从 linear 到 multilinear：Cauchy-Schwarz、分解与复杂度下降

旧博客原文

原题：Linear to Multi-linear

The technique that transform a problem which is in a linear setting to a multilinear setting is very powerful.

such like:

1.The renormalization technique in complex dynamic system, and the generalization

this is mainly the Ostrowoski representation,and something else.

2.Fouriour analysis

this can be view when it is difficult to investigate a quality about a function $f$ , it is always easier to take charge with some some part of $f$ , in this case is given by $\hat f(\xi),\xi \in R$ or $\hat f(k),k\in Z$ like cut $f$ into a lot of small parts,deal with every part and use some inequality(always the triangle inequality or similar thing) to glue it into a whole estimate of the quantity of $f$ .

3.Multi-scales theory

this is used in the improve of Minkowski dimension of 3-dim kakeya set by Katz-Tao.

4.The proof of Bourgain-Sarnak-Ziegler theorem

Theorem(B-S-Z). Let $F : N \to C$ with $|F| \leq 1$ and let $\nu$ be a multiplicative function with $|\nu| \leq 1$ . Let $\tau > 0$ be a small parameter and assume that for all primes $p_1, p_2 \leq e^{1/\tau}$ , $p_1 \neq p_2$ , we have that for $M$ large enough

$|\sum_{m\leq M} F(p_1m)\overline {F(p_2m)}| \leq \tau M$ .

Then for N large enough

$|\sum_{m\leq M} \nu(n)F(n) | \leq 2 \sqrt{\tau log(\frac{1}{\tau})}M$ .

this theorem is not difficult to prove by bilinear method and Cauchy-Schwarz,you can see the detail in https://arxiv.org/abs/1110.0992v1.
According to this theorem,to get a good approximation of $\sum_{1\leq k\leq x}\mu(k)f(k)$ we use need a good approximation on $\sum_{1\leq k\leq x}f(p_1x)\overline {f(p_2x)}$ .this will be much easier.but for the RHS $f(x)$ is very complicated so I do not have a non-trivial estimate for $\sum_{1\leq k\leq x}f(p_1x)\overline {f(p_2x)}$ until now.

5.The multiplier restriction theorem(Tao)

6.Some special construction in additive Combitriocs

Such like when we want to consider some set $A\subset Z_1$ satisfied $\frac{|A-A|}{|A+A|}>>1$ it is convenient to consider in a high dimensional linear space $Z^N$ rather than in $Z$ .

补充说明

以下是新整理的中文说明；上方旧博客原文保持不变。

很多问题直接估计一个线性量很难，但把它平方、分解或转成多线性形式后，结构反而清楚。这就是从 linear 到 multilinear 的基本哲学。

从 linear 到 multilinear：Cauchy-Schwarz、分解与复杂度下降 — 从 linear 到 multilinear 的基本动作是用 Cauchy-Schwarz 或分解降低复杂度，同时保留主要信息。

1. Cauchy-Schwarz 的真正作用

Cauchy-Schwarz 不只是让估计变粗。好的使用方式是：降低复杂度，同时几乎不损失主要信息。例如

如果右边的两个平方和更容易理解，这一步就是有效的。

2. Fourier 分解

在调和分析中，一个函数被切成频率块：

$$f=\sum_\theta f_\theta.$$

线性估计可能无法直接控制 $\sum_\theta f_\theta$，但双线性或多线性估计能利用不同 $\theta$ 之间的 transversal 结构。

3. Multiscale 理论

Kakeya 和 restriction 中，多尺度分解把一个集合或函数切成不同尺度上的 pieces。每一层可能只给出局部信息，但层与层之间的递归关系能产生全局维数或范数估计。

4. BSZ 准则

Bourgain-Sarnak-Ziegler 准则就是线性到双线性的典型例子。要证明

$$\sum_{n\le N}\mu(n)a_n=o(N),$$

可以转而控制

$$\sum_{n\le N}a_{pn}\overline{a_{qn}}$$

对不同素数 $p\ne q$ 的相关。这把乘法函数的线性相关转成序列自身的双线性相关。

5. 方法的边界

多线性化不是免费午餐。右边的量可能更复杂，分解后还要重新 glue 回整体估计。好的多线性方法，总是在“降低复杂度”和“保持信息”之间找到平衡。