A maximal oscillatory operator on compact manifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909133377110016 |
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| author | Liu, Ziyao Chen, Jiecheng Fan, Dashan |
| author_facet | Liu, Ziyao Chen, Jiecheng Fan, Dashan |
| contents | This is a continuation of our previous research about an oscillatory integral operator $T_{α, β}$ on compact manifolds $\mathbb{M}$. We prove the sharp $H^{p}$-$L^{p,\infty}$ boundedness on the maximal operator $T^{*}_{α, β}$ for all $0<p<1$. As applications, we first prove the sharp $H^{p}$-$L^{p,\infty}$ boundedness on the maximal operator corresponding to the Riesz means $I_{k,α}(|\mathcal{L}|)$ associated with the Schrödinger type group $e^{is\mathcal{L}^{α/2}}$ and obtain the almost everywhere convergence of $I_{k,α}(|\mathcal{L}|)f(x,t)\to f(x)$ for all $f\in H^{p}$. Also, we are able to obtain the convergence speed of a combination operator from the solutions of the Cauchy problem of fractional Schrödinger equations. All results are even new on the n-torus $T^{n}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_04996 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A maximal oscillatory operator on compact manifolds Liu, Ziyao Chen, Jiecheng Fan, Dashan Analysis of PDEs Classical Analysis and ODEs This is a continuation of our previous research about an oscillatory integral operator $T_{α, β}$ on compact manifolds $\mathbb{M}$. We prove the sharp $H^{p}$-$L^{p,\infty}$ boundedness on the maximal operator $T^{*}_{α, β}$ for all $0<p<1$. As applications, we first prove the sharp $H^{p}$-$L^{p,\infty}$ boundedness on the maximal operator corresponding to the Riesz means $I_{k,α}(|\mathcal{L}|)$ associated with the Schrödinger type group $e^{is\mathcal{L}^{α/2}}$ and obtain the almost everywhere convergence of $I_{k,α}(|\mathcal{L}|)f(x,t)\to f(x)$ for all $f\in H^{p}$. Also, we are able to obtain the convergence speed of a combination operator from the solutions of the Cauchy problem of fractional Schrödinger equations. All results are even new on the n-torus $T^{n}$. |
| title | A maximal oscillatory operator on compact manifolds |
| topic | Analysis of PDEs Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2403.04996 |