In this work, we prove the strong and weak estimates for a class of bilinear fiber-wise pseudo-differential operators. More precisely, we consider \(\begin{aligned} \int _{\mathbb {R}^{4}}a(x_2,\xi _1,\xi _2,\eta _2){\widehat{F}}(\xi ){\widehat{G}}(\eta )e^{2\pi ix\cdot (\xi +\eta )}d\xi d\eta ,\quad F,G \in \mathcal {S}(\mathbb {R}^2), \end{aligned}\) where the symbol a is assumed belonging to the Hörmander class \(S^0_{1,\delta }(\mathbb {R}\times \mathbb {R}^{3})\) ( \(0\le \delta <1\) ). The study of such specific structure of the symbol a is motivated by the work of Kovač V.: Boundedness of the twisted paraproduct. Rev. Mat. Iberoam. 28(4), 1143–1164 (2012), and Bernicot, F., Durcik, P.: Boundedness of somemulti-parameter fiber-wisemultiplier operators. Indiana 554 Univ. Math. J. 71(5), 2063–2098 (2022), where certain twisted and fiber-wise Fourier multipliers are investigated. Those Fourier multipliers may contain symbols acting on twisted fibers of two-dimensional functions, in contrast to the classical product-type bi-parameter structure. The above operator can be considered as the pseudo-differential analogue of a specific fiber-wise multiplier in Bernicot, F., Durcik, P.: Boundedness of somemulti-parameter fiber-wisemultiplier operators. Indiana 554 Univ. Math. J. 71(5), 2063–2098 (2022), and we will establish its strong and weak \(L^p\) estimates.
