Sparse Bounds for Oscillatory and Random Singular Integrals

Let $T_{P } f (x) = \int e ^{i P (y)} K (y) f (x-y) \; dy$, where $K (y)$ is a smooth Calder\'on-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.