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Bilinear multipliers of weighted Lebesgue spaces and variable exponent Lebesgue spaces
Journal of Inequalities and Applications volume 2013, Article number: 259 (2013)
Abstract
Let , and , , be weight functions on . Assume that , are slowly increasing functions.
We say that a bounded function defined on is a bilinear multiplier on of type (shortly ) if
is a bounded bilinear operator from to . We denote by (shortly ) the vector space of bilinear multipliers of type .
In this paper first we discuss some properties of the space . Furthermore, we give some examples of bilinear multipliers.
At the end of this paper, by using variable exponent Lebesgue spaces , and , we define the space of bilinear multipliers from to and discuss some properties of this space.
MSC:42A45, 42B15, 42B35.
1 Introduction
Throughout this paper , and denote the space of infinitely differentiable complex-valued functions with compact support on , the space of all continuous, complex-valued functions with compact support on and the space of infinitely differentiable complex-valued functions on rapidly decreasing at infinity, respectively. For , denotes the usual Lebesgue space. A continuous function ω satisfying and for will be called a weight function on . If for all , we say that . We say that a weight function is of polynomial type if for . Let f be a measurable function on . If there exist and such that
for all , then f is said to be a slowly increasing function. It is easy to see that polynomial-type weight functions are slowly increasing.
For , we set
It is known that is a Banach space under the norm
or
The dual of the space is the space , where and . For , the Fourier transform of f is denoted by . We know that is a continuous function on , which vanishes at infinity, and it has the inequality [3, 4]. Let f be a measurable function on . The translation, character and dilation operators , and are defined by , and respectively for , . With this notation out of the way one has, for and ,
We denote by the space of bounded regular Borel measures, by the space of μ in such that
If , the Fourier-Stieltjes transform of μ is denoted by [5]. In this paper, denotes the family of all measurable functions . We put
We shall also use the notation
The generalized Lebesgue space (or the variable exponent Lebesgue space) is defined to be a space of (equivalence classes) measurable functions f such that
for some . If , then
It is known by Theorem 2.5 in [6] that is a Banach space with the Luxemburg norm
If , then is dense in . Also, if is a constant function, then the above norm coincides with the usual norm . The vector space of locally integrable functions on is denoted by . The space is a solid space, that is, if is given and satisfies a.e., then and by [8]. In this paper we assume that .
2 The bilinear multipliers space
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 [9], we have . Hence, by (2.1), we obtain .
Now, we show that is dense . Let be given. Then . Since is dense by [6], 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 [10], 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 .
-
(a)
for each and
-
(b)
for each and
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
-
(b)
Let us rewrite the value as follows:
(2.21)
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)
Thus,
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 .
-
(a)
If , then and
-
(b)
If such that , then and
Proof (a) Let and . Since and , then by Proposition 2.5 in [11]
Also, since , we have by Theorem 2.4. So, we write
Hence . Finally, by (2.29), we obtain
-
(b)
Let . Take any and . It is known by Proposition 2.5 in [11] 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)
Hence, and
□
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
If we say , then we obtain
which means . □
The following theorem can be proved easily by using the technique of the proof in Theorem 2.10.
Theorem 2.11 Let , , and . If such that , and , then .
3 The bilinear multipliers space
Definition 3.1 Let and let , , . Assume that is a bounded function on . Define
for all .
m is said to be a bilinear multiplier on of type if there exists such that
for all , i.e., extends to a bounded bilinear operator from to . We denote by the space of bilinear multipliers of type and .
The following theorem can be proved easily by using the technique of the proof in Theorem 2.2.
Theorem 3.1 Let and for all . Then if and only if there exists such that
for all .
Theorem 3.2 Let . If , then .
Proof Take any . Then
Since the space is the Banach convolution module over such that , we write . Also, we have . Then by (3.1), we find such that
If we set , we obtain
and . □
Theorem 3.3 If , then and
for all .
Proof Let us take any . By the proof of (a) Theorem 2.4, we know that
By Lemma 5 in [8], we know and . Since , by (3.2), there exists such that
Thus . Moreover, by using the same technique as in the proof of Theorem 2.4, we obtain
□
Theorem 3.4 Let . If , then and there exists such that
for all .
Proof Take any . By Proposition 2.5 in [11], we know that
Since , then and
by Theorem 3.3. Using (3.3) and the Minkowski inequality for a variable exponent Lebesgue space [12], we find such that
Hence and by (3.4), we have
□
Theorem 3.5 Let .
-
(a)
If , and , then .
-
(b)
If and , then .
-
(c)
If and , then .
-
(d)
If and , then .
Proof (a) Let be given. Then
Since the spaces and are Banach convolution module over , we have and . Also, since , we write . Then, by the equality
the Hölder inequality and the inequality (3.5), we have
If we say , we obtain
Hence, .
-
(b)
Take any . By (a), we know that
Similarly, if we say , we obtain
which means .
In this theorem, (c) and (d) can be proved easily by using the technique of the proof in (a) and (b), respectively. □
Theorem 3.6 Let . If are bounded sets, then
Proof Let be given. By using the Fubini theorem, we can easily prove the following equality:
Also, by the Minkowski inequality and Theorem 3.3, we find such that
If we say , we obtain
Hence . □
Theorem 3.7
-
(a)
If and , then .
-
(b)
If , , and , then .
Proof (a) Take any . Since , there exists such that
Also, since , there exists such that
If we set and combine the inequalities (3.6) and (3.7), we have
Therefore .
-
(b)
Let us take any . Since , there exists such that
(3.8)
By using the inclusions , and , we find such that
and
If we set and combine the inequalities (3.8), (3.9), (3.10) and (3.11), we have
Then . □
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Acknowledgements
Dedicated to Professor Hari M Srivastava.
This research was supported by the Ondokuz Mayıs University (PYO.FEN.1904.13.002).
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Kulak, Ö., Gürkanlı, A.T. Bilinear multipliers of weighted Lebesgue spaces and variable exponent Lebesgue spaces. J Inequal Appl 2013, 259 (2013). https://doi.org/10.1186/1029-242X-2013-259
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DOI: https://doi.org/10.1186/1029-242X-2013-259