A remark on the relationship $1/p = 1/p_1 + 1/p_2$ for boundedness of bilinear pseudo-differential operators with exotic symbols

Abstract

We consider the bilinear pseudo-differential operators with symbols in the bilinear H\"ormander classes $BS_{\rho, \rho}^m$, $0 < \rho < 1$. In this paper, we show that the condition $1/p = 1/p_1 + 1/p_2$ is necessary when we consider the boudnedness from $H^{p_1} \times H^{p_2}$ to $L^p$ of those operators for the critical case.