Bochner-Riesz commutators for Grushin Operators
Md Nurul Molla, Joydwip Singh
TL;DR
The paper studies Bochner-Riesz commutators for the Grushin operator $\mathcal{L}$ on $\mathbb{R}^d$, establishing $L^q$-boundedness of $[b, S^{\alpha}(\mathcal{L})]$ under a $BMO^{\varrho}$ condition on $b$ for a sharp range of $p$ and a dimension-reducing threshold $\alpha> d(1/p-1/2)-1/2$, with $p<q<p'$. It further shows compactness of the commutator when $b\in CMO^{\varrho}$ in the same range. A key feature is the reduction from the homogeneous dimension $Q=d_1+2d_2$ to the topological dimension $d=d_1+d_2$, achieved by employing truncated restriction-type estimates, weighted Plancherel bounds, and a dyadic spectral decomposition adapted to the Grushin geometry. The analysis combines a dyadic decomposition of the multiplier, a careful separation into main and error parts, and a Kolmogorov–Riesz framework to handle compactness. These results advance the understanding of spectral multipliers and commutators in sub-elliptic settings and provide dimension-accurate bounds that are sharp within the Grushin framework.
Abstract
In this paper, we study the boundedness of Bochner-Riesz commutator $$[b, S^α(\mathcal{L})](f) = b S^α(\mathcal{L})(f) - S^α(\mathcal{L})(bf)$$ of a $BMO^{\varrho}(\mathbb{R}^d)$ function $b$ and the Bochner-Riesz operator $S^α(\mathcal{L})$ 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 $α> d(1/p - 1/2) - 1/2$, if $b \in BMO^{\varrho}(\mathbb{R}^d)$, then $[b, S^α(\mathcal{L})]$ 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 $[b, S^α(\mathcal{L})]$ is a compact operator on $L^q(\mathbb{R}^d)$ in the same range.