$L^p$ compactness for Calderón type commutators
Tom 237 / 2017
Studia Mathematica 237 (2017), 1-23
MSC: Primary 42B20; Secondary 42B25, 47G10.
DOI: 10.4064/sm8088-9-2016
Opublikowany online: 23 January 2017
Streszczenie
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.
