Let $f$ be a continuous $2\pi$ periodic function and $\Omega= [0,2\pi]\times [0,2\pi]$. Now if $u\in H^p(\Omega)$, Sobolev space, then can we prove that $fu\in H^p(\Omega)$ ?
I know an equivalent statement available in 1 D in Linear integral equation by Rainer Kress which states
For a nonnegative integer $k$ let $f\in C_{2\pi}^k$ and assume that $0\leq p\leq k.$ Then for all $\varphi\in H^s[0,2\pi]$ the product $fg\in h^p[0,2\pi]$ and $$ \Vert f\varphi \Vert_p \leq C\{ \Vert f \Vert_{\infty}+\Vert f^{(k)}\Vert_{\infty}\}\Vert \varphi\Vert_p $$
I would like to know does a similar result exists in $H^p(\Omega)$? Or any hint on proving it. Thanks in advance.