Weak Type Bound for Oscillatory 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. The maximal truncations of $T_P$ satisfy the weak $L ^{1}$ inequality, our proof simplifying and extending the argument of Chanillo and Christ for the weak type bound for $T_P$.