On the bilinear cone multiplier

For $f,g\in \mathcal{S}({\mathbb{R}}^n),n\ge 3$, consider the bilinear cone multiplier operator defined by $$T_R^{\lambda }(f,g)(x):={\int }_{{\mathbb{R}}^{2n}}{m}^{\lambda }(\frac{{\xi }^{\mathrm{\prime }}}{R{\xi }_n},\frac{{\eta }^{\mathrm{\prime }}}{R{\eta }_n})\widehat{f}(\xi )\widehat{g}(\eta )e^{2\pi \iota x\cdot (\xi +\eta )}d\xi d\eta ,$$ where $\lambda >0,R>0$ and $$m^{\lambda }(\frac{{\xi }^{\mathrm{\prime }}}{R{\xi }_n},\frac{{\eta }^{\mathrm{\prime }}}{R{\eta }_n})=(1-\frac{\mathrm{\mid }{\xi }^{\mathrm{\prime }}{\mathrm{\mid }}^2}{R^2{\xi }_n^2}-\frac{\mathrm{\mid }{\eta }^{\mathrm{\prime }}{\mathrm{\mid }}^2}{R^2{\eta }_n^2}{)}_{+}^{\lambda }\phi ({\xi }_n)\phi ({\eta }_n),$$ $({\xi }^{\mathrm{\prime }},{\xi }_n),({\eta }^{\mathrm{\prime }},{\eta }_n)\in {\mathbb{R}}^{n-1}\times \mathbb{R}$ and $\phi \in C_c^{\mathrm{\infty }}([\frac{1}{2},2])$. We investigate the problem of pointwise almost everywhere convergence of $T_R^{\lambda }(f,g)(x)$ as $R\to \mathrm{\infty }$ 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 $$T_{\ast }^{\lambda }(f,g)(x):=\underset{R>0}{\sup }\mathrm{\mid }{T}_R^{\lambda }(f,g)(x)\mathrm{\mid }\mathrm{.}$$