I am interested in studying the negative Sobolev norm of tensor product of functions. More specifically, for any function $g$ defined in a domain $D$ we define the norm $\dot{H}^{-1}(D)$ by duality as
$$ \|g\|_{\dot{H}^{-1}(D)} = \sup \left\lbrace \int_D g(x)\phi(x)dx : \|\phi\|_{\dot{H}^1(D)}=1 \right\rbrace, $$
where $\|\phi\|_{\dot{H}^1(D)} = \|\nabla\phi\|_{L^2(D)}$.
Is it true then that $(g\otimes g)(x,y) = g(x)g(y)$ is in $\dot{H}^{-1}(D\times D)$? I understand that this question might be related to this other question: Is $\dot{H}^1(D)\otimes \dot{H}^1(D)$ dense in $\dot{H}^1(D\times D)$?
Thank you!
PS: I write the homogeneous Sobolev norm $\dot{H}^1(D)$, but a similar result for the full Sobolev norm $H^1(D)$ would be good for me too.