Compactness criterions for certain commutators of oscillatory singular integrals

Abstract

Let Ω be homogeneous of degree zero, integrable and have mean zero on $S^{n-1}$, $P(x,y)$ be a real-valued polynomial on $R^n\times R^n$. Define the oscillatory singular integral $T_{\Omega }^P$ by$T_{\Omega }^{P}f\left(x\right)=p.v.\underset{R^n}{\int }{e}^{iP(x,y)}\frac{\Omega (x-y)}{|x-y{|}^n}f\left(y\right)dy.$ This paper is devoted to studying the compactness of $[b,T_{\Omega }^P]$, the commutators formed by $T_{\Omega }^P$ with $b\in L_{\text{loc}}^1\left(R^n\right)$. Several compactness criteria of $[b,T_{\Omega }^P]$ on Lebesgue spaces and Morrey spaces are given. As applications, the compactness of $[b,T_{\Omega }^P]$ with rough kernels on Lebesgue spaces and Morrey spaces is obtained.