Year: 2026
Type: preprint
Abstract: In their seminal work, F. Ricci and E. M. Stein [Proc. Nat. Acad. Sci. USA 83, 1--3 (1986)] introduced the oscillatory singular integral \[ T_Pf(x)=\mathrm{p.v.}\int_{\mathbb{R}^n} e^{iP(x,y)}\frac{Ω(x-y)}{|x-y|^n}f(y)\,dy, \] and S. Chanillo and M. Christ [Duke Math. J. 55, 141--155 (1987)] 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.
Source: arXiv (Cornell University)
Authors: Chen, Yanping, Ji, Zhengyang
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1096993535, G2606178647
OpenAlex ID: W7220439976