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Bilinear multipliers of weighted Wiener amalgam spaces and variable exponent Wiener amalgam spaces
Journal of Inequalities and Applications volume 2014, Article number: 476 (2014)
Abstract
Let , be slowly increasing weight functions, and let be any weight function on . Assume that is a bounded, measurable function on . We define
for all . We say that is a bilinear multiplier on of type if is a bounded operator from to , where , , . We denote by the vector space of bilinear multipliers of type . In the first section of this work, we investigate some properties of this space and we give some examples of these bilinear multipliers. In the second section, by using variable exponent Wiener amalgam spaces, we define the bilinear multipliers of type from to , where , , , for all . We denote by the vector space of bilinear multipliers of type . Similarly, we discuss some properties of this space.
MSC:42A45, 42B15, 42B35.
1 Introduction
Throughout this paper we will work on with Lebesgue measure dx. We denote by , and 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 that rapidly decrease at infinity, respectively. Let f be a complex-valued 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 ,
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 . 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 . 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 [1]. It is easy to see that polynomial-type weight functions are slowly increasing. 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 . 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 [2].
The space consists of classes of measurable functions f on such that for any compact subset , where is the characteristic function of K. Let us fix an open set with compact closure and . The weighted Wiener amalgam space consists of all elements such that belongs to ; the norm of is [3–5].
In this paper, denotes the family of all measurable functions . We put
We shall also use the notation
The variable exponent Lebesgue spaces (or generalized Lebesgue spaces) are defined as the set of all (equivalence classes) measurable functions f on such that for some , equipped with the Luxemburg norm
If , then if . The set is a Banach space with the norm . If is a constant function, then the norm coincides with the usual Lebesgue norm [6]. The spaces and have many common properties. A crucial difference between and the classical Lebesgue spaces is that is not invariant under translation in general. If , then is dense in . The space is a solid space, that is, if is given and satisfies a.e., then and by [6]. In this paper we will assume that .
The space consists of classes of measurable functions f on such that for any compact subset . Let us fix an open set with compact closure, and . The variable exponent amalgam space consists of all elements such that belongs to ; the norm of is [7].
2 The bilinear multipliers space
Lemma 2.1 Let and ω be a slowly increasing weight function. Then is dense in the Wiener amalgam space .
Proof Since [8], we have by a lemma in [9]. Also we have the inclusion
For the proof that is dense in , take any . For given , there exists such that
Also, since and is dense in , by Lemma 2.1 in [10], there exists such that
Furthermore, by using the inequality , we write
(see [11] and [5]). Combining (2.1) and (2.2), we obtain
This completes the proof. □
Definition 2.1 Let , , and , , be weight functions on . Assume that , are slowly increasing functions and is a bounded, measurable function on . Define
for all .
m is said to be a bilinear multiplier on of type if there exists such that
for all . That means that extends to a bounded bilinear operator from to .
We denote by the space of all bilinear multipliers of type and .
The following theorem is an example to a bilinear multiplier on of type .
Theorem 2.1 Let , and . If , then defines a bilinear multiplier and .
Proof We know by Theorem 2.1 in [10] that for ,
Also by Proposition 11.4.1 in [5], , . So, we write , .
Using the Minkowski inequality and the generalized Hölder inequality, we have
Again, by using Proposition 11.4.1 in [5] and the assumption , we write
From this result, we find . Hence by (2.4), (2.5) and the generalized Hölder inequality, we obtain
where . Then defines a bilinear multiplier. Finally, using (2.6), we obtain
□
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 .
Let ω be a weight function. The continuous function cannot be a weight function. But the following lemma can be proved easily by using the technique of the proof of Lemma 2.1.
Lemma 2.2 Let and ω be a slowly increasing continuous weight function. Then is dense in Wiener amalgam space.
Theorem 2.2 Let , , and be a continuous, symmetric slowly increasing weight function. Then if and only if there exists such that
for all .
Proof We assume that . By Theorem 2.2 in [10], we write, for all ,
where . If we set , we have
Since is a symmetric weight function, if we set , we have
We know from [3] and [5] that the definition of is independent of the choice of Q. Then there exists such that
Hence, by (2.9) and (2.10), we have
Since from the assumption the right-hand side of (2.11) is finite, thus . On the other hand, since , there exists such that
Combining (2.11) and (2.12), we have
If we apply the Hölder inequality to the right-hand side of inequality (2.7) and use inequality (2.13), we obtain
For the proof of converse, assume that there exists a constant such that
for all . From the assumption and (2.14), we write
Define a function ℓ from to ℂ such that
ℓ is linear and bounded by (2.15). Also, since , we have by Lemma 2.2. Thus ℓ extends to a bounded function from to ℂ. Then . Again, since the definition of is independent of the choice of Q, there exists such that
Combining (2.15) and (2.16), we obtain
This completes proof. □
The following theorem is a generalization of Theorem 2.1.
Theorem 2.3 Let , , and , , be a weight function. If and for , then . Moreover,
Proof Let . By Theorem 2.3 in [10], we have
Also by [5] we write the inequalities
and
From these inequalities, we have and . If we use the inequality and set , we obtain
and hence . Then by (2.17), (2.18), (2.19), (2.20) and the Hölder inequality, we have
Now, suppose that . Since is a slowly increasing weight function, there exist and such that
Then
Hence by (2.21)
Thus , and by (2.22) we obtain
Similarly, if , then
Therefore by (2.21) we have
Hence, we obtain , and by (2.23)
□
Now, we will give some properties of the space .
Theorem 2.4 Let .
-
(a)
for each and
-
(b)
for each and
Proof (a) Let . From Theorem 2.4 in [10], we write the equality
Also the equalities and are satisfied. Then, using equality (2.24) and the assumption , we have
for some . Thus . Moreover, we obtain
(b) For any , we write
by Theorem 2.4 in [10]. Also, the inequalities and are satisfied [5]. Since , by (2.25) we have
and hence . So, by (2.26) we obtain
□
Lemma 2.3 If ω is a slowly increasing weight function such that and , then . Moreover,
for some .
Proof Take any . If we get , we obtain
Again, if we say and use ω to be slowly increasing, then there exist and such that
by equation (2.27).
Let . Using inequality (2.28), we have
Since is a compact set and the definition of is independent of the choice of a compact set Q, then there exists such that
by [3, 5]. Then by (2.29) we write
Thus we have .
Now, assume that . Similarly, by (2.28) and (2.30), we get
Hence . □
Theorem 2.5 Let , , for , and let be a slowly increasing weight function. If , and , then . Moreover, then
Proof Let and be given. From Lemma 2.3, we have and . Also it is known by Theorem 2.5 in [10] that
If we use this equality, we write
If we say in the last equality, we have
On the other hand, since is a slowly increasing weight function, there exist , such that . If we make the substitution in equality (2.31), we obtain
We assume that . Then
Also, since yQ is a compact set, we have
Since , by Lemma 2.3 and inequality (2.32), we obtain
Then , and by (2.33) we have
Now let . Again, since , by Lemma 2.3 and inequality (2.32), we obtain
Thus , and by (2.34) we have
□
Theorem 2.6 Let .
-
(a)
If , then and
-
(b)
If such that , then and
Proof (a) Let be given. By Proposition 2.5 in [12]
If we use Theorem 2.4 and the assumption , we have
Hence , and by (2.35) we obtain
(b) Let . Take any . It is known by Proposition 2.5 in [12] that
Since , we have and
by Theorem 2.4. Then
Thus from (2.36) we obtain and
□
Theorem 2.7 Let . If , are bounded measurable sets in , then
Proof Let . We know by Theorem 2.9 in [10] that
From Theorem 2.4, we have
Hence, we obtain . □
Theorem 2.8 Let be a continuous, symmetric, slowly increasing weight function, , , , , , and . Assume that , and . If , then .
Proof Take any . Then, by Theorem 2.10 in [10], we write
On the other hand, we know that the inequalities
and
hold, where , by [3]. That means and . Also, every constant function is bilinear multiplier of type under the given conditions. So, by Theorem 2.6, we have . Now, if we say that , we have
In this here, we set and use to be symmetric. Then we have
by [5]. Using the Hölder inequality, inequalities (2.37), (2.38), (2.39) and , we find
If we say , then we obtain
From Theorem 2.2, we have . □
Theorem 2.9 Let , , , and such that and . Assume that is a continuous, bounded, symmetric weight function. If , then .
Proof Firstly, we show that . Take any . Let be a closed and bounded rectangle. Since the definition of is independent of the choice of a compact set Q, then, by using Fubini’s theorem, we get
for some . So, we have . By using the Hölder inequality, the Hausdorff-Young inequality and equality (2.40), we obtain
for some . Therefore by Theorem 2.2.
Now, we show that . Again, using Fubini’s theorem, we have
for some , . So, . Similarly, we have
for some , . That means . We set and . Then, by using the Hölder inequality, the Hausdorff-Young inequality, (2.41) and (2.42), we get
Thus, by Theorem 2.2, we obtain . Similarly, if we change the variables and , then
Hence .
We take , , and such that , , and . Since and , we have by the interpolation theorem in [13, 14] such that
for all . On the other hand, from equalities (2.43) and (2.44), we obtain the equalities and . Similarly, we take , , and such that , , and . Again, if we use and , we have by the interpolation theorem in [13, 14] such that
for all . So, from equalities (2.45) and (2.46), we have and .
Now, we choose , such that and . Let these numbers have the following conditions:
for . Again, we choose , such that and . Let these numbers have the following conditions:
for . If we use equalities (2.45), (2.46), (2.47), (2.48), (2.49) and (2.50), we write
Moreover, using the equalities , , (2.51) and the assumption , we obtain
Similarly, using the equalities , , (2.52) and the assumption , we obtain
Since , , then the bilinear multipliers and are bounded. Also, since , , , , by equalities (2.53) and (2.54) and by the interpolation theorem in [13], is bounded. That means . This completes the proof. □
3 The bilinear multipliers space
Lemma 3.1 Let .
-
(a)
If ω is a slowly increasing weight function, then is dense in the weighted variable exponent Wiener amalgam space .
-
(b)
If ω is a continuous, slowly increasing weight function, then is dense in the weighted variable exponent Wiener amalgam space .
This lemma can be proved easily by using the proof technique in Lemma 2.1.
Definition 3.1 Let , , , , , , and , , be weight functions on . Assume that , are slowly increasing functions and is a bounded, measurable function on . Define
for all . m is said to be a bilinear multiplier on of type if there exists such that
for all . That means extends to a bounded bilinear operator from to . We denote by the space of all bilinear multipliers of type and
The following theorem can be proved easily by using Lemma 3.1 and the technique of proof of Theorem 2.2.
Theorem 3.1 Let , , , and be a continuous, symmetric, slowly increasing weight function. Then if and only if there exists such that
for all .
Now we will give some properties of the space . Since some properties of usual Lebesgue spaces are not true in general in the variable exponent Lebesgue spaces, like translation invariance, then also some properties of the spaces do not hold true in general in the spaces .
Theorem 3.2 If , then and
for all .
Proof Let . Then we have
Similarly, the equality is written. So, by using these results and the assumption , we have
Thus . Moreover, by using the same technique as in the proof of Theorem 2.4, we obtain
□
Theorem 3.3 Let . Then , and there exists such that
for all .
Proof Take any . By Proposition 2.5 in [12], we know that
Since , then by Theorem 3.2, in the space and
Using (3.1) and the Minkowski inequality in [15], we find such that
Hence , and by (3.2) we have
□
Theorem 3.4 Let , , , and be a continuous, symmetric, slowly increasing weight function. If , and , then .
Proof Take any . Then, by Theorem 2.10 in [10], we write the following:
So, by Theorem 11.7.1 in [5] and inequalities (2.37), (2.38), there exists such that
Hence, by Theorem 3.1. □
Theorem 3.5 Let . If are bounded sets, then
Proof Let be given. The equality
is known by Theorem 2.9 in [10]. Using Theorem 3.2, there exists such that
Hence . □
Theorem 3.6 Let , , , , , , , , . Then
Proof Take any . Then there exists such that
On the other hand, by Proposition 2.5 in [7] we have , and . So, there exist , and such that
and
Combining (3.3), (3.4), (3.5) and (3.6), we get
That means . Hence, we obtain . □
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Acknowledgements
This work was supported by the Ondokuz Mayıs University, Project number PYO.FEN.1904.13.002.
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Kulak, Ö., Gürkanlı, A.T. Bilinear multipliers of weighted Wiener amalgam spaces and variable exponent Wiener amalgam spaces. J Inequal Appl 2014, 476 (2014). https://doi.org/10.1186/1029-242X-2014-476
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DOI: https://doi.org/10.1186/1029-242X-2014-476