Endpoint regularity of general Fourier integral operators
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929477518360576 |
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| author | Zhu, Xiangrong Li, Wenjuan |
| author_facet | Zhu, Xiangrong Li, Wenjuan |
| contents | Let $n\geq 1,0<ρ<1, \max\{ρ,1-ρ\}\leq δ\leq 1$ and $$m_1=ρ-n+(n-1)\min\{\frac 12,ρ\}+\frac {1-δ}{2}.$$ If the amplitude $a$ belongs to the Hörmander class $S^{m_1}_{ρ,δ}$ and $ϕ\in Φ^{2}$ satisfies the strong non-degeneracy condition, then we prove that the following Fourier integral operator $T_{ϕ,a}$ 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*} is bounded from the local Hardy space $h^1(\mathbb{R}^n)$ to $L^1(\mathbb{R}^n)$. As a corollary, we can also obtain the corresponding $L^p(\mathbb{R}^n)$-boundedness when $1<p<2$.
These theorems are rigorous improvements on the recent works of Staubach and his collaborators. When $0\leq ρ\leq 1,δ\leq \max\{ρ,1-ρ\}$, by using some similar techniques in this note, we can get the corresponding theorems which coincide with the known results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_15280 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Endpoint regularity of general Fourier integral operators Zhu, Xiangrong Li, Wenjuan Classical Analysis and ODEs 42B20, 35S30 Let $n\geq 1,0<ρ<1, \max\{ρ,1-ρ\}\leq δ\leq 1$ and $$m_1=ρ-n+(n-1)\min\{\frac 12,ρ\}+\frac {1-δ}{2}.$$ If the amplitude $a$ belongs to the Hörmander class $S^{m_1}_{ρ,δ}$ and $ϕ\in Φ^{2}$ satisfies the strong non-degeneracy condition, then we prove that the following Fourier integral operator $T_{ϕ,a}$ 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*} is bounded from the local Hardy space $h^1(\mathbb{R}^n)$ to $L^1(\mathbb{R}^n)$. As a corollary, we can also obtain the corresponding $L^p(\mathbb{R}^n)$-boundedness when $1<p<2$. These theorems are rigorous improvements on the recent works of Staubach and his collaborators. When $0\leq ρ\leq 1,δ\leq \max\{ρ,1-ρ\}$, by using some similar techniques in this note, we can get the corresponding theorems which coincide with the known results. |
| title | Endpoint regularity of general Fourier integral operators |
| topic | Classical Analysis and ODEs 42B20, 35S30 |
| url | https://arxiv.org/abs/2408.15280 |