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.


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