

Détail

DOI : 10.4064/sm8088-9-2016

Keywords: discuss compactness calder type commutators defined begin equation* text int mathbb frac varomega x y x y y end equation* where y a nabla cdot x y beta mathrm bmo mathbb beta moreover varomega homogeneous degree zero has vanishing moment order mathbb n prove its maximal operator * compact operators mathbb infty satisfying conditions moreover compactness fractional operators alpha alpha proved