Strichartz estimates to fractional Schrödinger equations

Abstract

In this paper, we firstly study Strichartz estimates $\Vert e^{it{D}^{\alpha }}{u}_0{\Vert }_{L_t^q(\mathbb{R};L_x^r({\mathbb{R}}^d))}\le C(d,\alpha ,q,r,s)\Vert u_0{\Vert }_{{\dot{H}}^s}.$ We show some counterexamples for $(q,r)=(2,\infty )$. Then we consider the embedding $X_{\alpha }^{s,b}\hookrightarrow L_t^q(\mathbb{R};L_x^r({\mathbb{R}}^d))$, where $\Vert u{\Vert }_{X_{\alpha }^{s,b}}:=\Vert ⟨\xi {⟩}^s⟨\tau -|\xi {|}^{\alpha }{⟩}^b\hat{u}(\tau ,\xi ){\Vert }_{L_{\tau ,\xi }^2}$. We present the necessary and sufficient conditions for this embedding.