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 $\mathcal{L}=-{\Delta }_{x^{\prime }}-|x^{\prime }{|}^2{\Delta }_{x^{\prime \prime }}$ 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.

arXiv comment44 pages, 2 figures, Few changes made on section 2 and 11