Almost orthogonality：Cotlar-Stein lemma、Schur test 与奇异积分

旧博客原文

原题：Almost orthogonality

Motivation and Cotlar’s lemma

We always need to consider a transform $T$ on Hilbert space $l^2(\mathbb Z)$ (this is a discrete model), or a finite dimensional space $V$ . If under a basis $T$ is given by a diagonal matrix this story is easy,

$\displaystyle A = \begin{pmatrix} \Lambda_1 & 0 & \ldots & 0 \\ 0 & \Lambda_2 & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & \Lambda_n \end{pmatrix} \ \ \ \ \ (5)$

Then $||T||=\max_{i}\lambda_i$ .

In fact, for $T$ is a transform of a finite dimensional space, $T$ is given by $(a_{ij})_{n\times n}$ by duality we have $||T||=||TT^*||$ , so we have,

$||T||=||TT^*||=|(\sum a_{ij}x_j)y_i|\leq |\sum_{i,j}\frac{1}{2}(|a_{ij}(|x_i|^2+|y_j|^2)|\leq M$

If we have given $\sum_{i}|a_{ij}|\leq M$ and $\sum_{j}|a_{ij}|\leq M$ $\forall i,j\in \{1,2,...,n\}$ .

But in application of this idea, the orthogonal condition always seems to be too restricted and due too this we have the following lemma which is follow the idea but change the orthogonal condition by almost orthogonal.

Lemma(Catlar-Stein)

Let $\{T_j\}_{j=1}^N$ be finitely many operators on some Hilbert space $H$ . Such that for some function $\gamma : \mathbb Z\to R^+$ one has,

$||T_j^*T_k||\leq \gamma^2(j-k),||T_jT_k^*||\leq \gamma^2(j-k)$

for any $1\leq j,k\leq N$ . Let $\sum_{l=-\infty}^{\infty}\gamma(l)=A<\infty$ . then ,

$||\sum_{j=1}^NT_j||\leq A$

Pf:

tensor power trick + duality $||T||=||TT^*||^{\frac{1}{2}}$ .

Singular integrals on $L^2$

Lemma(Schur)

Define $T$ is a operator on measure space $X\times Y$ equipped positive product measure $\mu\wedge \nu$ , via,

$(Tf)(x)=\int_YK(x,y)f(y)\nu(dy)$

$K$ is a measurable kernel, then,

1). $||T||_{1\to 1}\leq \sup_{y\in Y}\int_{X}|K(x,y)|\mu(dx)=:A$ .

2). $||T||_{\infty\to \infty}\leq \sup_{x\in X}\int_{Y}|K(x,y)|\nu(dy)=:B$ .

3). $||T||_{p\to p}\leq A^{\frac{1}{p}}B^{\frac{1}{p'}}$ , $\forall 1\leq p\leq \infty$ .

4). $||T||_{1\to \infty}\leq ||K||_{L^{\infty}(X\times Y)}$ .

Pf:

1),2),4) merely due to Fubini theorem and Bath lemma.

3) proof by the interpolation and combine 1) and 2).

Theorem

Let K be a Calderon-Zegmund operator, with the additional assumption

that $|\nabla K(x)|\leq B|x|^{-d-1}$ . Then

$||T||_{2\to 2} \leq CB$

with $C = C(d)$ .

Caldero ́n–Vaillancourt theorem

Hardy’s inequality

Theorem(Hardy inequality)

For any $0 \leq s < \frac{d}{2}$ there is a constant $C(s, d)$ with the prop-

arty that,

$|||x|^{-s} f||_2 \leq C(s,d)||f||_{H^s(R^d)}$

for all $f \in H^s(R^d)$ .

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。

正交性是 Hilbert space 中最强的简化机制；almost orthogonality 则是在真实分析问题中更常见的替代品。频率块、空间块和算子族通常不完全正交，但交互足够小。

Almost orthogonality：Cotlar-Stein lemma、Schur test 与奇异积分 — almost orthogonality 用交互衰减替代严格正交，是奇异积分和伪微分算子估计的基础。

1. 从正交到几乎正交

若 $T_j$ 的像彼此正交，则

$$\left\|\sum_j T_j f\right\|_2^2=\sum_j\|T_jf\|_2^2.$$

实际中通常只有 $T_i^\ast T_j$ 和 $T_iT_j^\ast$ 随 $|i-j|$ 衰减。

2. Cotlar-Stein lemma

若

$$\|T_i^\ast T_j\|+\|T_iT_j^\ast\|\le a(i-j)$$

且 $\sum_k a(k)^{1/2}<\infty$，则

$$\left\|\sum_jT_j\right\|_{2\to2}<\infty.$$

证明可用 tensor power trick 和对偶性。

3. Schur test

对 kernel operator

$$Tf(x)=\int K(x,y)f(y)\,dy,$$

若

$$\sup_x\int |K(x,y)|dy<\infty,\qquad \sup_y\int |K(x,y)|dx<\infty,$$

则 $T$ 在 $L^2$ 上有界。

4. 奇异积分中的用途

Calderon-Zygmund theory 中，常把算子分解成不同尺度的 pieces。尺度相隔很远时，kernel 的光滑性带来交互衰减；相近尺度则只需有限重叠。

5. Calderon-Vaillancourt 方向

伪微分算子的 $L^2$ 有界性也可以看成 almost orthogonality 的结果：把相空间切成小块后，不同块之间的交互由 symbol 的导数控制。

Almost orthogonality：Cotlar-Stein lemma、Schur test 与奇异积分

旧博客原文

补充说明

1. 从正交到几乎正交

2. Cotlar-Stein lemma

3. Schur test

4. 奇异积分中的用途

5. Calderon-Vaillancourt 方向

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