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 $BMO^{ \varrho}(\mathbb{ R}^d)$ 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 $1\leq p \leq \min \{ 2d_1/(d_1 +2), 2(d_2 +1)/(d_2+3)\}$ and $\alpha > d(1/p - 1/2) - 1/2$, if $b \in BMO^{ \varrho}(\mathbb{ R}^d)$, then $[b, S^{ \alpha}(\mathcal{ L})]$ is bounded on $L^q(\mathbb{ R}^d)$ whenever $p < q < p'$. Moreover, if $b \in CMO^{ \varrho}(\mathbb{ R}^d)$, then we show that $[b, S^{ \alpha}(\mathcal{ L})]$ is a compact operator on $L^q(\mathbb{ R}^d)$ in the same range.
Categories: math.AP