Bilinear pseudodifferential operators with forbidden symbols on Lipschitz and Besov spaces. (Q1406521)

The author considers bilinear pseudodifferential operators of the form \[ T_\sigma(f,g)(x)= \int_{{\mathbb R}^n} \int_{{\mathbb R}^n} \sigma(x,\xi,\eta) \widehat f(\xi)\widehat g(\eta)\,e^{ix\cdot(\xi+\eta)}\,d\xi\,d\eta \] with (Friedrichs-Kumano-go double) symbols in the class \(BS_{\rho,\delta}^m\) satisfying the inequalities \[ | \partial_x^\alpha \partial_\xi^\beta\partial_\eta^\gamma \sigma(x,\xi,\eta)| \leq C_{\alpha\beta\gamma} (1+| \xi| +| \eta| )^{m+\delta| \sigma| -\rho(| \beta| +| \gamma| )} \] for all \((x,\xi,\eta)\in{\mathbb R}^{3n}\). It is shown that \(T_\sigma\) with symbols in the (so-called exotic or forbidden) class \(BS_{1,1}^0\) are bounded on products of Lipschitz and Besov spaces. The proof relies on decomposition techniques related to Littlewood-Paley theory and Coifman-Mayer time-frequency analysis, thus reducing the problem to the study of bilinear pseudodifferential operators with elementary double symbols.