Notes on the spaces of bilinear multipliers

A locally integrable function $m(\xi,\eta)$ defined on $\mathbb R^n\times \mathbb R^n$ is said to be a bilinear multiplier on $\mathbb R^n$ of type $(p_1,p_2, p_3)$ if $$B_m(f,g)(x)=\int_{\mathbb R^n} \int_{\mathbb R^n}\hat f(\xi)\hat g(\eta)m(\xi,\eta)e^{2\pi i(<\xi +\eta,x>} d\xi d\eta$$ defines a bounded bilinear operator from $L^{p_1}(\mathbb R^n)\times L^{p_2}(\mathbb R^n)$ to $L^{p_3}(\mathbb R^n)$. The study of the basic properties of such spaces is investigated and several methods of constructing examples of bilinear multipliers are provided. The special case where $m(\xi,\eta)= M(\xi-\eta)$ for a given $M$ defined on $\mathbb R^n$ is also addressed.