Sparse bounds for maximal oscillatory rough singular integral operators

Abstract

We prove sparse bounds for maximal oscillatory rough singular integral operator $$T_{\Omega ,\ast }^{P}f(x):=\underset{\epsilon >0}{\sup }\left|{\int }_{|x-y|>\epsilon }{e}^{\iota P(x,y)}\frac{\Omega ((x-y)/|x-y|)}{|x-y{|}^n}f(y)dy\right|,$$ 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 $T_{\Omega ,\ast }^P$. Moreover, the norm $\Vert T_{\Omega ,\ast }^P{\Vert }_{L^p\to L^p}$ 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 $T_{\Omega }^P$ for $\Omega \in L^q({\mathbb{S}}^{n-1})$, $1<q\le \infty$.