On the bilinear cone multiplier
Saurabh Shrivastava, Kalachand Shuin
TL;DR
This work analyzes the pointwise convergence of the bilinear cone multiplier ${ m T}^{\,oldsymbol{\lambda}}_{R}$ as $R o\infty$ for functions in $L^{p_1} imes L^{p_2}$, establishing convergence to the product $f g$ under explicit threshold conditions on $oldsymbol{\lambda}$ governed by the exponents $p_1,p_2,p$. The authors reduce the problem to proving weighted $L^{2} imes L^{2} o L^{1}$ estimates for the maximal operator ${ m T}^{oldsymbol{\lambda}}_{*}$ via a detailed multiplier decomposition and a bilinear, Stein–Weiss–type representation, building on square-function and maximal-function techniques from CHLY and related works. Key contributions include a robust decomposition of the bilinear cone multiplier into $j eq 1$ and $j=1$ components, precise weighted $L^{2}$ bounds for associated square-functions and maximal operators, and a direct appendix confirming the essential $L^{2} imes L^{2} o L^{1}$ boundedness. The results extend pointwise convergence theory to bilinear cone multipliers and illustrate how weighted bilinear estimates underpin almost everywhere convergence in higher dimensions. These methods advance the toolkit for bilinear multiplier problems and connect cone multiplier theory with bilinear Bochner–Riesz frameworks.
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 \[m^λ\left(\frac{ξ'}{Rξ_n},\frac{η'}{Rη_n}\right)=\Big(1-\frac{|ξ'|^2}{R^2ξ^2_n}-\frac{|η'|^2}{R^2η^2_n}\Big)^λ_{+}\varphi(ξ_n)\varphi(η_n),\] $(ξ',ξ_n), (η',η_n)\in\mathbb{R}^{n-1}\times \mathbb{R}$ and $\varphi\in C_{c}^{\infty}([\frac{1}{2},2])$. 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 \[{T}^λ_{*}(f,g)(x):=\sup_{R>0}|{T}^λ_{R}(f,g)(x)|.\]