Notes on the spaces of bilinear multipliers

Abstract

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.

13 pages, Notes on the course given in La Falda (Argentina)2008