Sharp Variational Estimates of Stein--Wainger Type Operators
Renhui Wan
TL;DR
The paper develops sharp $L^p$ bounds for the $r$-variation of Stein--Wainger type oscillatory operators with general radial phases, covering $n\ge 2$ and phases $\Gamma\in\mathcal{U}$, including the challenging case $\Gamma(|t|)=|t|^\alpha$ with $\alpha\in(0,1)$. The authors decompose the variational problem into long and short variation components, proving a jump inequality for the long part and a Besov-space–based short-variation bound that relies on stationary-phase analysis, square-function estimates, and localization; they also address endpoint behavior under a homogeneous phase condition via Bourgain’s interpolation trick. Their results establish sharp $L^p$ variation inequalities for $r>p'/n$ and a restricted weak-type endpoint when a homogeneous scaling is present, thereby extending and answering questions raised in prior work (notably GRY20). The work integrates techniques from Carleson-type operators, stationary phase, Stein–Tomas restriction theory, and Besov space methods to handle generic radial phases and to obtain endpoint results, highlighting the broader relevance for convergence questions and discrete analogues in oscillatory harmonic analysis.
Abstract
For any integer $n \geq 2$, we establish $L^p(\R^n)$ inequalities for the $r$-variations of Stein-Wainger type oscillatory integral operators with general radial phase functions. These inequalities closely related to Carleson's theorem are sharp, up to endpoints. In particular, when the phase function is chosen as $|t|^\A$ with $\A\in (0,1)$, our results provide an affirmative answer to a question posed in Guo-Roos-Yung (Anal. PDE, 2020). Furthermore, we obtain the restricted weak type estimates for endpoints in the specific case of homogeneous phase functions.