Sparse Bounds for Maximally Truncated Oscillatory Singular Integrals

For polynomial $P (x,y)$, and any Calder\'{o}n-Zygmund kernel, $K$, the operator below satisfies a $(1,r)$ sparse bound, for $1< r \leq 2$. $$\sup _{\epsilon >0} \Bigl\lvert \int_{|y| > \epsilon} f (x-y) e ^{2 \pi i P (x,y) } K(y) \; dy \Bigr\rvert$$ The implied bound depends upon $P (x,y)$ only through the degree of $P$. We derive from this a range of weighted inequalities, including weak type inequalities on $L ^{1} (w)$, which are new, even in the unweighted case. The unweighted weak-type estimate, without maximal truncations, is due to Chanillo and Christ (1987).