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 $\Omega\in L^{ q}(\mathbb{ S}^{ n-1})$, $1<q\leq\infty$.
Categories: math.CA math.AP