Strichartz estimates in Lorentz space

Statement

Let $d \ge 1$. For $f \in L^2(\R^d)$, let $e^{it\Delta} f$ be the solution at time $t$ of the free Schrödinger equation with data $f$. It is given by the Fourier transform as $\widehat{e^{it\Delta} f}(\xi) = e^{-it\abs{\xi}^2} \hat f(\xi)$.

Call a pair $(q, r)$ admissible if $2 \le q \le \infty$, $2 \le r < \infty$, $(q, r, d) \ne (2, \infty, 2)$ and $$\frac{2}{q} + \frac{d}{r} = \frac{d}{2}.$$ For $1 \le s \le \infty$, let $s'$ be the conjugate exponent, with $\frac{1}{s} + \frac{1}{s'} = 1$. For an interval $I \subseteq \R$, let $\norm{u}_{L^q_t L^{r,2}_x(I \times \R^d)}$ be the $L^q$ norm over $t \in I$ of the function $t \mapsto \norm{u(t)}_{L^{r,2}(\R^d)}$. Here $L^{r,2}$ is the Lorentz space with Lebesgue measure, and $L^q$ is a Lebesgue space.

Then the following two estimates hold, where the implied constants depend only on $d$ and the exponents.

1. Let $(q, r)$ be admissible. Then for all $f \in L^2(\R^d)$, $$\norm{e^{it\Delta} f}_{L^q_t L^{r,2}_x(\R \times \R^d)} \lesssim \norm{f}_{L^2(\R^d)}.$$

2. Let $(q_1, r_1)$ and $(q_2, r_2)$ be admissible. Let $I \subseteq \R$ be an interval and let $t_0 \in I$. Then for all $F \in L^{q_2'}_t L^{r_2',2}_x(I \times \R^d)$, $$\Big\| \int_{t_0}^t e^{i(t-\tau)\Delta} F(\tau) \, d\tau \Big\|_{L^{q_1}_t L^{r_1,2}_x(I \times \R^d)} \lesssim \norm{F}_{L^{q_2'}_t L^{r_2',2}_x(I \times \R^d)}.$$