For $\xi \in \mathbb{R}^n$ we have $(1+|\xi|)^{-t}$ is integrable in $\mathcal{L}^1(\mathbb{R}^n)$ for $t>n$. Now, for $\xi,\eta \in \mathbb{R}^n$, i need to prove that for every $d\in \mathbb{R}$ exist $s\in \mathbb{R}$ such that
\begin{equation} \int \int (1 + |\xi|)^{d}(1 + |\xi - \eta|)^{-k}(1 + |\eta|)^{-k} \; d(\xi \times \eta) <\infty \end{equation}
is integrable in $\mathcal{L}^1(\mathbb{R}^n\times\mathbb{R}^n)$ for $k>s$.
I've tried use Peetre's Inequality, where
$(1 + |\xi|)^{d}(1 + |\xi - \eta|)^{-k}(1 + |\eta|)^{-k}$
$= (1 + |\xi - \eta + \eta|)^{d}(1 + |\xi - \eta|)^{-k}(1 + |\eta|)^{-k}$
$\leq (1 + |\xi - \eta|)^{d}(1 + |\eta|)^{|d|}(1 + |\xi - \eta|)^{-k}(1 + |\eta|)^{-k}$
$= (1 + |\xi - \eta|)^{d-k}(1 + |\eta|)^{|d|-k}$
So
\begin{equation} \int \int (1 + |\xi - \eta|)^{d-k}(1 + |\eta|)^{|d|-k} \; d\eta \; d\xi = \end{equation}
\begin{equation}= \int (f\star g)(\xi) \; d\xi = ||f\star g||_{1}. \end{equation}
Where is integrable if $-d+k>n$ and $-|d|+k>n$ ; but iterated integral is not sufficent for say the first integral is finite.
Because this is take from Spin Geomtry's book,page 180. He have
\begin{equation} \int \int e^{i\langle x,\xi\rangle}\hat{a}(x,\xi-\eta,\xi)\hat{u}(\eta) \; d\eta \; d\xi \end{equation} and need interchange of integration, and say
\begin{equation} |\hat{a}(x,\xi-\eta,\xi)||\hat{u}(\eta)|\leq (1 + |\xi|)^{d}(1 + |\xi - \eta|)^{-k}(1 + |\eta|)^{-k} \end{equation}
so is integrable for $k$ large.