# Sparse bounds for maximal oscillatory rough singular integral operators

## TL;DR

This paper establishes sparse bounds for maximal oscillatory rough singular integrals with polynomial phase, leading to weighted L^p estimates and extending techniques to operators with less regular angular parts.

## Contribution

It introduces novel sparse bounds for maximal oscillatory rough singular integrals with polynomial phases, independent of polynomial coefficients, and extends results to rough singular integrals with less regular angular functions.

## Key findings

- Established sparse bounds for maximal oscillatory rough singular integrals.
- Derived weighted L^p estimates based on these bounds.
- Extended techniques to rough singular integrals with less regular angular functions.

## 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 $\Omega\in L^{q}(\mathbb{S}^{n-1})$, $1<q\leq\infty$.