Theorem 1(Strichartz esitmates for Schrödinger).Fix and and call a pair of exponents admissible if , and . Then for any admissible exponents and , we have the homogeneous Strichartz estimate

(1)

the dual homogeneous Strichartz estimate

(2)

and the inhomogeneous (or retarded) Strichartz estimate

(3)

We illustrate some application of Strichartz estimate. If is the solution to an inhomogeneous Schrödinger equation

where , is given by Duhamel’s formula

on some time interval containing . Apply to both sides and using Strichartz’s estimate, we have

for any admissible and .

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

*Proof of Theorem 1.* For , by Minkowski’s integral inequality, we have

For any Schwarz function in spacetime,

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

Since and satisfying we have .

Recall the Hardy-Littlewood-Sobolev inequality:

whenever

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

Now by Hölder’s inequality, we have

On the left-hand side,

Thus,

This proves the second part (2).

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

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

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

Lemma(Christ-Kiselev).Let be Banach spaces, let be a time interval, and let be a kernel taking values in the space of bounded operators from to . Suppose that satisfying

for all and some . Then we have

By (1)

Hence by Christ-Kiselev’s lemma,