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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2406.03076 |
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| _version_ | 1866913482057711616 |
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| author | Ye, Xiaofeng Zhang, Chunjie Zhu, Xiangrong |
| author_facet | Ye, Xiaofeng Zhang, Chunjie Zhu, Xiangrong |
| contents | In this note, we consider a Fourier integral operator defined by \begin{align*} T_{ϕ,a}f(x) = \int_{\mathbb{R}^{n}}e^{iϕ(x,ξ)}a(x,ξ)\widehat{f} ξ)dξ, \end{align*}here $a$ is the amplitude, and $ϕ$ is the phase. Let $0\leqρ\leq 1,n\geq 2$ or $0\leqρ<1,n=1$ and $$m_p=\frac{ρ-n}{p}+(n-1)\min\{\frac 12,ρ\}.$$ If $a$ belongs to the forbidden Hörmander class $S^{m_p}_{ρ,1}$ and $ϕ\in Φ^{2}$ satisfies the strong non-degeneracy condition, then for any $\frac {n}{n+1}<p\leq 1$, we can show that the Fourier integral operator $T_{ϕ,a}$ is bounded from the local Hardy space $h^p$ to $L^p$. Furthermore, if $a$ has compact support in variable $x$, then we can extend this result to $0<p\leq 1$. As $S^{m_p}_{ρ,δ}\subset S^{m_p}_{ρ,1}$ for any $0\leq δ\leq 1$, our result supplements and improves upon recent theorems proved by Staubach and his collaborators for $a\in S^{m}_{ρ,δ}$ when $δ$ is close to 1. As an important special case, when $n\geq 2$, we show that $T_{ϕ,a}$ is bounded from $H^1$ to $L^1$ if $a\in S^{(1-n)/2}_{1,1}$ which is a generalization of the well-known Seeger-Sogge-Stein theorem for $a\in S^{(1-n)/2}_{1,0}$. This result is false when $n=1$ and $a\in S^{0}_{1,1}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_03076 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fourier integral operators on Hardy spaces with Hormander class Ye, Xiaofeng Zhang, Chunjie Zhu, Xiangrong Differential Geometry 42B20, 35S30 In this note, we consider a Fourier integral operator defined by \begin{align*} T_{ϕ,a}f(x) = \int_{\mathbb{R}^{n}}e^{iϕ(x,ξ)}a(x,ξ)\widehat{f} ξ)dξ, \end{align*}here $a$ is the amplitude, and $ϕ$ is the phase. Let $0\leqρ\leq 1,n\geq 2$ or $0\leqρ<1,n=1$ and $$m_p=\frac{ρ-n}{p}+(n-1)\min\{\frac 12,ρ\}.$$ If $a$ belongs to the forbidden Hörmander class $S^{m_p}_{ρ,1}$ and $ϕ\in Φ^{2}$ satisfies the strong non-degeneracy condition, then for any $\frac {n}{n+1}<p\leq 1$, we can show that the Fourier integral operator $T_{ϕ,a}$ is bounded from the local Hardy space $h^p$ to $L^p$. Furthermore, if $a$ has compact support in variable $x$, then we can extend this result to $0<p\leq 1$. As $S^{m_p}_{ρ,δ}\subset S^{m_p}_{ρ,1}$ for any $0\leq δ\leq 1$, our result supplements and improves upon recent theorems proved by Staubach and his collaborators for $a\in S^{m}_{ρ,δ}$ when $δ$ is close to 1. As an important special case, when $n\geq 2$, we show that $T_{ϕ,a}$ is bounded from $H^1$ to $L^1$ if $a\in S^{(1-n)/2}_{1,1}$ which is a generalization of the well-known Seeger-Sogge-Stein theorem for $a\in S^{(1-n)/2}_{1,0}$. This result is false when $n=1$ and $a\in S^{0}_{1,1}$. |
| title | Fourier integral operators on Hardy spaces with Hormander class |
| topic | Differential Geometry 42B20, 35S30 |
| url | https://arxiv.org/abs/2406.03076 |