Strichartz estimates for Schrodinger equation

By | January 27, 2018

Theorem 1 (Strichartz esitmates for Schrödinger). Fix n\ge1 and \hbar=m=1 and call a pair \left(q,r\right) of exponents admissible if 2\le q,r\le\infty, \frac{2}{q}+\frac{n}{r}=\frac{n}{2} and \left(q,r,n\right)\neq\left(2,\infty,2\right). Then for any admissible exponents \left(q,r\right) and \left(\tilde{q},\tilde{r}\right), we have the homogeneous Strichartz estimate

(1)   \begin{equation*} \Norm{e^{it\triangle/2}u_{0}}_{\Leb q_{t}\Leb r_{x}\left(\mathbb{R}\times\mathbb{R}^{n}\right)}\apprle_{n,q,r}\Norm{u_{0}}_{\Leb 2_{x}\left(\mathbb{R}^{n}\right)} \end{equation*}

the dual homogeneous Strichartz estimate

(2)   \begin{equation*} \Norm{\int_{\mathbb{R}}e^{-is\triangle/2}F\left(s\right)ds}_{\Leb 2_{x}\left(\mathbb{R}^{n}\right)}\apprle_{n,\tilde{q},\tilde{r}}\Norm F_{\Leb{\tilde{q}^{\prime}}_{t}\Leb{\tilde{r}^{\prime}}_{x}\left(\mathbb{R}\times\mathbb{R}^{n}\right)} \end{equation*}

and the inhomogeneous (or retarded) Strichartz estimate

(3)   \begin{equation*} \Norm{\int_{t'<t}e^{i\left(t-t'\right)\triangle/2}F\left(t'\right)ds}_{\Leb q_{t}\Leb r_{x}\left(\mathbb{R}\times\mathbb{R}^{n}\right)}\apprle_{n,q,r,\tilde{q},\tilde{r}}\Norm F_{\Leb{\tilde{q}^{\prime}}_{t}\Leb{\tilde{r}^{\prime}}_{x}\left(\mathbb{R}\times\mathbb{R}^{n}\right)}. \end{equation*}

We illustrate some application of Strichartz estimate. If u:I\times\mathbb{R}^{n}\rightarrow\mathbb{C} is the solution to an inhomogeneous Schrödinger equation

    \begin{align*} i\partial_{t}u+\frac{1}{2}\triangle u & =F, \\ u\left(0,x\right) & =u_{0}\left(x\right), \end{align*}

where u_{0}\in\tSob s\left(\mathbb{R}^{n}\right), is given by Duhamel’s formula

    \[ u\left(t\right)=e^{\left(t-t_{0}\right)\frac{\triangle}{2}}u_{0}\left(t_{0}\right)+\int_{t_{0}}^{t}e^{i\left(t-s\right)\frac{\triangle}{2}}F\left(s\right)ds \]

on some time interval I containing 0. Apply \left\langle \nabla^{s}\right\rangle to both sides and using Strichartz’s estimate, we have

    \[ \Norm u_{\Leb q_{t}\Sob sr_{x}\left(I\times\mathbb{R}^{n}\right)}\apprle_{n,q,\tilde{q},\tilde{r},s}\Norm{u_{0}}_{\tSob s_{x}\left(\mathbb{R}^{n}\right)}+\Norm F_{\Leb{\tilde{q}^{\prime}}_{t}\Sob s{\tilde{r}^{\prime}}_{x}\left(I\times\mathbb{R}^{n}\right)} \]

for any admissible \left(q,r\right) and \left(\tilde{q},\tilde{r}\right).

Now we begin the proof of Theorem 1 when admissible pairs are non-endpoint.

Proof of Theorem 1. For G:\mathbb{R}\times\mathbb{R}\times\mathbb{R}^{n}\rightarrow\mathbb{C}, by Minkowski’s integral inequality, we have

    \begin{align*} \Norm{\int_{\mathbb{R}}G\left(t,s,x\right)ds}_{\Leb q_{t}\Leb r_{x}}^{q} & =\int_{\mathbb{R}}\left[\int_{\mathbb{R}^{n}}\left(\int_{\mathbb{R}}G\left(t,s,x\right)ds\right)^{r}dx\right]^{\frac{q}{r}}dt\\ & \le\int_{\mathbb{R}}\left[\int_{\mathbb{R}}\Norm{G\left(t,s,\cdot\right)}_{\Leb r_{x}}ds\right]^{q}dt\\ & =\Norm{\int_{\mathbb{R}}\Norm{G\left(t,s,\cdot\right)}_{\Leb r_{x}}ds}_{\Leb q_{t}}^{q}. \end{align*}

For any Schwarz function F in spacetime,

    \begin{align*} \Norm{\int_{\mathbb{R}}e^{i\left(t-s\right)\triangle/2}F\left(s\right)ds}_{\Leb q_{t}\Leb r_{x}\left(\mathbb{R}\times\mathbb{R}^{n}\right)} & \le\Norm{\int_{\mathbb{R}}\Norm{e^{i\left(t-s\right)\triangle/2}F\left(s\right)}_{\Leb r_{x}\left(\mathbb{R}^{n}\right)}ds}_{\Leb q_{t}\left(\mathbb{R}\right)} \end{align*}

Then by \eqref{eq:dispersive-estimiate-Schrodinger}, we have

    \begin{align*} & \apprle\Norm{\Norm F_{\Leb{r^{\prime}}_{x}\left(\mathbb{R}^{n}\right)}*\frac{1}{\left|t\right|^{n\left(\frac{1}{2}-\frac{1}{r}\right)}}}_{\Leb q_{t}\left(\mathbb{R}\right)}. \end{align*}

Since 2<r\le\infty and 2<q\le\infty satisfying \frac{2}{q}+\frac{n}{r}=\frac{n}{2}, we have n\left(\frac{1}{2}-\frac{1}{r}\right)=\frac{2}{q}.

Recall the Hardy-Littlewood-Sobolev inequality:

    \[ \Norm{F*\frac{1}{\left|t\right|^{\alpha}}}_{\Leb q_{t}}\le C\Norm F_{\Leb p_{t}} \]

whenever

    \[ \frac{1}{p}=\frac{1}{q}+\frac{n-\alpha}{n}. \]

So by the Hardy-Littlewood-Sobolev inequalty, we have

    \[ \Norm{\Norm F_{\Leb{r^{\prime}}_{x}\left(\mathbb{R}^{n}\right)}*\frac{1}{\left|t\right|^{n\left(\frac{1}{2}-\frac{1}{r}\right)}}}_{\Leb q_{t}\left(\mathbb{R}\right)}\apprle\Norm{\Norm F_{\Leb{r^{\prime}}_{x}\left(\mathbb{R}^{n}\right)}}_{\Leb{q'}_{t}\left(\mathbb{R}\right)}=\Norm F_{\Leb{q'}_{t}\Leb{r^{\prime}}_{x}\left(\mathbb{R}\times\mathbb{R}^{n}\right)}. \]

Now by Hölder’s inequality, we have

    \[ \left|\int_{\mathbb{R}}\int_{\mathbb{R}}\left\langle e^{i\left(t-s\right)\triangle/2}F\left(s\right),F\left(t\right)\right\rangle dsdt\right|\apprle_{n,q,r}\Norm F_{\Leb{q'}_{t}\Leb{r'}_{x}\left(\mathbb{R}\times\mathbb{R}^{n}\right)}^{2}. \]

On the left-hand side,

    \[ \int_{\mathbb{R}}\int_{\mathbb{R}}\left\langle e^{i\left(t-s\right)\triangle/2}F\left(s\right),F\left(t\right)\right\rangle dsdt=\Norm{\int_{\mathbb{R}}e^{-is\triangle/2}F\left(s\right)ds}_{\Leb 2_{x}\left(\mathbb{R}^{n}\right)}^{2}. \]

Thus,

    \[ \Norm{\int_{\mathbb{R}}e^{-is\triangle/2}F\left(s\right)ds}_{\Leb 2_{x}\left(\mathbb{R}^{n}\right)}\apprle\Norm F_{\Leb{q^{\prime}}_{t}\Leb{r'}_{x}\left(\mathbb{R}\times\mathbb{R}^{n}\right)}. \]

This proves the second part (2).

Now we show the first part (1) by duality argument. Note

    \begin{align*} & \int_{\mathbb{R}}\int_{\mathbb{R}^{n}}e^{it\triangle/2}u_{0}\overline{F\left(t\right)}dxdt\\ & =\int_{\mathbb{R}^{n}}u_{0}\int_{\mathbb{R}}\overline{e^{-it\triangle/2}F\left(t\right)}dtdx\\ & \le\Norm{u_{0}}_{\Leb 2\left(\mathbb{R}^{n}\right)}\Norm{\int_{\mathbb{R}}e^{-it\triangle/2}F\left(t\right)dt}_{\Leb 2\left(\mathbb{R}^{n}\right)}\\ & \apprle\Norm{u_{0}}_{\Leb 2\left(\mathbb{R}^{n}\right)}\Norm F_{\Leb{q^{\prime}}_{t}\Leb{r^{\prime}}_{x}}. \end{align*}

The last part comes from (2). Hence by duality, we get

    \[ \Norm{e^{it\triangle/2}u_{0}}_{\Leb q_{t}\Leb r_{x}}\apprle\Norm{u_{0}}_{\Leb 2\left(\mathbb{R}^{n}\right)}. \]

For the last part, we use some abstract lemma: the Chirst-Kiselev lemma.

Lemma (Christ-Kiselev). Let X,Y be Banach spaces, let I be a time interval, and let K\in C^{0}\left(I\times I;B\left(X;Y\right)\right) be a kernel taking values in the space of bounded operators from X to Y. Suppose that 1\le p\le q\le\infty satisfying

    \[ \Norm{\int_{I}K\left(t,s\right)f\left(s\right)ds}_{\Leb q_{t}\left(I;Y\right)}\le A\Norm f_{\Leb p_{t}\left(I;X\right)} \]

for all f\in\Leb p_{t}\left(I;X\right) and some A>0. Then we have

    \[ \Norm{\int_{s\in I:s<t}K\left(t,s\right)f\left(s\right)ds}_{\Leb q_{t}\left(I;Y\right)}\apprle_{p,q}A\Norm f_{\Leb p_{t}\left(I;X\right)}. \]

By (1)

    \begin{align*} \Norm{\int_{\mathbb{R}}e^{i\left(t-s\right)\triangle/2}F\left(s\right)ds}_{\Leb q_{t}\Leb r_{x}\left(\mathbb{R}\times\mathbb{R}^{n}\right)} & \Norm{e^{it\triangle/2}\int_{\mathbb{R}}e^{-is\triangle/2}F\left(s\right)ds}_{\Leb q_{t}\Leb r_{x}\left(\mathbb{R}\times\mathbb{R}^{n}\right)}\\ & \apprle\Norm{\int_{\mathbb{R}}e^{-is\triangle/2}F\left(s\right)ds}_{\Leb 2_{x}}\\ & \apprle\Norm F_{\Leb{q'}_{t}\Leb{r'}_{x}}. \end{align*}

Hence by Christ-Kiselev’s lemma,

    \[ \Norm{\int_{s<t}e^{i\left(t-s\right)\triangle/2}F\left(s\right)ds}_{\Leb q_{t}\Leb r_{x}\left(\mathbb{R}\times\mathbb{R}^{n}\right)}\apprle\Norm F_{\Leb{q'}_{t}\Leb{r'}_{x}}.\qedhere \]

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