Let $\mathbb{T}^n:=(\mathbb{R}/\mathbb{Z})^n$ be the $n$-dimensional torus.
Let us fix a nonempty open set $A \subset (0,1)^n$ and define $A_p:=\cup_{ \overrightarrow{m} \in \mathbb{Z}^n} (A+\overrightarrow{m}) \subset \mathbb{R}^n$. Then, $A_p$ may in fact be identified as an open subset of $\mathbb{T}^n$.
Now, consider the Sobolev space $H^1(\mathbb{T}^n)=W^{1,2}(\mathbb{T}^n)$. Then for any $f \in H^1(\mathbb{T}^n)$, I would like to find the space in which the function \begin{equation} \chi_{A_p}(x) \cdot f(x) \end{equation} belongs to. Here, $\chi_{A_p}(x)$ is the characteristic function at $A_p$.
I am tempted to say that $\chi_{A_p}(x) \cdot f(x)$ belongs $H^1(A_p)$ but what exactly is $H^1(A_p)$?
Could anyone please clarify for me?