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## ap.evaluation of pdes – Proof of Taylor’s Schwartz kernel appraise of pseudodifferential operators

I’m within the proof of the next end result which supplies an appraise on the Schwartz Kernel of a $Psi$assassinate. There is one side the proof that’s not limpid to me which I’d affection to query the Mathoverflow group about:

**The theorem**

Suppose that $q(x,D)$ is a pseudodifferential operator performing on distributions in $mathbb{R}^n$ with character $q$ in $S^{-s}_{1,0}$ and outline

$displaystyle tilde Phi(x,z) = int q(x,xi) e^{izcdotxi},textual content{d}xi,.$.

In his bespeak “Pseudodifferential Operators” from 1981, Michael E. Taylor claims in Lemma 3.1 of Chapter XII that for $|xi| leq C$ and $s < n$, we maintain the appraise

$displaystyle |tilde Phi(x,z)| leq C |z|^{-n + s},. $

*level to:* $C$ is a common ceaseless, so the acceptation of $C$ might change from one line to the subsequent.

**The beginning of the proof and my query**

Firstly, Taylor observes that it suffices to assume symbols that assassinate not cipher on $x$. Furthermore, he observes that $q in S^{-s}_{1,0}$ implies that the household of features $q_r(xi) = r^sq(rxi)$ the place $r$ runs from $1$ to $infty$ types a bounded clique in $C^infty(1 leq |xi| leq 2)$. I occupy he means right here that every one the derivatives are uniformly bounded. What I assassinate not win is the next sever:

Taylor claims that the above details suggest that $q$ can breathe written as

$displaystyle q(xi) = q_0(xi) + int_{0}^infty p_tau(e^{-tau}xi),textual content{d}tau$

the place $q_0(xi) in C^infty_c$ and $e^{stau}p_tau(xi)$ is bounded within the Schwartz house $mathcal{S}(mathbb{R}^n)$ for $0 leq tau < infty$. *Unfortunately, I assassinate not graze how this follows.*

**The leisure of the proof**

I give a sketch of the comfort of the proof right here for completeness sake:

Firstly, we maintain

$displaystyle hat q(z) = hat q_0(z) + int_{0}^infty e^{ntau} hat p_tau(e^{-tau}z),textual content{d}tau$

We retain that $e^{stau} hat p_tau(z)$ is just too bounded in $mathcal{S}$, which ends up in the estimates

$e^{stau} |hat p_tau(z)| leq C_N(1 + |z|)^{-N}$

with the constants unbiased of $tau$. Then, we are able to appraise

$displaystyle |hat q(z)| leq C + C_N |z|^{s-n}int_0^infty e^tau|z|^{n-s}(1 + |e^tau z|)^{-n},textual content{d}tau$.

Using a change of the design $tau mapsto tau + log(|z)$, one can sure the ultimate integral and quick the proof.

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