Sharp maximal function estimates for Hilbert transforms along monomial curves in higher dimensions
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arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866911990926016512 |
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| author | Wan, Renhui |
| author_facet | Wan, Renhui |
| contents | For any nonempty set $U\subset\R^+$, we consider the maximal operator $\h^U$ defined as $\h^Uf=\sup_{u\in U}|H^{(u)} f|$, where $H^{(u)}$ represents the Hilbert transform along the monomial curve $uγ(s)$. We focus on the $L^p(\mathbb{R}^d)$ operator norm of $\h^U$ for $p\in (p_\circ(d),\infty)$, where $p_\circ(d)$ is the optimal exponent known for the $L^p$ boundedness of the maximal averaging operator obtained by Ko-Lee-Oh \cite{KLO22,KLO23} and Beltran-Guo-Hickman-Seeger \cite{BGHS}. To achieve this goal, we employ a novel bootstrapping argument to establish a maximal estimate for the Mihlin-Hörmander-type multiplier, along with utilizing the local smoothing estimate for the averaging operator and its vector-valued extension to obtain crucial decay estimates. Furthermore, our approach offers an alternative means for deriving the upper bound established in \cite{Guo20}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_09110 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Sharp maximal function estimates for Hilbert transforms along monomial curves in higher dimensions Wan, Renhui Classical Analysis and ODEs 42B25 For any nonempty set $U\subset\R^+$, we consider the maximal operator $\h^U$ defined as $\h^Uf=\sup_{u\in U}|H^{(u)} f|$, where $H^{(u)}$ represents the Hilbert transform along the monomial curve $uγ(s)$. We focus on the $L^p(\mathbb{R}^d)$ operator norm of $\h^U$ for $p\in (p_\circ(d),\infty)$, where $p_\circ(d)$ is the optimal exponent known for the $L^p$ boundedness of the maximal averaging operator obtained by Ko-Lee-Oh \cite{KLO22,KLO23} and Beltran-Guo-Hickman-Seeger \cite{BGHS}. To achieve this goal, we employ a novel bootstrapping argument to establish a maximal estimate for the Mihlin-Hörmander-type multiplier, along with utilizing the local smoothing estimate for the averaging operator and its vector-valued extension to obtain crucial decay estimates. Furthermore, our approach offers an alternative means for deriving the upper bound established in \cite{Guo20}. |
| title | Sharp maximal function estimates for Hilbert transforms along monomial curves in higher dimensions |
| topic | Classical Analysis and ODEs 42B25 |
| url | https://arxiv.org/abs/2305.09110 |