$L^p$-Boundedness of a Class of Bi-Parameter Pseudo-Differential Operators

In this paper, we explore a specific class of bi-parameter pseudo-differential operators characterized by symbols $\sigma (x_1,x_2,{\xi }_1,{\xi }_2)$ falling within the product-type Hörmander {class} ${\mathbf{S}}_{\rho ,\delta }^m$. This classification imposes constraints on the behavior of partial derivatives of $\sigma$ with respect to both spatial and frequency variables. Specifically, we demonstrate that for each multi-index $\alpha ,\beta$, the inequality $\mathrm{\mid }{\mathrm{\partial }}_{\xi }^{\alpha }{\mathrm{\partial }}_x^{\beta }\sigma (x_1,x_2,{\xi }_1,{\xi }_2)\mathrm{\mid }\le C_{\alpha ,\beta }(1+\mathrm{\mid }\xi \mathrm{\mid }{)}^m{\prod }_{i=1}^2(1+\mathrm{\mid }{\xi }_i\mathrm{\mid }{)}^{-\rho \mathrm{\mid }{\alpha }_i\mathrm{\mid }+\delta \mathrm{\mid }{\beta }_i\mathrm{\mid }}$ is satisfied. Our investigation culminates in a rigorous analysis of the $L^p$-boundedness of such pseudo-differential operators, thereby extending the seminal findings of C. Fefferman from 1973 concerning pseudo-differential operators within the Hörmander class.