Bochner-Riesz commutators for Grushin Operators

In this paper, we study the boundedness of Bochner-Riesz commutator $$[b,S^{\alpha }(\mathcal{L})](f)=b{S}^{\alpha }(\mathcal{L})(f)-S^{\alpha }(\mathcal{L})(bf)$$ of a $BM{O}^{\varrho }({\mathbb{R}}^d)$ function $b$ and the Bochner-Riesz operator $S^{\alpha }(\mathcal{L})$ associated to the Grushin operator $\mathcal{L}$ on ${\mathbb{R}}^d$ with $d:=d_1+d_2$. We prove that for $1\le p\le \min \{2d_1\mathrm{/}(d_1+2),2(d_2+1)\mathrm{/}(d_2+3)\}$ and $\alpha >d(1\mathrm{/}p-1\mathrm{/}2)-1\mathrm{/}2$, if $b\in BM{O}^{\varrho }({\mathbb{R}}^d)$, then $[b,S^{\alpha }(\mathcal{L})]$ is bounded on $L^q({\mathbb{R}}^d)$ whenever $p<q<p^{\mathrm{\prime }}$. Moreover, if $b\in CM{O}^{\varrho }({\mathbb{R}}^d)$, then we show that $[b,S^{\alpha }(\mathcal{L})]$ is a compact operator on $L^q({\mathbb{R}}^d)$ in the same range.