Notes on Regularity of Fourier integral operators with symbol in $S^{m}_{0,δ}$
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| Format: | Preprint |
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2024
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| _version_ | 1866912713965305856 |
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| author | Wang, Guangqing He, Suixin |
| author_facet | Wang, Guangqing He, Suixin |
| contents | Let $T_{a,φ}$ be a Fourier integral operator defined with $a\in S^{m}_{0,δ}(0\leqδ<1)$ and $φ\in Φ^{2}$ satisfying the strong non-degenerate condition. We demonstrate that when the order satisfies $$m\leq-\frac{n}{2}-\frac{n}{p}δ+\frac{n}{p},$$ the operator $T_{a,φ}$ becomes bounded on $L^{p}(\mathbb{R}^n)$ for $2< p<\infty$ and maps $L^{\infty}(\mathbb{R}^n)$ to $BMO(\mathbb{R}^n)$ when $p=\infty$. Furthermore, the derived bound on $m$ is sharp for $L^{p}$ estimates in the case $δ=0$, and for $(L^{\infty},BMO)$ when $0\leqδ<1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_00409 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Notes on Regularity of Fourier integral operators with symbol in $S^{m}_{0,δ}$ Wang, Guangqing He, Suixin Classical Analysis and ODEs Analysis of PDEs Let $T_{a,φ}$ be a Fourier integral operator defined with $a\in S^{m}_{0,δ}(0\leqδ<1)$ and $φ\in Φ^{2}$ satisfying the strong non-degenerate condition. We demonstrate that when the order satisfies $$m\leq-\frac{n}{2}-\frac{n}{p}δ+\frac{n}{p},$$ the operator $T_{a,φ}$ becomes bounded on $L^{p}(\mathbb{R}^n)$ for $2< p<\infty$ and maps $L^{\infty}(\mathbb{R}^n)$ to $BMO(\mathbb{R}^n)$ when $p=\infty$. Furthermore, the derived bound on $m$ is sharp for $L^{p}$ estimates in the case $δ=0$, and for $(L^{\infty},BMO)$ when $0\leqδ<1$. |
| title | Notes on Regularity of Fourier integral operators with symbol in $S^{m}_{0,δ}$ |
| topic | Classical Analysis and ODEs Analysis of PDEs |
| url | https://arxiv.org/abs/2412.00409 |