How can we bound this integral:
$${\displaystyle \int_{-1}^{1}2\left[\dfrac{1}{4}-\dfrac{1}{4\left(1-\xi^{2}\right)}\left(1-\dfrac{\xi^{2}}{2}\right)^{2}\right]\left(\hat{f}\left(\xi\right)\right)^{2}e^{-2t}d\xi} $$
I've tried to use Holder inequality but when I compute $\left\Vert 2\left[\dfrac{1}{4}-\dfrac{1}{4\left(1-\xi^{2}\right)}\left(1-\dfrac{\xi^{2}}{2}\right)^{2}\right]\right\Vert _{L_{\xi}^{2}\left(\left[-1,1\right]\right)} $, it is not defined since we have some kinds of $\log\left(\xi\pm1\right) $ in the anti-derivative.
Thank you very much.
The absolute value of the term in brackets is on the order of $1/(1-\xi)$ near $1;$ similar problem near $-1.$ Thus it's not in $L^2.$ It's not even in $L^1.$