Continuity properties of multi-parameter pseudodifferential operators on Bony class

It is well-known that the pseudodifferential operator with the symbol in Bony class, a subset of \(S_{1,1}^0(\mathbb R^n)\) $S_{1,1}^0({\mathbb{R}}^n)$ , is bounded on \(L^{2}(\mathbb {R}^{n})\) $L^2({\mathbb{R}}^n)$ . The main purpose of this paper is to extend the classical results to multi-parameter case, i.e., to discuss the boundedness on \(L^2(\mathbb {R}^{n_1+n_2})\) $L^2(\mathbb{R}_{}^{n_1+n_2})$ and on \(h^{p}(\mathbb {R}^{n_{1}}\times \mathbb {R}^{n_{2}}) (0<p\le 1)\) $h^p({\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2})(0<p\le 1)$ of multi-parameter pseudodifferential operator with symbol satisfying multi-parameter Bony conditions.