Bi-parameter and bilinear Calderón–Vaillancourt theorem with critical order

Abstract In this paper, we establish the sharp Calderón–Vaillancourt theorem on L p L^{p} spaces for bi-parameter and bilinear pseudo-differential operators with symbols of critical order by deriving a sufficient and necessary condition on its symbol. This sharpens the result of [G. Lu and L. Zhang, Bi-parameter and bilinear Calderón–Vaillancourt theorem with subcritical order, Forum Math. 28 2016, 6, 1087–1094] which was only proved for symbols of subcritical order.