Bochner-Riesz commutators for Grushin Operators
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908374457647104 |
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| author | Molla, Md Nurul Singh, Joydwip |
| author_facet | Molla, Md Nurul Singh, Joydwip |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_16593 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bochner-Riesz commutators for Grushin Operators Molla, Md Nurul Singh, Joydwip Analysis of PDEs 42B15, 43A85 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. |
| title | Bochner-Riesz commutators for Grushin Operators |
| topic | Analysis of PDEs 42B15, 43A85 |
| url | https://arxiv.org/abs/2505.16593 |