Bochner-Riesz means for critical magnetic Schrödinger operators in ${\mathbb R^2}$
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909189222170624 |
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| author | Miao, Changxing Yan, Lixin Zhang, Junyong |
| author_facet | Miao, Changxing Yan, Lixin Zhang, Junyong |
| contents | We study $L^p$-boundedness of the Bochner-Riesz means for critical magnetic Schrödinger operators $\mathcal{L}_{\bf A}$ in ${\mathbb{R}^2}$, which involve the physcial Aharonov-Bohm potential. We show that for $1\leq p\leq +\infty$ and $p\neq 2$, the Bochner-Riesz operator $S_λ^δ(\mathcal{L}_{\bf A})$ of order $δ$ is bounded on $L^p(\mathbb{R}^2)$ if and only if $δ>\max\big\{0, 2\big|1/2-1/p\big|-1/2\big\}$. The new ingredient of the proof is to obtain the localized $L^4(\mathbb R^2)$ estimate of $S_λ^δ(\mathcal{L}_{\bf A})$, whose kernel is heavily affected by the physical magnetic diffraction, and more singular than the classical Bochner-Riesz means $S_λ^δ(Δ)$ for the Laplacian $Δ$ in $\mathbb{R}^2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02531 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bochner-Riesz means for critical magnetic Schrödinger operators in ${\mathbb R^2}$ Miao, Changxing Yan, Lixin Zhang, Junyong Analysis of PDEs We study $L^p$-boundedness of the Bochner-Riesz means for critical magnetic Schrödinger operators $\mathcal{L}_{\bf A}$ in ${\mathbb{R}^2}$, which involve the physcial Aharonov-Bohm potential. We show that for $1\leq p\leq +\infty$ and $p\neq 2$, the Bochner-Riesz operator $S_λ^δ(\mathcal{L}_{\bf A})$ of order $δ$ is bounded on $L^p(\mathbb{R}^2)$ if and only if $δ>\max\big\{0, 2\big|1/2-1/p\big|-1/2\big\}$. The new ingredient of the proof is to obtain the localized $L^4(\mathbb R^2)$ estimate of $S_λ^δ(\mathcal{L}_{\bf A})$, whose kernel is heavily affected by the physical magnetic diffraction, and more singular than the classical Bochner-Riesz means $S_λ^δ(Δ)$ for the Laplacian $Δ$ in $\mathbb{R}^2$. |
| title | Bochner-Riesz means for critical magnetic Schrödinger operators in ${\mathbb R^2}$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2405.02531 |