On the bilinear cone multiplier

Abstract

For $f,g\in \mathcal{S}({\mathbb{R}}^n),n\ge 3$, consider the bilinear cone multiplier operator defined by \[{T}^λ_{R}(f,g)(x):=\int_{\mathbb{R}^{2n}}m^λ\left(\frac{ξ'}{Rξ_n},\frac{η'}{Rη_n}\right)\hat{f}(ξ)\hat{g}(η)e^{2πιx\cdot(ξ+η)}~dξdη,\] where $\hat{I}»>0,R>0$ and \[m^λ\left(\frac{ξ'}{Rξ_n},\frac{η'}{Rη_n}\right)=\Big(1-\frac{|ξ'|^2}{R^2ξ^2_n}-\frac{|η'|^2}{R^2η^2_n}\Big)^λ_{+}φ(ξ_n)φ(η_n),\] $(\hat{I}{¾}^{\prime },\hat{I}{¾}_n),(\hat{I}{\cdot }^{\prime },\hat{I}{\cdot }_n)\in {\mathbb{R}}^{n-1}\times \mathbb{R}$ and $\ddot{I}†\in C_c^{\infty }([\frac{1}{2},2])$. We investigate the problem of pointwise almost everywhere convergence of $T^{\hat{I}}{»}_R(f,g)(x)$ as $R\to \infty$ for $(f,g)\in L^{p_1}\times L^{p_2}$ for a wide range of exponents $p_1,p_2$ satisfying the Hölder relation $\frac{1}{p_1}+\frac{1}{p_2}=\frac{1}{p}$. This assertion is proved by establishing suitable weighted $L^2\times L^2\to L^1$--estimates of the maximal bilinear cone multiplier operator \[{T}^λ_{*}(f,g)(x):=\sup_{R>0}|{T}^λ_{R}(f,g)(x)|.\]

25 pages