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.



Controlling the tensor product of functions in $H^1$ with lower derivatives
Assuming the inequality you hoped for is $\| \phi \otimes \phi \|\_{\dot{H}^1} \lesssim \|\phi\|\_{\dot{H}^{1/2}}^2$, I claim that this is still impossible.


Let $D = [0,2\pi]$ again. Set
$$ \phi\_N(x) = \sum\_{k = 1}^N k^{-1/2} \sin(kx) $$
Then
$$ \|\phi\_N\|\_{\dot{H}^{1/2}}^2 \approx \sum\_{k = 1}^N 1 = N $$
But
$$ \|\phi\_N\otimes \phi\_N\|\_{\dot{H}^1} \approx \left( \sum\_{k = 1}^N k^{-1} \right)^{\frac12} \left( \sum\_{k = 1}^N k \right)^{\frac12} \approx \sqrt{\ln(1+N)} N $$
showing that the desired inequality is impossible.


