Definition 2.1 For two integers and , we say that is an admissible pair if the following condition is satisfied:
We have the following Strichartz estimate (see [6]): For any admissible pair
(2.1)
Lemma 2.1 Assume that is an admissible pair. Let . We have
(2.2)
For any , we have
(2.3)
(2.4)
where .
Proof Firstly, we prove the inequality (2.2).
For any , we have
where .
Noting that , using the Strichartz estimate, we obtain
Next we prove (2.3), we only need to prove that for we have
(2.5)
(2.6)
Changing the variable to , we obtain
(2.7)
From (2.6) and (2.7), we can obtain
so that (2.5) holds.
Similarly, (2.4) follows from the following inequality:
This inequality has been proved in Theorem 4.1 of [7]. We omit its detailed proof here. □
Lemma 2.2 If for . Then we have
(2.8)
where is as same as in Lemma 2.1.
Proof Noting that is an admissible pair, so we have
(2.9)
Therefore, using (2.7), Minkowski’s inequality, (2.9), and taking , we can obtain
(2.10)
By interpolation [8] between (2.10) and the following relation:
we immediately obtain for all
□
We take a function with on and . We denote .
Similar to the proof of Lemma 3.2 in [9] (or see [[10], Lemmas 3.1-3.3], [[11], Lemma 2.3], [[12], Lemma 2.7]), we have the following estimates.
Lemma 2.3 For any real s, , , and . We have
Lemma 2.4 [12]
If f, , , and belong to a Schwartz space on , then we have
where
Lemma 2.5 Let , , . Then we have
(2.11)
Proof By the definition of and duality, the inequality (2.11) is reduced to the following estimate:
for all , where
Let , , .
Without loss of generality, we can assume that , for .
We split the domain of integration into two cases and .
Case I. Assume that .
Noting that , by Lemma 2.4, the Hölder inequality, and Lemma 2.1, we obtain
Case II. Assume that .
Subcase 1. Assume that . Then we have .
Noting that , by Lemma 2.4, the Hölder inequality, and Lemma 2.2, we obtain
Subcase 2. Assume that . We split this domain of integration in several pieces.
Assume that .
Similar to the proof of subcase 1, noting that , , by Lemma 2.4, the Hölder inequality, and Lemma 2.2, we obtain
Assume that .
In this case, the proof is similar to that of the above case, so here we omit the detailed proof.
Assume that .
In this case, we easily get
By a straightforward calculation, we can obtain
(2.13)
3.0: Assume that , so holds.
Noting that , by Lemma 2.4, the Hölder inequality, and Lemma 2.1, we obtain
3.1: Assume that . By (2.10) and (2.11), we immediately obtain .
Noting that , , by Lemma 2.4, the Hölder inequality, and Lemma 2.1, we obtain
3.2: Assume that .
The proof is similar to that of the case 3.1, so here omit the detailed proof.
3.3: Assume that . By (2.10) and (2.11), we immediately obtain .
Noting that , , by Lemma 2.4, the Hölder inequality, and Lemma 2.1, we obtain
□
Lemma 2.6 Let , . Then we have the following inequality:
(2.14)
Proof Firstly, we prove (2.14) holds for .
By the definition, the Hölder inequality, and the Sobolev inequality, we have
(2.15)
That means
(2.16)
Again using the Hölder inequality, the Sobolev inequality in the variable x, and Lemma 2.1, we obtain
(2.17)
where , , , , is admissible pair.
Combining (2.16) and (2.17), we obtain
(2.18)
Secondly, using the Leibniz rule for fractional power, in a similar way, we obtain
(2.19)
where , , , , is admissible pair for .
Taking , , ; and for some , , , ; , , , for in (2.19), noting and , we obtain
(2.20)
which completes the proof. □