Let \(\vec{f}= (f_1, f_2)\). Consider the bilinear square-function Fourier multiplier operator \(\operatorname{Im}_{\lambda, m}\) associated with the multilinear \(g_{\lambda}^\ast\)-function due to S. Shi et al. [J. Math. Pures Appl. (9) 101, No. 3, 394–413 (2014; Zbl 1287.42017)] \[ \operatorname{Im}_{\lambda, m}(\vec{f})(x) = {\bigg(} \int\!\!\!\int_{\mathbb{R}_{+}^{n+1}} \ {\bigg(} \frac{t}{t+|x-z|} {\bigg)}^{n\lambda} \ |T_{m}(\vec{f})(z)|^2 \ \frac{dz \, dt}{t^{n+1}} {\bigg)}^{1/2} \] where \[T_{m}(\vec{f})(z) = \frac{1}{t^{2n}}\int_{(\mathbb{R}^{2n})} K{\bigg(}\frac{z-y_1}{t}, \frac{z-y_2}{t} {\bigg)} f_1(y_1)f_2(y_2) \, dy_1\, dy_2. \]
For the main results, we assume that the weight \(\vec{\omega}=(\omega_1, \omega_2)\) satisfies multiple weights classes \(A_{\vec{P}}\) condition introduced in [A. K. Lerner et al., Adv. Math. 220, No. 4, 1222–1264 (2009; Zbl 1160.42009)], namely, \[ \sup_{Q} \bigg( \frac{1}{|Q|} \int_{Q} \prod_{i=1}^{2} \omega_i^{1/p_i} \bigg)^{1/p} \prod_{i=1}^{2} \bigg( \frac{1}{|Q|} \int_{Q} \omega_i^{1-p_i^\prime} \bigg)^{1/p^\prime} < \infty, \] where \(1 \leq p_1\), \(p_2 < \infty\), \(1/p = 1/p_1 + 1/p_2\), \(\vec{P} = (p_1, p_2)\), and denote \(\nu_{\vec{\omega}} = \prod_{i=1}^{2} \omega_i^{p/p_i}\) for given \(\vec{\omega}\).
We also assume that \(m \in L^{\infty}((\mathbb{R} )^2)\) satisfies the following conditions \[ \big{|}\partial^{\alpha}m(\xi_1, \xi_2) \big{|} \leq C \, \frac{(|\xi_1|+|\xi_2|)^{-|\alpha|+\epsilon_1}}{(1+|\xi_1|+|\xi_2|)^{\epsilon_1+\epsilon_2}} \tag{1}\]
and
\[\big{|}m(\xi_1, \xi_2) \big{|} \leq C \, \frac{(|\xi_1|+|\xi_2|)^{-s+\epsilon_1}}{(1+|\xi_1|+|\xi_2|)^{\epsilon_1+\epsilon_2}}\tag{2} \] for some \(\epsilon_1, \epsilon_2 > 0\), \(|\alpha| \leq s\) and \(s \in [n+1,2n]\) for some integer \(s\).
In this paper the authors prove that if \(p_0 < p_1\), \(p_2 < \infty\), \(1/p = 1/p_1 + 1/p_2\), \(\nu_{\vec{\omega}} \in A_{\vec{P}/p_0}\) and \(m \in L^{\infty}((\mathbb{R} )^2)\) satisfies (1) and (2), then Fourier multiplier operator \(\operatorname{Im}_{\lambda, m}\) maps from \(L^{p_1}(\omega_1) \times L^{p_2}(\omega_2)\) into \(L^{p}(\nu_{\vec{\omega}})\) where \(s\) is an integer with \(s \in [n+1,2n]\) and \(\lambda > 2s/n + 1\) and \(2n/s \leq p_0 \leq 2\). Also if \( p_0 > 2n/s\) and \(p_1 = p_0\) or \(p_2 = p_0\), then \(\operatorname{Im}_{\lambda, m}\) satisfies a weighted weak type \((p,p)\) estimate from \(L^{p_1}(\omega_1) \times L^{p_2}(\omega_2)\) into \(L^{p, \infty}(\nu_{\vec{\omega}})\). Moreover, they treat the commutators of \(\operatorname{Im}_{\lambda, m}\) under the same condition as above to obtain a strong \(L^{p}(\nu_{\vec{\omega}})\)-bound on products of weighted Lebesgue spaces \(L^{p_1}(\omega_1) \times L^{p_2}(\omega_2)\) and the weighted endpoint \(L\log L\)-type estimate with \(\vec{\omega} \in A_{(1,1)}\).