The spaces of bilinear multipliers of weighted Lorentz type modulation spaces

Abstract

AbstractFix 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.