On the bilinear cone multiplier

Abstract

For $f,g \in \mathscr{S}(\R^n), n\geq 3$, 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 $λ>0, R>0$ and \[m^λ\left(\frac{ξ'}{Rξ_n},\frac{η'}{Rη_n}\right)=\Big(1-\frac{|ξ'|^2}{R^2ξ^2_n}-\frac{|η'|^2}{R^2η^2_n}\Big)^λ_{+}φ(ξ_n)φ(η_n),\] $(ξ',ξ_n), (η',η_n)\in\mathbb{R}^{n-1}\times \mathbb{R}$ and $φ\in C_{c}^{\infty}([\frac{1}{2},2])$. We investigate the problem of pointwise almost everywhere convergence of ${T}^λ_{R}(f,g)(x)$ as $R\rightarrow \infty$ 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 $L^{2}\times L^{2}\rightarrow L^{1}$--estimates of the maximal bilinear cone multiplier operator \[{T}^λ_{*}(f,g)(x):=\sup_{R>0}|{T}^λ_{R}(f,g)(x)|.\]