Cfrgtkky

I want to understand a certain integration by parts argument in Treves' book Introduction to Pseudodifferential and Fourier Integral Operators (It appears in the proof of Theorem 4.1). Here is a self-contained description of the result:

Let $b:Omegatimes Omega times mathbb{R^n}rightarrow mathbb{C}$ be an amplitude of order $min mathbb{R}$ on an open subset $Omegasubset mathbb{R}^n$. That means $b$ is smooth and for every compact set $Ksubset Omegatimes Omega$ there is a constant $C>0$ such that
$$
vert D^alpha_xD^beta_yD^gamma_{xi}b(x,y,xi)vert le C (1+vert xivert^2)^{(m-vert gammavert )/2} quad text{for } (x,y)in K, xi in mathbb{R^n},
$$
where $D^alpha=i^{-vert alphavert}partial_1^{alpha_1}dots partial_n^{alpha_n}$. Now fix some multiindex $alpha$ and consider
$$
b_alpha(x,y,xi)=(y-x)^alpha b(x,y,xi)quad text{and} quad tilde b_alpha(x,y,xi)=D_xi^alpha b(x,y,xi),
$$
both of which are amplitudes having orders $m$ and $m -vert alphavert$ respectively. Treves claimes that both amplitudes give rise to the same pseudodifferential operators and proves this by by the following integration by parts for the respective Schwartz kernels $k_alpha$ and $tilde k_alpha$:

$$
begin{align}
k_alpha(x,y)
&:=int_{mathbb{R^n}}e^{i(x-y)cdot xi} b_alpha(x,y,xi)frac{dxi}{(2pi)^n} \
&= int_{mathbb{R^n}}(y-x)^alpha e^{i(x-y)cdot xi} b(x,y,xi)frac{dxi}{(2pi)^n} \
&= int_{mathbb{R^n}}(-D_xi)^alpha e^{i(x-y)cdot xi} b(x,y,xi)frac{dxi}{(2pi)^n} \
&= int_{mathbb{R^n}} e^{i(x-y)cdot xi} D_xi^alpha b(x,y,xi)frac{dxi}{(2pi)^n} \
&= int_{mathbb{R^n}} e^{i(x-y)cdot xi} tilde b_alpha(x,y,xi)frac{dxi}{(2pi)^n} =: tilde k_alpha(x,y).
end{align}
$$
Note that all the integrals are to be understood as oscillatory integrals, so the integration by parts needs further justification.

Here is how I want to justify the integration by parts: Take $chi:mathbb{R}^nrightarrow[0,1]$ smooth, $chi equiv 1$ in a neighbourhood of $0$ and $chi(xi)=0$ for $vert xivert ge 1$. Define
$$
k_{alpha,epsilon}(x,y):=int_{mathbb{R^n}}e^{i(x-y)cdot xi} (y-x)^alpha b(x,y,xi)chi(epsilon xi)frac{dxi}{(2pi)^n},
$$
then $k_{alpha,epsilon}in C^infty(Omegatimes Omega)$ and $k_{alpha,epsilon} rightarrow k_alpha$ as distributions on $Omegatimes Omega$ (by the definition of oscillatory integrals). Now for each of the regularised integrals we can perform the desired integration by parts and use the Leibniz rule to find that
$$
k_{alpha,epsilon}(x,y)=sum_{0le nule alpha } k_{alpha,epsilon}^nu(x,y),
$$
where
$$
k_{alpha,epsilon}^nu(x,y) = int_mathbb{R^n}e^{i(x-y)cdot xi} epsilon^{vert nuvert} D^nuchi(epsilon xi) a^nu(x,y,xi) frac{dxi}{(2pi)^n},
$$
where $a^nu$ is an amplitude of order $m-vert alphavert + nu$. In particular, for $nu=0$ we have $a^nu = tilde b_alpha$ and thus $k_{alpha,epsilon}^nu rightarrow tilde k_alpha$ as distributions (again by the very definition of oscillatory integrals).

Question 1: In order to complete the justification, we need to prove that $k_{alpha,epsilon}^nu rightarrow 0$ as distribution on $OmegatimesOmega$, when $nuneq 0$. Is this correct and are my considerations so far OK?

Here are my thoughts so far: Upon the change of variable $eta = epsilon xi$ we have
$$
k_{alpha,epsilon}^nu(x,y) = epsilon^{vert nuvert - n } int_mathbb{R^n}e^{iepsilon^{-1} (x-y)cdot eta} D^nuchi(eta) a^nu(x,y,epsilon^{-1} eta) frac{deta}{(2pi)^n}
$$
and after taking absolute values of the integrand and using the amplitude estimate we find (for $(x,y)$ lying in a compact set $K$)
$$
vert k_{alpha,epsilon}^nu(x,y) vert lesssim_K epsilon^{nu - n } cdot epsilon^{-(m-vert alphavert + nu)} = epsilon^{vert alpha vert -(m+n)},
$$
which tends to $0$ for large enough $alpha$. This is in general not enough and further it strikes me as odd that this result seems to be independent of $nu$ (wouldn't this imply that $tilde k_alpha = 0$ for high enough $alpha$?). To get better estimates, do we maybe need some kind of stationary phase argument? (But how to use it here: The symbol we are integrating against the oscillation depends on $epsilon$ as well.)

Question 2: My estimates seem to be faulty because I don't believe in the consequence it would have for high $alpha$. Where is my mistake? How to save the argument?