The spaces of bilinear multipliers of weighted Lorentz type modulation spaces

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Abstract Fix a nonzero window g ∈ 𝒮 ⁢ ( ℝ n ) ${g\in\mathcal{S}(\mathbb{R}^{n})}$ , a weight function w on ℝ 2 ⁢ n ${\mathbb{R}^{2n}}$ and 1 ≤ p , q ≤ ∞ ${1\leq p,q\leq\infty}$ . The weighted Lorentz type modulation space M ⁢ ( p , q , w ) ⁢ ( ℝ n ) ${M(p,q,w)(\mathbb{R}^{n})}$ consists of all tempered distributions f ∈ 𝒮 ′ ⁢ ( ℝ n ) ${f\in\mathcal{S}^{\prime}(\mathbb{R}^{n})}$ such that the short time Fourier transform V g ⁢ f ${V_{g}f}$ is in the weighted Lorentz space L ⁢ ( p , q , w ⁢ d ⁢ μ ) ⁢ ( ℝ 2 ⁢ n ) ${L(p,q,w\,d\mu)(\mathbb{R}^{2n})}$ . The norm on M ⁢ ( p , q , w ) ⁢ ( ℝ n ) ${M(p,q,w)(\mathbb{R}^{n})}$ is