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.

中文翻译：

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双线性傅立叶乘法器的具有混合范数的有限平滑条件

在本文中，双线性傅立叶乘子算子的\（L ^ 2 \ times L ^ {\ infty} \ rightarrow L ^ 2 \）和\（L ^ 2 \ times L ^ 2 \ rightarrow L ^ 1 \）的估计是在弱平滑条件下对乘数进行了讨论。作为一个应用，我们证明了双线性算子的\（L ^ 2 \ times BMO \ rightarrow L ^ 2 \）和\（L ^ 2 \ times L ^ 2 \ rightarrow H ^ 1 \）有界且有限光滑度的乘积满足消失的条件。