Strichartz estimate

Consider the linear Schrödinger equation in ${\displaystyle \mathbb {R} ^{d}}$ with h = m = 1. Then the solution for initial data ${\displaystyle u_{0}}$ is given by ${\displaystyle e^{it\Delta /2}u_{0}}$. Let q and r be real numbers satisfying ${\displaystyle 2\leq q,r\leq \infty }$; ${\displaystyle {\frac {2}{q}}+{\frac {d}{r}}={\frac {d}{2}}}$; and ${\displaystyle (q,r,d)\neq (2,\infty ,2)}$.

In this case the homogeneous Strichartz estimates take the form:^[2]

${\displaystyle \|e^{it\Delta /2}u_{0}\|_{L_{t}^{q}L_{x}^{r}}\leq C_{d,q,r}\|u_{0}\|_{L_{x}^{2}}.}$

Further suppose that ${\displaystyle {\tilde {q}},{\tilde {r}}}$ satisfy the same restrictions as ${\displaystyle q,r}$ and ${\displaystyle {\tilde {q}}',{\tilde {r}}'}$ are their dual exponents, then the dual homogeneous Strichartz estimates take the form:^[2]

${\displaystyle \left\|\int _{\mathbb {R} }e^{-is\Delta /2}F(s)\,ds\right\|_{L_{x}^{2}}\leq C_{d,{\tilde {q}},{\tilde {r}}}\|F\|_{L_{t}^{{\tilde {q}}'}L_{x}^{{\tilde {r}}'}}.}$

The inhomogeneous Strichartz estimates are:^[2]

${\displaystyle \left\|\int _{s<t}e^{i(t-s)\Delta /2}F(s)\,ds\right\|_{L_{t}^{q}L_{x}^{r}}\leq C_{d,q,r,{\tilde {q}},{\tilde {r}}}\|F\|_{L_{t}^{{\tilde {q}}'}L_{x}^{{\tilde {r}}'}}.}$