Spectral Multipliers II: Elliptic and Parabolic Operators and Bochner-Riesz Means
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| Format: | Preprint |
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2023
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| _version_ | 1866916704175521792 |
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| author | Beceanu, Marius Goldberg, Michael |
| author_facet | Beceanu, Marius Goldberg, Michael |
| contents | We establish estimates for the Poisson kernel, the heat kernel, and Bochner--Riesz means defined in terms of $H=-Δ+V$, where $V$ is a possibly large rough real-valued scalar potential and $H$ can have negative eigenvalues. All results are in three space dimensions.
We eliminate several unnecessary conditions on $V$, leaving just $V \in \mathcal K_0$, meaning that $V$ is locally integrable and $(-Δ)^{-1}|V|$ is bounded.
For the spectral multiplier bounds, we assume that $H$ has no zero or positive energy bound states. For $V \in \mathcal K_0$, we prove that $H$ has at most a finite number of negative bound states. If in addition $V \in \dot W^{-1/4, 4/3}$, then by [GoSc] and [KoTa] there are no positive energy bound states. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_09606 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Spectral Multipliers II: Elliptic and Parabolic Operators and Bochner-Riesz Means Beceanu, Marius Goldberg, Michael Analysis of PDEs Spectral Theory 35J08, 35J10, 35J25, 35K10, 35K15, 37J11, 42B08, 42B15, 42B37, 47A25, 47D60 We establish estimates for the Poisson kernel, the heat kernel, and Bochner--Riesz means defined in terms of $H=-Δ+V$, where $V$ is a possibly large rough real-valued scalar potential and $H$ can have negative eigenvalues. All results are in three space dimensions. We eliminate several unnecessary conditions on $V$, leaving just $V \in \mathcal K_0$, meaning that $V$ is locally integrable and $(-Δ)^{-1}|V|$ is bounded. For the spectral multiplier bounds, we assume that $H$ has no zero or positive energy bound states. For $V \in \mathcal K_0$, we prove that $H$ has at most a finite number of negative bound states. If in addition $V \in \dot W^{-1/4, 4/3}$, then by [GoSc] and [KoTa] there are no positive energy bound states. |
| title | Spectral Multipliers II: Elliptic and Parabolic Operators and Bochner-Riesz Means |
| topic | Analysis of PDEs Spectral Theory 35J08, 35J10, 35J25, 35K10, 35K15, 37J11, 42B08, 42B15, 42B37, 47A25, 47D60 |
| url | https://arxiv.org/abs/2308.09606 |