It is known that the bilinear Strichartz estimate $$\Vert e^{it\Delta}u_0\cdot \overline{e^{it\Delta}v_0}\Vert_{L_t^2L^2(\mathbb{R}^d)}\lesssim_{\delta} \Vert u_0\Vert_{H^{-1/2+\delta}(\mathbb{R}^d)}\Vert v_0\Vert_{H^{(d-1)/{2}-\delta}(\mathbb{R}^d)},\quad\delta>0$$ for the Schrodinger equation in dimension $d\geq 2$, fails for $\delta=0$ (it holds true with $\delta=0$ for dyadically localised data, but the dyadic contributions do not lead to a finite sum in general).

I'm wondering what happens in the case $u_0=v_0$, as in principle one may expect some cancellations when summing up the dyadic contributions.

More precisely, one can consider the following inequality:
$$ \Vert e^{it\Delta}u_0\Vert_{L_t^4L^4(\mathbb{R}^d)}^2\lesssim\Vert u_0\Vert_{H^{-1/2}(\mathbb{R}^d)}\Vert u_0\Vert_{H^{(d-1)/{2}}(\mathbb{R}^d)}\qquad(*)$$
**Is estimate (*) actually true?** Thank you for any suggestion.