Bilinear Multipliers on Banach Function Spaces

Let X1,X2,X3 be Banach spaces of measurable functions in L0(R) and let m(ξ,η) be a locally integrable function in R2. We say that m∈BM(X1,X2,X3)(R) if Bm(f,g)(x)=∫R∫Rf^(ξ)g^(η)m(ξ,η)e2πi<ξ+η,x>dξdη, defined for f and g with compactly supported Fourier transform, extends to a bounded bilinear operator from X1×X2 to X3. In this paper we investigate some properties of the class BM(X1,X2,X3)(R) for general spaces which are invariant under translation, modulation, and dilation, analyzing also the particular case of r.i. Banach function spaces. We shall give some examples in this class and some procedures to generate new bilinear multipliers. We shall focus on the case m(ξ,η)=M(ξ-η) and find conditions for these classes to contain nonzero multipliers in terms of the Boyd indices for the spaces.

The spaces of bilinear multipliers of weighted Lorentz type modulation spaces

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.