# Strichartz estimates for fractional Schrodinger equations

A pair $$(q,r)$$ is $$\alpha-$$fractional admissible if $$q\geq 2, r\geq 2$$ and $$\frac{\alpha}{q} = d \left( \frac{1}{2} - \frac{1}{r} \right).$$

We take fractional Schrodinger propagator $$U(t)=e^{it (-\Delta)^{\alpha/2}}.$$ I state the following result from the paper, see Corollary 3.10.

Result: Let $$d\ge 2$$ and $$\frac{2d}{2d-1} < \alpha \leq 2.$$ Assume that $$\phi$$ and $$F$$ are radial. Then

(1) For any $$\alpha-$$fractional admissible pair $$(p,q),$$ there exists $$C_q$$ such that $$\|U(t) \phi \|_{L^{p,q}_{t,x}} \leq C_q \|\phi \|_{L^2}, \ \forall \phi \in L^2 (\mathbb R^d).$$ (2) Define $$DF(t,x) = \int_0^t U(t-\tau )F(\tau,x) d\tau.$$ For all $$\alpha-$$fractional admissible pair $$(p_i,q_i), (i=1, 2)$$, there exists $$C$$ (constant) such that for all intervals $$I \ni 0,$$ $$\|D(F)\|_{L^{p_1, q_1}_{t,x}} \leq C \|F\|_{L^{p_2', q_2'}_{t,x}}, \ \forall F \in L^{p_2'} (I, L^{q_2'})$$ where $$p_i'$$ and $$q_i'$$ are H\"older conjugates of $$p_i$$ and $$q_i'$$ respectively.

My Questions are: (1) When $$\alpha>2,$$ the analogue of above result available? If so, can you suggests some references? (2)Why do we need radial assumption when $$\alpha \neq 2$$?

Edit: I guess, Proposition 2.2 in the paper might be useful? Though I'm not sure.