Endpoint eigenfunction bounds for the Hermite operator
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866910282659397632 |
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| author | Jeong, Eunhee Lee, Sanghyuk Ryu, Jaehyeon |
| author_facet | Jeong, Eunhee Lee, Sanghyuk Ryu, Jaehyeon |
| contents | We establish the optimal $L^p$, $p=2(d+3)/(d+1),$ eigenfunction bound for the Hermite operator $\mathcal H=-Δ+|x|^2$ on $\mathbb R^d$. Let $Π_λ$ denote the projection operator to the vector space spanned by the eigenfunctions of $\mathcal H$ with eigenvalue $λ$. The optimal $L^2$--$L^p$ bounds on $Π_λ$, $2\le p\le \infty$, have been known by the works of Karadzhov and Koch-Tataru except $p=2(d+3)/(d+1)$. For $d\ge 3$, we prove the optimal bound for the missing endpoint case. Our result is built on a new phenomenon: improvement of the bound due to asymmetric localization near the sphere $\sqrtλ\mathbb S^{d-1}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_03036 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Endpoint eigenfunction bounds for the Hermite operator Jeong, Eunhee Lee, Sanghyuk Ryu, Jaehyeon Classical Analysis and ODEs Analysis of PDEs 42B99 (primary), 42C10 (secondary) We establish the optimal $L^p$, $p=2(d+3)/(d+1),$ eigenfunction bound for the Hermite operator $\mathcal H=-Δ+|x|^2$ on $\mathbb R^d$. Let $Π_λ$ denote the projection operator to the vector space spanned by the eigenfunctions of $\mathcal H$ with eigenvalue $λ$. The optimal $L^2$--$L^p$ bounds on $Π_λ$, $2\le p\le \infty$, have been known by the works of Karadzhov and Koch-Tataru except $p=2(d+3)/(d+1)$. For $d\ge 3$, we prove the optimal bound for the missing endpoint case. Our result is built on a new phenomenon: improvement of the bound due to asymmetric localization near the sphere $\sqrtλ\mathbb S^{d-1}$. |
| title | Endpoint eigenfunction bounds for the Hermite operator |
| topic | Classical Analysis and ODEs Analysis of PDEs 42B99 (primary), 42C10 (secondary) |
| url | https://arxiv.org/abs/2205.03036 |