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 $1<r\le 2$. $$\underset{ϵ>0}{\sup }\mid {\int }_{\mathrm{\mid }y\mathrm{\mid }>ϵ}f(x-y)e^{2\pi iP(x,y)}K(y)dy\mid$$ 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).