Bilinear Bochner-Riesz Means for Grushin Operators

This paper is devoted to the study of $L^{p_1}\times L^{p_2}$ to $L^p$ boundedness of the bilinear Bochner-Riesz mean ${\mathcal{B}}^{\alpha }$ associated with the Grushin operator $\mathcal{L}=-{\mathrm{\Delta }}_{x^{\mathrm{\prime }}}-\mathrm{\mid }{x}^{\mathrm{\prime }}{\mathrm{\mid }}^2{\mathrm{\Delta }}_{x^{\mathrm{\prime }\mathrm{\prime }}}$ on ${\mathbb{R}}^{d_1}\times {\mathbb{R}}^{d_2}$. Our result almost resembles the corresponding Euclidean results, where the Euclidean dimension in the smoothness threshold is replaced by the topological dimension $d$ of the underlying space, except at few cases.