what Sobolev space does this function belong to?

Question

If $u(x,y)$ satisfies $$\int_{\mathbb{R}^2}|u(x,y)|^2dxdy<\infty,\tag{1}$$ then we say $u\in L^2(\mathbb{R}^2)$

If $f(x,y)$ satisfies $$\int_{\mathbb{R}^2}|y f(x,y)|^2dxdy<\infty.\tag{2}$$

Then $f(x,y)\in ?$