Lp-Boundedness of a Class of Bi-Parameter Pseudo-Differential Operators

In this paper, I 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 Hormander class 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, I demonstrate that for each multi-index alpha,beta, the inequality vertical bar partial derivative(alpha)(xi)partial derivative(beta)(x)sigma(x(1),x(2),xi(1),xi(2))vertical bar <= C-alpha,C-beta(1+vertical bar xi vertical bar)(m)Pi(2)(i=1)(1+vertical bar xi(i)vertical bar)(-rho|alpha i vertical bar delta|beta i vertical bar) is satisfied. My 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 Hormander class.