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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2303.17721 |
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| _version_ | 1866909323533221888 |
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| author | Sharma, Himani Sikora, Adam |
| author_facet | Sharma, Himani Sikora, Adam |
| contents | We consider the setting of manifolds with ends which are obtained by compact perturbation (gluing) of ends of the form $\mathbb{R}^{n_i}\times \mathcal{M}_i$. We investigate family of vertical resolvent $\{\sqrt{t}\nabla(1+tΔ)^{-m}\}_{t>0}$ where $m\geq1$. We show that the family is uniformly continuous on all $L^p$ for $1\le p \le \min_{i}n_i$. Interestingly this is a closed-end condition in the considered setting. We prove that the corresponding Maximal function is bounded in the same range except that it is only weak-type $(1,1)$ for $p=1$. The Fefferman-Stein vector-valued maximal function is again of weak-type $(1,1)$ but bounded if and only if $1<p<\min_{i}n_i$, and not at $p=\min_{i}n_i$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_17721 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Vertical Maximal Functions on Manifolds with Ends Sharma, Himani Sikora, Adam Analysis of PDEs 42B25, 42B37 We consider the setting of manifolds with ends which are obtained by compact perturbation (gluing) of ends of the form $\mathbb{R}^{n_i}\times \mathcal{M}_i$. We investigate family of vertical resolvent $\{\sqrt{t}\nabla(1+tΔ)^{-m}\}_{t>0}$ where $m\geq1$. We show that the family is uniformly continuous on all $L^p$ for $1\le p \le \min_{i}n_i$. Interestingly this is a closed-end condition in the considered setting. We prove that the corresponding Maximal function is bounded in the same range except that it is only weak-type $(1,1)$ for $p=1$. The Fefferman-Stein vector-valued maximal function is again of weak-type $(1,1)$ but bounded if and only if $1<p<\min_{i}n_i$, and not at $p=\min_{i}n_i$. |
| title | Vertical Maximal Functions on Manifolds with Ends |
| topic | Analysis of PDEs 42B25, 42B37 |
| url | https://arxiv.org/abs/2303.17721 |