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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2408.02180 |
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| _version_ | 1866909883578712064 |
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| author | Chen, Peng Shen, Minxing Wang, Yunxiang Yan, Lixin |
| author_facet | Chen, Peng Shen, Minxing Wang, Yunxiang Yan, Lixin |
| contents | In this article we investigate $L^p$ boundedness of the spherical maximal operator $\mathfrak{m}^α$ of (complex) order $α$ on the $n$-dimensional hyperbolic space $\mathbb{H}^n$, which was introduced and studied by El Kohen. We prove that when $n\geq 2$, for $α\in\mathbb{R}$ and $1<p<\infty$, if $\mathfrak{m}^α$ is bounded on $L^p(\mathbb{H}^n)$, then we must have $α>1-n+n/p$ for $1<p\leq 2$; or $α\geq \max\{1/p-(n-1)/2,(1-n)/p\}$ for $2<p<\infty$. Furthermore, we improve El Kohen's result [J. Operator Theory 3 (1980)] on $L^p$ boundedness of $\mathfrak{m}^α$ by showing that $\mathfrak{m}^α$ is bounded on $L^p(\mathbb{H}^n)$ provided that $\mathop{\mathrm{Re}}α> \max \{{(2-n)/p}-{1/(p p_n)},{(2-n)/p}- (p-2)/[p p_n(p_n-2)]\} $ for $2\leq p\leq \infty$, with $p_n=2(n+1)/(n-1)$ for $n\geq 3$ and $p_n=4$ for $n=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_02180 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Spherical Maximal Operators on Hyperbolic Spaces Chen, Peng Shen, Minxing Wang, Yunxiang Yan, Lixin Functional Analysis 42B25, 22E30, 35S30, 43A90 In this article we investigate $L^p$ boundedness of the spherical maximal operator $\mathfrak{m}^α$ of (complex) order $α$ on the $n$-dimensional hyperbolic space $\mathbb{H}^n$, which was introduced and studied by El Kohen. We prove that when $n\geq 2$, for $α\in\mathbb{R}$ and $1<p<\infty$, if $\mathfrak{m}^α$ is bounded on $L^p(\mathbb{H}^n)$, then we must have $α>1-n+n/p$ for $1<p\leq 2$; or $α\geq \max\{1/p-(n-1)/2,(1-n)/p\}$ for $2<p<\infty$. Furthermore, we improve El Kohen's result [J. Operator Theory 3 (1980)] on $L^p$ boundedness of $\mathfrak{m}^α$ by showing that $\mathfrak{m}^α$ is bounded on $L^p(\mathbb{H}^n)$ provided that $\mathop{\mathrm{Re}}α> \max \{{(2-n)/p}-{1/(p p_n)},{(2-n)/p}- (p-2)/[p p_n(p_n-2)]\} $ for $2\leq p\leq \infty$, with $p_n=2(n+1)/(n-1)$ for $n\geq 3$ and $p_n=4$ for $n=2$. |
| title | The Spherical Maximal Operators on Hyperbolic Spaces |
| topic | Functional Analysis 42B25, 22E30, 35S30, 43A90 |
| url | https://arxiv.org/abs/2408.02180 |