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I'm trying to learn more tools from microlocal analysis by reading Grigis and Sjostrand's Microlocal analysis for differential operators. In the first chapter, they define the standard $C^\infty$ symbol class $S^m_{\rho , \delta}(X \times \mathbb{R}^N)$ for $X \subseteq \mathbb{R}^n$ open, $m \in \mathbb{R}$, and $0 \le \rho, \delta \le 1$:

$$ a \in S^m_{\rho , \delta} \iff \forall \text{ compact $K \subseteq X$, $\alpha \in \mathbb{N}^n$, and $\beta \in \mathbb{N}^N$, $\exists C_{K, \alpha, \beta}>0$ such that } \\ |D_x^\alpha D^\beta_\theta a(x, \theta)| \le C_{K, \alpha, \beta}(1 + |\theta|)^{m - \rho |\beta| + \delta|\alpha|}, \qquad \forall (x, \theta) \in K \times \mathbb{R}^N. $$

The class $S^{-\infty}(X \times \mathbb{R}^N)$ is defined by:

$$ a \in S^{-\infty} \iff \forall \text{ compact $K \subseteq X$, $\alpha \in \mathbb{N}^n$, $\beta \in \mathbb{N}^N$, and $M \in \mathbb{R}$, $\exists C_{K, \alpha, \beta,M}>0$ such that } \\ |D_x^\alpha D^\beta_\theta a(x, \theta)| \le C_{K, \alpha, \beta,M}(1 + |\theta|)^M, \qquad \forall (x, \theta) \in K \times \mathbb{R}^N.$$

After these definitions they make the following comment (what follows is a direct quote from the text).

"There is no point in introducing spaces $S^m_{\rho, \delta}$ with $\rho > 1$ or with $\delta < 0$. For instance, if $a \in S^m_{\rho, \delta}$ with $m < 0$ and $\rho > 1$, then applying $|\theta| \partial_{|\theta|} = \sum_j \theta_j \partial_{\theta_j}$ (working in polar coordinates) many times and integrating, we see that $a \in S^{-\infty}$."

This is a very interesting observation. I have been trying to figure out how to turn this hint into a formal argument. It's probably useful for me to know how to make this type of "differentiate a lot and then integrate" trick work.

Here's what I have so far. We assume $a \in S^m_{\rho, \delta}$ with $\rho > 1$ and for any $M \in \mathbb{R}$ we need get an estimate of the form $$ |D_x^\alpha D^\beta_\theta a(x, \theta)| \le C_{K, \alpha, \beta,M}(1 + |\theta|)^M, \qquad \forall (x, \theta) \in K \times \mathbb{R}^N. $$

I think the idea is that, we want to differentiate $D_x^\alpha D^\beta_\theta a(x, \theta)$ with respect to $\theta$ many times, and they integrate many times to sort of "undo" the differentiation. But the fact that $\rho > 1$ means that, in doing this two-step process, we will have gained all the negative powers of $(1 + |\theta|)$ that we need. I am very unsure how to make this argument formal and particularly how to use the hint that is given.

Any suggestions or solutions are greatly appreciated!

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    $\begingroup$ Use induction. If you can reduce the exponent a little, then you can reduce it as much as you want $\endgroup$
    – Fan Zheng
    Commented Jan 9, 2017 at 2:01
  • $\begingroup$ @FanZheng---Thank you, I see where you are going with this idea. It is enough to just show that we can get $|a(x, \theta) | \le C(1 + |\theta|)^{m - \varepsilon}$. But I'm having trouble deciding how to appropriately express $a(x,\theta)$ in terms of one if its partial derivatives. The best I have got so far is setting $\theta_0 = |\theta_0| \omega_0$ with $|\omega_0| = 1$ and then writing $|\theta_0| a(x, \theta_0) = \int_0^{|\theta_0|} \partial_r(r a(x, r \omega_0) )dr$, but when I use the product rule and estimate the integral, it doesn't appear that my estimates will work out. $\endgroup$
    – JZS
    Commented Jan 9, 2017 at 14:23
  • $\begingroup$ Why do you multiply a by $\theta_0$? You don't need to do that. $\endgroup$
    – Fan Zheng
    Commented Jan 9, 2017 at 18:09

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