For $f,g \in \mathscr{ S}(\R^n), n\geq 3$, consider the bilinear cone multiplier operator defined by \[{ T}^{ \lambda}_{ R}(f,g)(x):=\int_{ \mathbb{ R}^{ 2n}}m^{ \lambda}\left(\frac{ \xi'}{ R\xi_n},\frac{ \eta'}{ R\eta_n}\right)\hat{ f}(\xi)\hat{ g}(\eta)e^{ 2\pi\iota x\cdot(\xi+\eta)}~d\xi d\eta,\] where $\lambda>0, R>0$ and \[m^{ \lambda}\left(\frac{ \xi'}{ R\xi_n},\frac{ \eta'}{ R\eta_n}\right)=\Big(1-\frac{ |\xi'|^2}{ R^2\xi^2_n}-\frac{ |\eta'|^2}{ R^2\eta^2_n}\Big)^{ \lambda}_{ +}\varphi(\xi_n)\varphi(\eta_n),\] $(\xi',\xi_n), (\eta',\eta_n)\in\mathbb{ R}^{ n-1}\times \mathbb{ R}$ and $\varphi\in C_{ c}^{ \infty}([\frac{ 1}{ 2},2])$. We investigate the problem of pointwise almost everywhere convergence of ${ T}^{ \lambda}_{ R}(f,g)(x)$ 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 H\"{ o}lder 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 \[{ T}^{ \lambda}_{ *}(f,g)(x):=\sup_{ R>0}|{ T}^{ \lambda}_{ R}(f,g)(x)|.\]
Categories: math.CA