Notes on the spaces of bilinear multipliers

Abstract

A locally integrable function $m(ξ,η)$ 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 $$ B_m(f,g)(x)=\int_{\mathbb R^n} \int_{\mathbb R^n}\hat f(ξ)\hat g(η)m(ξ,η)e^{2πi(<ξ+η,x>} dξdη$$ 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 $m(ξ,η)= M(ξ-η)$ for a given $M$ defined on $\mathbb R^n$ is also addressed.