In this section, we will give an important bilinear estimate.
We give an important relation before proving the bilinear estimate.
where
(3.1)
Lemma 3.1 Let , , where , . Then
(3.2)
Proof Let
To establish (3.2), it is sufficient to derive the following inequality:
(3.3)
where
(3.4)
Without loss of generality, we assume that , (). To derive (3.3), it suffices to prove that
(3.5)
By using the symmetry between and , without loss of generality, we can assume that . Obviously,
where
We will denote the integrals in (3.5) corresponding to () by (), respectively. Let , , .
-
(1)
Subregion . Since , we have , which yields
Then, by the Plancherel identity, the Hölder inequality, and , we derive
-
(2)
Subregion . In this subregion, obviously, .
It is easily checked that
Consequently, by the Cauchy-Schwarz inequality and Lemma 2.2, we have
-
(3)
Subregion . In this subregion, we derive . Thus,
-
(i)
Case . By (3.1), we derive
If , then
If , then
This case can be proved similarly to .
-
(ii)
Case . Since , we have
If , we have
consequently, by using the Cauchy-Schwarz inequality and (2.5) and (2.4), we have
If , since , we have
This case can be proved similarly to the above case.
-
(iii)
Case . This case is similar to (ii) case .
-
(4)
Subregion . In this subregion, , and it is easy to obtain
-
(i)
Case . By using, , when , we have
This case can be proved similarly to Subregion . When , we have
If , since , then
By using the Cauchy-Schwarz inequality, we have
-
(ii)
Case . Since , by using , we have
This case can be proved similarly to .
-
(iii)
Case .
This case can be proved similarly to .
-
(5)
Subregion . In this region , thus, we have
-
(i)
If , by using (3.1) and , we have
By using the Plancherel identity, the Hölder inequality, and as well as (2.5), we have
-
(ii)
If , then . By using (3.1), we have
By using the Plancherel identity, the Hölder inequality, (2.5) and , we have
-
(iii)
If .
This case can be proved similarly to the case .
-
(6)
Subregion . In this region, we have .
This case can be proved similarly to the Subregion .
We have completed the proof of Lemma 3.1. □