Sparse Bounds for Maximal Monomial Oscillatory Hilbert Transforms

For each $d \geq 2$, the Hilbert transform with a polynomial oscillation as below satisfies a $(1, r )$ sparse bound, for all $r>1$ $$H _{ \ast } f (x) = \sup _{\epsilon } \Bigl\lvert \int_{|y| > \epsilon} f (x-y) \frac { e ^{2 \pi i y ^d }} y\; dy \Bigr\rvert.$$ This quickly implies weak-type inequalities for the maximal truncations, which hold for $A_1$ weights, but are new even in the case of Lebesgue measure. The unweighted weak-type estimate \emph{without maximal truncations} but with arbitrary polynomials, is due to Chanillo and Christ (1987).