Bilinear Bochner-Riesz Means for Grushin Operators

TL;DR

This paper investigates the boundedness of bilinear Bochner-Riesz means related to Grushin operators, revealing results similar to Euclidean cases but with some exceptions, advancing understanding of harmonic analysis on these operators.

Contribution

It establishes boundedness results for bilinear Bochner-Riesz means associated with Grushin operators, extending Euclidean harmonic analysis results to this non-Euclidean setting.

Findings

Almost matches Euclidean results in smoothness thresholds

Replaces Euclidean dimension with topological dimension in results

Identifies exceptions where results differ from Euclidean case

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.