Given some function $\phi:D\to\mathbb{R}$, with $D\subseteq \mathbb{R}^d$. I was wondering whether we can find an estimate of the form

$$ \|\phi\otimes\phi\|_{\dot{H}^1(D\times D)} \lesssim \|\phi\|_{\dot{H}^{1/2}(D)}^2. $$

Many thanks.