Strichartz estimates to fractional Schrödinger equations

In this paper, we firstly study Strichartz estimates $\|e^{it D^α} u_0\|_{L_t^q(\mathbb{R};L_x^r(\mathbb{R}^d))}\leq C(d,α,q,r,s)\|u_0\|_{\dot{H}^s}.$ We show some counterexamples for $(q,r) = (2,\infty)$. Then we consider the embedding $X^{s,b}_α\hookrightarrow L_t^q(\mathbb{R};L_x^r(\mathbb{R}^d))