Sparse bounds for maximal oscillatory rough singular integral operators

Abstract

In this paper, we prove sparse bounds for the maximal oscillatory rough singular integral operator$T_{\Omega ,⁎}^{P}f\left(x\right):=\underset{ϵ>0}{\sup }\left|\underset{|x-y|>ϵ}{\int }{e}^{\iota P(x,y)}\frac{\Omega \left(\right(x-y)/|x-y\left|\right)}{|x-y{|}^n}f\right(y\left)dy\right|,$ where $P(x,y)$ is a real-valued polynomial on $R^n\times R^n$ and $\Omega \in L^{\infty }\left(S^{n-1}\right)$ is a homogeneous function of degree zero with ${\int }_{S^{n-1}}\Omega \left(\theta \right)d\theta =0$. This allows us to conclude weighted $L^p$-estimates for the operator $T_{\Omega ,⁎}^P$. Moreover, the norm ${\Vert T_{\Omega ,⁎}^P\Vert }_{L^p\left(\omega \right)\to L^p\left(\omega \right)}$ depends only on the total degree of the polynomial $P(x,y)$, but not on the coefficients of $P(x,y)$. Finally, we will show that these techniques also apply to obtain sparse bounds for oscillatory rough singular integral operator $T_{\Omega }^P$ for $\Omega \in L^q\left(S^{n-1}\right)$, $1<q\le \infty$.