On Multi-linear Maximal Operators Along Homogeneous Curves
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866914514053627904 |
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| author | Becker, Lars Krause, Ben |
| author_facet | Becker, Lars Krause, Ben |
| contents | Suppose that \[ \vecγ(t) := (γ_1(t),\dots,γ_n(t)) = (a_1 t^{d_1},\dots,a_n t^{d_n}), \; \; \; 1\leq d_1 < \dots < d_n, \ a_i \neq 0\] is a homogeneous polynomial curve. We prove that whenever $p_1,\dots,p_n > 1$ and $\frac{1}{p} = \sum_{j=1}^n \frac{1}{p_j} \leq 1$, there exists an absolute constant $0 < C = C_{p_1,\dots,p_n;\vecγ} < \infty$ so that \[ \| \sup_{r > 0} \ \frac{1}{r} \int_{0}^r \prod_{i=1}^n |f_i(x-γ_i(t))| \ dt \|_{L^p(\mathbb{R})} \leq C \cdot \prod_{i=1}^n \| f_j \|_{L^{p_j}(\mathbb{R})}. \] Our main tool is a smoothing estimate, adapted from work of Kosz-Mirek-Peluse-Wright. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_09080 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Multi-linear Maximal Operators Along Homogeneous Curves Becker, Lars Krause, Ben Classical Analysis and ODEs Suppose that \[ \vecγ(t) := (γ_1(t),\dots,γ_n(t)) = (a_1 t^{d_1},\dots,a_n t^{d_n}), \; \; \; 1\leq d_1 < \dots < d_n, \ a_i \neq 0\] is a homogeneous polynomial curve. We prove that whenever $p_1,\dots,p_n > 1$ and $\frac{1}{p} = \sum_{j=1}^n \frac{1}{p_j} \leq 1$, there exists an absolute constant $0 < C = C_{p_1,\dots,p_n;\vecγ} < \infty$ so that \[ \| \sup_{r > 0} \ \frac{1}{r} \int_{0}^r \prod_{i=1}^n |f_i(x-γ_i(t))| \ dt \|_{L^p(\mathbb{R})} \leq C \cdot \prod_{i=1}^n \| f_j \|_{L^{p_j}(\mathbb{R})}. \] Our main tool is a smoothing estimate, adapted from work of Kosz-Mirek-Peluse-Wright. |
| title | On Multi-linear Maximal Operators Along Homogeneous Curves |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2508.09080 |