- Open Access
Bilinear multipliers of weighted Wiener amalgam spaces and variable exponent Wiener amalgam spaces
© Kulak and Gürkanli licensee Springer. 2014
- Received: 9 September 2014
- Accepted: 12 November 2014
- Published: 27 November 2014
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.
- bilinear multipliers
- weighted Wiener amalgam space
- variable exponent Wiener amalgam space
If , the Fourier-Stieltjes transform of μ is denoted by .
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].
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 . 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 . 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 .
Lemma 2.1 Let and ω be a slowly increasing weight function. Then is dense in the Wiener amalgam space .
This completes the proof. □
for all .
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 .
Also by Proposition 11.4.1 in , , . So, we write , .
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.
for all .
This completes proof. □
The following theorem is a generalization of Theorem 2.1.
Now, we will give some properties of the space .
- (a)for each and
- (b)for each and
for some .
by equation (2.27).
Thus we have .
Hence . □
- (a)If , then and
- (b)If such that , then and