Limited smoothness conditions with mixed norms for bilinear Fourier multipliers

Abstract

In this paper, the $$L^2 \times L^{\infty } \rightarrow L^2$$ and $$L^2 \times L^2 \rightarrow L^1$$ estimates of bilinear Fourier multiplier operators are discussed under weak smoothness conditions on multipliers. As an application, we prove the $$L^2 \times BMO \rightarrow L^2$$ and $$L^2 \times L^2 \rightarrow H^1$$ boundedness of bilinear operators with multipliers of limited smoothness satisfying vanishing conditions.