Bilinear Bochner-Riesz Means for Grushin Operators

Abstract

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 $\mathcal{L} = -\Delta_{x'} - |x'|^2 \Delta_{x''}$ 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.