Let ω i be weight functions on R, (i=1,2,3,4). In this work, we define CW p,q,r,s,τ ω 1 ,ω 2 ,ω 3 ,ω 4 (R) to be vector space of (f, g) ∈ L p ω 1 × L q ω 2 (R) such that the τ −Wigner transforms Wτ (f, .) and Wτ (., g) belong to L r ω 3 R 2 and L s ω 4 R 2 respectively for 1 ≤ p, q, r, s < ∞, τ ∈ (0, 1). We endow this space with a sum norm and prove that CW p,q,r,s,τ ω 1 ,ω 2 ,ω 3 ,ω 4 (R) is a Banach space. We also show that CW p,q,r,s,τ ω 1 ,ω 2 ,ω 3 ,ω 4 (R) becomes an essential Banach module over L 1 ω 1 × L 1 ω 2 (R). We then consider approximate identities.2 f t s for all x, ω ∈ R, 0 = s ∈ R, respectively. The parameters in wavelet theory are "time"x and "scale" s. Dilation operator D s preserves the shape of f , but it changes the scale, [7].Given any fixed 0 = g ∈ L 2 (R)(called the window function), the short-time Fourier transform (STFT) of a function f ∈ L 2 (R) with respect to g is defined by

UDC 512.5 In this paper, we introduce the notion of multiplication Krasner hypermodules over commutative hyperrings and topologize the collection of all multiplication Кrasner hypermodules. In addition, we investigate some properties of this topological space.

Let ω 1 , ω 2 be slowly increasing weight functions, and let ω 3 be any weight function. We say that m(ξ , η) is a bilinear multiplier on R n of type

Let 1 ≤ p 1 , p 2 < ∞, 0 < p 3 ≤ ∞ and ω 1 , ω 2 , ω 3 be weight functions on R n. Assume that ω 1 , ω 2 are slowly increasing functions. We say that a bounded function m(ξ , η) defined on R n × R n is a bilinear multiplier on R n of type (p 1 , ω 1 ; p 2 , ω 2 ; p 3 , ω 3) (shortly (ω 1 , ω 2 , ω 3)) if B m (f , g)(x) = R n R nf (ξ)ĝ(η)m(ξ , η)e 2π i ξ +η,x dξ dη is a bounded bilinear operator from L p 1 ω 1 (R n) × L p 2 ω 2 (R n) to L p 3 ω 3 (R n). We denote by BM(p 1 , ω 1 ; p 2 , ω 2 ; p 3 , ω 3) (shortly BM(ω 1 , ω 2 , ω 3)) the vector space of bilinear multipliers of type (ω 1 , ω 2 , ω 3). In this paper first we discuss some properties of the space BM(ω 1 , ω 2 , ω 3). Furthermore, we give some examples of bilinear multipliers. At the end of this paper, by using variable exponent Lebesgue spaces L p 1 (x) (R n), L p 2 (x) (R n) and L p 3 (x) (R n), we define the space of bilinear multipliers from L p 1 (x) (R n) × L p 2 (x) (R n) to L p 3 (x) (R n) and discuss some properties of this space.

Fix a nonzero window ${g\in\mathcal{S}(\mathbb{R}^{n})}$, a weight function w on ${\mathbb{R}^{2n}}$ and ${1\leq p,q\leq\infty}$. The weighted Lorentz type modulation space ${M(p,q,w)(\mathbb{R}^{n})}$ consists of all tempered distributions ${f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})}$ such that the short time Fourier transform ${V_{g}f}$ is in the weighted Lorentz space ${L(p,q,w\,d\mu)(\mathbb{R}^{2n})}$. The norm on ${M(p,q,w)(\mathbb{R}^{n})}$ is ${\|f\/\|_{M(p,q,w)}=\|V_{g}f\/\|_{pq,w}}$. This space was firstly defined and some of its properties were investigated for the unweighted case by Gürkanlı in [9] and generalized to the weighted case by Sandıkçı and Gürkanlı in [16]. Let ${1<p_{1},p_{2}<\infty}$, ${1\leq q_{1},q_{2}<\infty}$, ${1\leq p_{3},q_{3}\leq\infty}$, ${\omega_{1},\omega_{2}}$ be polynomial weights and ${\omega_{3}}$ be a weight function on ${\mathbb{R}^{2n}}$. In the present paper, we define the bilinear multiplier operator from ${M(p_{1},q_{1},\omega_{1})(\mathbb{R}^{n})\times M(p_{2},q_{2},\omega_{2})(% \mathbb{R}^{n})}$ to ${M(p_{3},q_{3},\omega_{3})(\mathbb{R}^{n})}$ in the following way. Assume that ${m(\xi,\eta)}$ is a bounded function on ${\mathbb{R}^{2n}}$, and define$B_{m}(f,g)(x)=\int_{\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\hat{f}(\xi)\hat{g}(% \eta)m(\xi,\eta)e^{2\pi i\langle\xi+\eta,x\rangle}\,d\xi\,d\eta\quad\text{for % all ${f,g\in\mathcal{S}(\mathbb{R}^{n})}$. }$The function m is said to be a bilinear multiplier on ${\mathbb{R}^{n}}$ of type ${(p_{1},q_{1},\omega_{1};p_{2},q_{2},\omega_{2};p_{3},q_{3},\omega_{3})}$ if ${B_{m}}$ is the bounded bilinear operator from ${M(p_{1},q_{1},\omega_{1})(\mathbb{R}^{n})\times M(p_{2},q_{2},\omega_{2})(% \mathbb{R}^{n})}$ to ${M(p_{3},q_{3},\omega_{3})(\mathbb{R}^{n})}$. We denote by ${\mathrm{BM}(p_{1},q_{1},\omega_{1};p_{2},q_{2},\omega_{2})(\mathbb{R}^{n})}$ the space of all bilinear multipliers of type ${(p_{1},q_{1},\omega_{1};p_{2},q_{2},\omega_{2};p_{3},q_{3},\omega_{3})}$, and define ${\|m\|_{(p_{1},q_{1},\omega_{1};p_{2},q_{2},\omega_{2};p_{3},q_{3},\omega_{3})% }=\|B_{m}\|}$. We discuss the necessary and sufficient conditions for ${B_{m}}$ to be bounded. We investigate the properties of this space and we give some examples.