Bilinear Pseudo-Differential Operators with Exotic Class Symbols of Limited Smoothness

Abstract

We consider the bilinear pseudo-differential operators with symbols in the bilinear Hörmander class \(BS_{\rho , \rho }^m ({\mathbb {R}}^n) \) for \(m \in {\mathbb {R}}\) and \(0 \le \rho < 1\). The aim of this paper is to discuss smoothness conditions for symbols to assure the boundedness from \(h^p \times h^q\) to \(h^r\) for \(1 \le r \le 2 \le p,q \le \infty \) satisfying \(1/p+1/q=1/r\), where \(h^\infty \) is replaced by the space bmo.