---
abstract: 'Let $ T _{P } f (x) =   \int e ^{i P (y)} K (y) f (x-y) \; dy $, where $ K (y)$ is a smooth Calderón-Zygmund kernel on $ \mathbb R ^{n}$, and $ P$ be a polynomial. We show that there is a sparse bound for the bilinear form $ \langle T_P f, g \rangle$. This in turn easily implies $ A_p $ inequalities. The method of proof is applied in a random discrete setting, yielding the first weighted inequalities for operators defined on sparse sets of integers.'
address:
- ' School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332, USA'
- ' School of Mathematics, Georgia Institute of Technology, Atlanta GA 30332, USA'
author:
- 'Michael T. Lacey'
- Scott Spencer
title: |
    Sparse Bounds for Oscillatory\
    and Random Singular Integrals
---

Introduction
============

Singular integral operators can be pointwise dominated by sparse operators, which are positive localized operators, something that singular integrals are not. This paper extends this theme to the settings of (a) oscillatory singular integrals, and (b) discrete random operators. In both cases, we easily derive weighted inequalities. In the latter case, these are the first such weighted inequalities known. 