Lemma 2.1 Let and let ω be a slowly increasing weight function. Then is dense in .
Proof Let be given. Since ω is a slowly increasing weight function, there exist and such that
for all . Also, since m is a polynomial, then by Proposition 19.2.2 in , we have . Hence, by (2.1), we obtain .
Now, we show that is dense . Let be given. Then . Since is dense by , for given , there exists such that
Therefore, by using the inequality (2.2), we write
Also, since and m is a polynomial, we have . Thus, we have . By using the inclusion , we obtain .
Now, take any . Since , there exists such that
Furthermore, since is dense , there exists such that
Combining the inequalities (2.3) and (2.4), we have
which means . □
Definition 2.1 Let , and , , be weight functions on . Assume that , are slowly increasing functions and is a bounded function on . Define
for all .
m is said to be a bilinear multiplier on of type (shortly ) if there exists such that
for all . That means extends to a bounded bilinear operator from to .
We denote by (shortly ) the space of all bilinear multipliers of type and .
Theorem 2.1 Let and . If , then defines a bilinear multiplier and .
Proof For , we have and . Thus, by the Fubini theorem, we write
Since , we have . Hence, by (2.5), we obtain
Since , then
and hence . Therefore from (2.6) and the Minkowski inequality, we write
Hence, using the generalized Hölder inequality and combining (2.7), (2.8), we have
If we set , we obtain
Then . Consequently, using (2.9), we have
Definition 2.2 Let , and , , be weight functions on . Suppose that , are slowly increasing functions. We denote by the space of measurable functions such that , that is to say,
extends to a bounded bilinear map from to . We denote .
Theorem 2.2 Let and . Then if and only if there exists such that
for all , where .
Proof Let . We take any . From the Fubini theorem, we write
where . On the other hand, since , then . Thus we obtain . Also, . Hence, using the Hölder inequality and the inequality (2.10), we write
Moreover, since , there exists such that
If we combine (2.11) and (2.12), we write
For the proof of converse, assume that there exists a constant such that
for all . From the assumption and (2.10), we write
Define a function l from to ℂ such that
It is clear that the function ℓ is linear and bounded by (2.13). By using in , it is easy to show that . So, by the inclusion , we have . Thus ℓ extends to a bounded function from to ℂ. Then and by (2.13), we have
Hence, we obtain . □
Theorem 2.3 Let , and , be a weight function of polynomial type such that . If and for , then . Moreover,
Proof Let . Then
On the other hand, by the assumption , it is easy to see that and
Also, . Then, by (2.14), (2.15) and the generalized Hölder inequality, we have
Now, suppose that . Then we write
Hence by (2.16)
Thus and by (2.17), we have
Similarly, if , then we write
Again, by (2.16) we have
Hence, we obtain and by (2.18)
Theorem 2.4 Let .
Proof (a) Let us take any and . If we say that and , then
By (2.19), we have
Since , and are satisfied for all and . Hence, by (2.20), we have
for some . Thus . Also, we obtain
Let us rewrite the value as follows:
Also, the inequalities and are satisfied for all , . Hence, since , by (2.21) we have
Then , and by (2.22) we obtain
Lemma 2.2 If is a polynomial-type weight function and , then . Moreover,
Proof Let be a polynomial-type weight function and . Assume that . If we get ,
Thus we have and .
Now, assume that . Similarly by (2.23)
Hence , and we also have . □
Theorem 2.5 Let , , be weight functions of polynomial type and let . If and , then . Moreover, then
Proof Let and be given. We know by Lemma 2.2 that and . If we get and , we obtain
Hence, from the equality , we have
Assume that . Since , by Lemma 2.2 and using equality (2.24), we obtain
Then , and by (2.25)
Now let . Again, since , by Lemma 2.2 and using equality (2.24), we obtain
Thus and by (2.26)
Theorem 2.6 Let , , be weight functions of polynomial type and let such that for any , where . Then
Proof Take any , . It is known by Theorem 2.5 that
On the other hand, using and changing the variables , , we note that
Hence by (2.27) and (2.28), we have
Since for , we let . Assume first that . Also, since , by Theorem 2.5 we have and . Then by (2.28)
Hence . Since , we have . Thus, we write .
Assume now that . Again, by Theorem 2.5, we have and . Similarly,
Thus, we have
Hence Since , we have . Thus, we write . □
Theorem 2.7 Let and .
If , then and
If such that , then and
Proof (a) Let and . Since and , then by Proposition 2.5 in 
Also, since , we have by Theorem 2.4. So, we write
Hence . Finally, by (2.29), we obtain
Let . Take any and . It is known by Proposition 2.5 in  that the equality
Since , by Theorem 2.4 we have and
Then we write
Thus from (2.30), we obtain and
Theorem 2.8 Let , , be weight functions of polynomial type and let . If such that , then . Moreover,
Proof Let us take . Then
Since , by Theorem 2.5, thus we observe that
Also, since for , and for , , by (2.31)
Theorem 2.9 Let and . If , are bounded measurable sets in , then
Proof Take any . Then we write
By using Theorem 2.4, we have
Hence, we obtain . □
Theorem 2.10 Let , , and . Assume that , and . If , then .
Proof For the proof we will use Theorem 2.2. Take any . Then
Since the spaces and are Banach convolution module over the spaces , respectively, we write and . Also, by Theorem 2.7, . Therefore we obtain . By using the Hölder inequality and the inequality (2.32), we find