Inequalities are established for certain trilinear scalar-valued functionals. These functionals act on measurable functions of one real variable, are defined by integration over two- or three-dimensional spaces, and are controlled in terms of Lebesgue space norms of the functions, and of negative powers of large parameters describing a degree of oscillation. Related sublevel set inequalities are a central element of the analysis.
The main results and the main lines of their proofs are largely unchanged in this draft, but some details have been corrected. The analysis has already been extended in work of the author, Durcik, and Roos.
Sparse Bounds for Oscillatory and Random Singular Integrals
令$ T_{P } f (x) = \int e ^{i P (y)} K (y) f (x-y) \; dy $，其中$ K (y)$是$ \mathbb R ^{n}$上的光滑Calderón-Zygmund核，$ P$是多项式。我们证明了双线性型$ \langle T_P f, g \rangle$存在一个稀疏界。这反过来又容易地暗示$ A_p $不等式。证明的方法应用在一个随机离散设置，产生的第一个加权不等式的运营商定义在稀疏集的整数。
Sparse Bounds for Maximally Truncated Oscillatory Singular Integrals
对于多项式$ P (x,y)$和任何Calderón-Zygmund核$K$，下面的算子满足$ (1,r)$稀疏界，对于$ 1< r \leq 2$。$$ \sup _{ε>0} \Bigl\lvert \int_{|y| > ε} f (x-y) e ^{2 πi P (x,y) } K(y) \; dy \Bigr\rvert $$隐含的界限仅通过$ P$的次数依赖于$ P (x,y)$。我们由此得到一系列加权不等式，包括弱型不等式$ L ^{1} (w)$，这是新的，即使在未加权的情况下。未加权的弱型估计，没有最大截断，是由于Chanillo和Christ（1987）。
Sharp variational estimates of Stein-Wainger type operators
Ministry of Education Key Laboratory of NSLSCS, School of Mathematical Sciences and Institute of Mathematical Sciences, Nanjing Normal University（南京师范大学数学科学学院和数学科学研究院）
