Sparse bounds for maximal oscillatory rough singular integral operators

We prove sparse bounds for maximal oscillatory rough singular integral operator $$T_{\mathrm{\Omega },\ast }^{P}f(x):=\underset{\epsilon >0}{\sup }\mid {\int }_{\mathrm{\mid }x-y\mathrm{\mid }>\epsilon }{e}^{\iota P(x,y)}\frac{\mathrm{\Omega }((x-y)\mathrm{/}\mathrm{\mid }x-y\mathrm{\mid })}{\mathrm{\mid }x-y{\mathrm{\mid }}^n}f(y)dy\mid ,$$ where $P(x,y)$ is a real-valued polynomial on ${\mathbb{R}}^n\times {\mathbb{R}}^n$ and $\mathrm{\Omega }\in L^{\mathrm{\infty }}({\mathbb{S}}^{n-1})$ is a homogeneous function of degree zero with ${\int }_{{\mathbb{S}}^{n-1}}\mathrm{\Omega }(\theta )d\theta =0$. This allows us to conclude weighted $L^p-$estimates for the operator $T_{\mathrm{\Omega },\ast }^P$. Moreover, the norm $\mathrm{\parallel }{T}_{\mathrm{\Omega },\ast }^P{\mathrm{\parallel }}_{L^p\to L^p}$ 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 $T_{\mathrm{\Omega }}^P$ for $\mathrm{\Omega }\in L^q({\mathbb{S}}^{n-1})$, $1<q\le \mathrm{\infty }$.