I have this example as an application to Bernsteins Lemma, and Im confused because I`m actually not able to find the lemma in any book that I have. So I would appreciate it if you at least give me a reference to the lemma and explain this to me.
$$ g(\xi)=\left\{
\begin{array}{c} (1-\vert \xi \vert^2)^{\alpha} , \vert \xi \vert \le1\\
0 , \vert \xi \vert >1 \end{array}
\right. $$
when is $g \in \mathscr F L^1 $?
similarly $h(\xi)=(1+\vert \xi \vert ^2 )^{-\epsilon}$ when this is a fourier transform of some function in $L^1$ if $\epsilon >0$ ?