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

Abstract

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 H\"ormander {class} $\mathbf{S}^m_{\rho, \delta}$. 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 $| \partial_\xi^\alpha \partial_x^\beta \sigma(x_1,x_2,\xi_1,\xi_2)| \le C_{\alpha, \beta}(1+|\xi|)^m\prod_{i=1}^2 (1+|\xi_i|)^{-\rho|\alpha_i|+\delta|\beta_i|} $ 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 H\"ormander class.