Boundedness of bilinear pseudo-differential operators of $$S_{0,0}$$-type on $$L^2 \times L^2$$

Abstract

We extend the known result that the bilinear pseudo-differential operators with symbols in the bilinear Hörmander class $$BS^{-n/2}_{0,0}(\mathbb {R}^n)$$ are bounded from $$L^2 \times L^2$$ to $$h^1$$ . We show that those operators are also bounded from $$L^2 \times L^2$$ to $$L^r $$ for every $$1< r\le 2$$ . Moreover we give similar results for symbol classes wider than $$BS^{-n/2}_{0,0}(\mathbb {R}^n)$$ . We also give results for symbols of limited smoothness.