Sparse bounds for maximal oscillatory rough singular integral operators

Abstract

We prove sparse bounds for maximal oscillatory rough singular integral operator $$T^{P}_{\Omega,*}f(x):=\sup_{\epsilon>0} \left|\int_{|x-y|>\epsilon}e^{\iota P(x,y)}\frac{\Omega\big((x-y)/|x-y|\big)}{|x-y|^{n}}f(y)dy\right|,$$ where $P(x,y)$ is a real-valued polynomial on $\mathbb{R}^{n}\times \mathbb{R}^{n}$ and $\Omega\in L^{\infty}(\mathbb{S}^{n-1})$ is a homogeneous function of degree zero with $\int_{\mathbb{S}^{n-1}}\Omega(\theta)~d\theta=0$. This allows us to conclude weighted $L^p-$estimates for the operator $T^{P}_{\Omega,*}$. Moreover, the norm $\|T^P_{\Omega,*}\|_{L^p\rightarrow L^p}$ 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 $T^{P}_{\Omega}$ for $\Omega\in L^{q}(\mathbb{S}^{n-1})$, $1<q\leq\infty$.