Thank you very much. Could you please elaborate a bit on the last estimte for the $\dot{H}^1$ norm of the tensor product? If I try to compute that myself with your example function I still obtain something of order $N$. If it's related to your first comment under my question, how do you know that $\|\phi\otimes\phi\|_{\dot{H}^1} \sim \|\phi\|_{L^2}\|\phi\|_{\dot{H}^1}$? I try to derive an estimate like that I always need to use some higher $L^p$ norms for $\phi$ and/or $\nabla\phi$.

@WillieWong You're totally right, I want the RHS squared, thank you very much for pointing that out!