Decoupling for certain quadratic surfaces of low co-dimensions

Decoupling inequalities for quadratic forms

We prove sharp $\ell^q L^p$ decoupling inequalities for $p,q \in [2,\infty)$ and arbitrary tuples of quadratic forms. Connections to prior results on decoupling inequalities for quadratic forms are also explained. We also include some applications of our results to exponential sum estimates and to Fourier restriction estimates. The proof of our main result is based on scale-dependent Brascamp--Lieb inequalities.

三变量中两个二次型的解耦：完整刻画

Decoupling for two quadratic forms in three variables: a complete characterization

该研究证明了ℝ⁵中三变量两个二次型对应的余维数2退化曲面的精确解耦不等式，结合已有非退化情形的研究，完成了三变量二次型对的解耦不等式分类。

Optimal mean value estimates for the quadratic Weyl sums in two dimensions

We use decoupling theory to prove a sharp (up to $N^ε$ losses) estimate for Vinogradov's mean value theorem in two dimensions

ℝ⁴中某些退化曲面的ℓ²解耦

${l}^2$ Decoupling for certain degenerate surfaces in $\mathbb{R}^4$

该研究针对ℝ⁴中由三次参数与四次幂图定义的退化超曲面𝒮³₄，建立相应的ℓ²解耦估计。

Mean value estimates for Weyl sums in two dimensions

We use decoupling theory to estimate the number of solutions for quadratic and cubic Parsell--Vinogradov systems in two dimensions.

Small cap decoupling inequalities: Bilinear methods

We obtain sharp small cap decoupling inequalities associated to the moment curve for certain range of exponents $p$. Our method is based on the bilinearization argument due to Bourgain and Bourgain-Demeter. Our result generalizes theirs to all higher dimensions.