Weak type $(1,1)$ bounds for oscillatory singular integrals with rough kernels

In their seminal work, F. Ricci and E. M. Stein introduced the oscillatory singular integral $T_{P}f(x)=\mathrm{p.v.}{\int }_{{\mathbb{R}}^n}{e}^{iP(x,y)}\frac{\mathrm{\Omega }(x-y)}{\mathrm{\mid }x-y{\mathrm{\mid }}^n}f(y)dy$, and S. Chanillo and M. Christ established its weak-type $(1,1)$ inequality for smooth $\mathrm{\Omega }$. For the rough variant, however, the weak-type $(1,1)$ inequality remained a long-standing open question. In the present paper, we settle this problem: specifically, we prove that the rough oscillatory singular integral is of weak type $(1,1)$ with $\mathrm{\Omega }\in L{\log }^{+}\text{\hspace{0.17em}⁣}L({\mathbb{S}}^{n-1})$. Moreover, we establish weak-type $(1,1)$ bounds for the rough maximal oscillatory operator associated with $T_P$, a result of independent interest.