Let $\Omega=[0,2\pi]\times [0,2\pi]$ and $H^{p}(\Omega)$ be sobolev space such that $\psi\in H^{p}(\Omega)$. Now if $f$ is a smooth periodic function on $\Omega$ then does $f\psi \in H^{p}(\Omega)$? I know such result holds in $1D $ that is when $\Omega=[0,2\pi]$.
My attempt:
I tried a particular case when $f=e^{2\pi i k\cdot x}$ (I need this)
Consider $\Vert e^{2\pi i k\cdot x}\psi \Vert _{H^p(\Omega)}^2=\sum_{m\in Z^2}(1+\vert m\vert^2)^p\vert \widehat{e^{2\pi i k\cdot x} \psi}_m \vert^2 =\sum_{m\in Z^2}(1+\vert m\vert^2)^p)\vert \widehat{\psi}_{k+m} \vert^2$
Now how can I show that the series is convergent to prove the claim?
Thanks