The boundedness of multi-linear and multi-parameter pseudo-differential operators (Q2658872)

The authors prove \(L^p\) estimates for bilinear pseudo-differential operators associated with certain bi-parameter symbols. This extends previous work by \textit{W. Dai} and \textit{G. Lu} [Bull. Soc. Math. Fr. 143, No. 3, 567--597 (2015; Zbl 1335.47033)], which in turn is a generalization of well-known work by \textit{C. Muscalu} et al. [Acta Math. 193, No. 2, 269--296 (2004; Zbl 1087.42016)] on multilinear Fourier multipliers with multi-parameter symbols. The main result of the paper reads as follows. Let \(\sigma\) be a smooth function on \((\mathbb{R}^{n_1}\times \mathbb{R}^{n_2})^3\) satisfying the condition \[ |\partial^{\alpha}_x \partial^{\beta}_{\xi} \partial^{\gamma}_{\eta} \sigma(x,\xi,\eta)|\lesssim \prod_{i=1}^2 (1+|\xi_i|+|\eta_i|)^{-|\beta_i|-|\gamma_i|+\delta_i |\alpha_i|} \] with \(x=(x_1,x_2)\), \(\xi=(\xi_1,\xi_2)\), \(\eta=(\eta_1,\eta_2)\in \mathbb{R}^{n_1+n_2}\) and \(\delta_1,\delta_2\in [0,1)\). Then the bilinear operator \[ T_\sigma (f,g) (x) = \int_{(\mathbb{R}^{n_1+n_2})^2} \sigma(x,\xi,\eta) \widehat{f}(\xi)\widehat{g}(\eta) e^{2\pi i x\cdot(\xi+\eta)}\,d\xi\,d\eta \] maps \(L^p(\mathbb{R}^{n_1+n_2})\times L^q(\mathbb{R}^{n_1+n_2})\to L^r(\mathbb{R}^{n_1+n_2})\) for \(p,q\in (1,\infty)\), \(r\in (0,\infty)\) with \(\frac1p+\frac1q=\frac1r\). The case \(n_1=n_2=1\), \(\delta_1=\delta_2=0\) was studied by Dai and Lu [loc. cit.].