spaces and Schatten-von Neumann classes. The proofs are in general omitted.

Weight functions
----------------

We start by discussing general properties on weight functions. A *weight* on ${\mathbf R^{d}}$ is a positive function $\omega
\in  L^\infty _{loc}({\mathbf R^{d}})$ such that $1/\omega \in  L^\infty _{loc}({\mathbf R^{d}})$. We usually assume that $\omega$ is *moderate*, or *$v$-moderate* for some positive function $v \in
 L^\infty _{loc}({\mathbf R^{d}})$. This means that $$\label{moderate}
\omega (x+y) \lesssim \omega (x)v(y),\qquad x,y\in {\mathbf R^{d}}.$$ Here $A\lesssim B$ means that $A\le cB$ for a suitable constant $c>0$, and for future references, we write $A\asymp B$ when $A\lesssim B$ and $B\lesssim A$. We note that implies that $\omega$ fulfills the estimates $$\label{moderateconseq}
v(-x)^{-1}\lesssim \omega (x)\lesssim v(x),\quad x\in {\mathbf R^{d}}.$$ Furthermore, if $v$ in can be chosen as a polynomial, then $\omega$ is called a weight of *polynomial type*. We let ${\mathscr P}({\mathbf R^{d}})$ and ${\mathscr P}_E({\mathbf R^{d}})$ be the sets of all weights of polynomial type and moderate weights on ${\mathbf R^{d}}$, respectively.

It can be proved that if $\omega \in {\mathscr P}_E({\mathbf R^{d}})$, then $\omega$ is $v$-moderate for some $v(x) = e^{r|x|}$, provided the positive constant $r>0$ is chosen large enough (cf. [@Gc2.5]). In particular, shows that for any $\omega \in 