In the book The Analysis of Linear Partial Differential Operators I by Hormander Section 5.2 page 130.
For operator $T:\mathcal{D}(\mathbb{R}^m)\to\mathcal{D}'(\mathbb{R}^n)$. I have known that $$\exists N_1,N_2\in\mathbb{Z}_{\geq 0},s.t.\;\left|\left<T\varphi,\psi\right>\right|\lesssim p_{N_1}(\psi) p_{N_2}(\varphi),\; \forall \psi\in\mathcal{D}(K_1),\varphi\in\mathcal{D}(K_2).$$ Where $K_1\Subset \mathbb{R}^n,\; K_2\Subset \mathbb{R}^m$ and $p_{N}(\varphi):=\sum_{|\alpha|\leq N}\sup\left|\partial^\alpha \varphi\right|$.
To prove there exist a distribution $K\in\mathcal{D}'(\mathbb{R}^ {n+m})$, we use $K_\varepsilon\in C^\infty(\mathbb{R}^{n+m})$ to appromix K.
$$K_\varepsilon(\bar{x},x^*):=\left<T\left(\frac{1}{\varepsilon^m}\varphi_2\left(\frac{\bar{x}-\cdot}{\varepsilon}\right)\right),\frac{1}{\varepsilon^n}\varphi_1\left(\frac{x^*-\cdot}{\varepsilon}\right) \right>.$$ Here $\bar{x}\in \mathbb{R}^n,\; x^*\in\mathbb{R}^m$, $\varphi_1\in\mathcal{D}(K_1)$.
Using the upper inequality we have $\left|K_\varepsilon\right|\leq C\varepsilon^{-\mu},\quad \mu=n+m+N_1+N_2$
For fixed $\bar{x},x^*$, it's clear that $$\partial_\varepsilon K_\varepsilon=\left<T\left(\partial_\varepsilon\left(\varepsilon^{-m}\varphi_2\left(\frac{x^*-\cdot}{\varepsilon}\right)\right)\right),\varepsilon^{-n}\varphi_1\left(\frac{\bar{x}-\cdot}{\varepsilon}\right)\right>+\left<T\left(\varepsilon^{-m}\varphi_1\left(\frac{x^*-\cdot}{\varepsilon}\right)\right),\partial_\varepsilon\left(\varepsilon^{-n}\varphi_1\left(\frac{\bar{x}-\cdot}{\varepsilon}\right)\right)\right>.$$
The book claim that$$\partial_\varepsilon\left(\varepsilon^{-n}\varphi_1\left(\frac{\bar{z}}{\varepsilon}\right)\right)=\varepsilon^{-n}\sum_{i=1}^{n}\partial_j\left(\varphi_{1,j}\left(\frac{\bar{z}}{\varepsilon}\right)\right),\quad \varphi_{1,j}(\bar{z})=-\bar{z}_j\varphi_1(\bar{z}).$$ This is correct. But he says it can drive $\left|\partial_\varepsilon K_\varepsilon\right|\leq C\varepsilon^{-\mu}$.
I can only get that $|\partial_\varepsilon K_\varepsilon|\leq C\varepsilon^{-(\mu+2)}$.
And the integral remainder is $\int_0^1(1-t)^\mu \partial_\varepsilon^{j} K_{\varepsilon_0+t(\varepsilon-\varepsilon_0)}{\operatorname{d}}t$. By my estimate it will be infinity when $\varepsilon\to 0$ thus useless.
Is there anyone tell me how he get the estimate needed.