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

Abstract

In their seminal work, F. Ricci and E. M. Stein introduced the oscillatory singular integral $T_Pf(x)=\operatorname{p.v.}\int_{\mathbb{R}^n}e^{iP(x,y)}\frac{Ω(x-y)}{|x-y|^n}f(y)\,dy$, and S. Chanillo and M. Christ established its weak-type $(1,1)$ inequality for smooth $Ω$. 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 $Ω\in L\log^+\!L(\mathbb{S}^{n-1})$. Moreover, we establish weak-type $(1,1)$ bounds for the rough maximal oscillatory operator associated with $T_P$, a result of independent interest.