Bochner-Riesz commutators for Grushin Operators

TL;DR

This paper investigates the boundedness and compactness of Bochner-Riesz commutators associated with Grushin operators on certain Lebesgue spaces, extending understanding of their behavior in harmonic analysis.

Contribution

It establishes boundedness and compactness criteria for these commutators on Lebesgue spaces, considering functions in BMO and CMO spaces, for the first time in this context.

Findings

Boundedness of commutators on L^q for specific p and alpha ranges.

Compactness of commutators when b is in CMO^{ ho}.

Extension of results to Grushin operators and associated Bochner-Riesz operators.

Abstract

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.