Damping oscillatory Integrals of convex analytic functions
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866911004353363968 |
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| author | Lee, Sanghyuk Oh, Sewook |
| author_facet | Lee, Sanghyuk Oh, Sewook |
| contents | Let $H\subset \R^{d+1}$ be a compact, convex, analytic hypersurface of finite type with a smooth measure $σ$ on $H$. Let $κ$ denote the Gaussian curvature on $H$. We consider the oscillatory integral $(κ^{1/2} σ)^\wedge$ with the damping factor $κ^{1/2}$ and prove the optimal decay estimate
\[ |(κ^{1/2} σ)^\wedge(ξ)|\le C|ξ|^{-d/2}\]
for $d=2,3,$ and with an extra logarithmic factor for $d=4$. Our result provides an essentially complete answer, since such decay estimates generally fail for $d \ge 5$, even for convex analytic hypersurfaces, as shown by Cowling--Disney--Mauceri--Müller. Furthermore, we prove the same estimates for $(κ^{1/2+it} σ)^\wedge$ with $C$ growing polynomially in $|t|$. As consequences, we obtain the best possible estimates for the convolution, maximal, and adjoint restriction operators associated with $H$, incorporating the mitigating factors of optimal orders. In particular, for $d=2, 3$, we prove the $L^2$--$L^{2(d+2)/(d+4)}$ restriction estimate with respect to the affine surface measure $κ^{1/(d+2)} σ$. This work was inspired by the stationary set method due to Basu--Guo--Zhang--Zorin-Kranich. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15492 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Damping oscillatory Integrals of convex analytic functions Lee, Sanghyuk Oh, Sewook Classical Analysis and ODEs Primary 42B20, Secondary 42B10 Let $H\subset \R^{d+1}$ be a compact, convex, analytic hypersurface of finite type with a smooth measure $σ$ on $H$. Let $κ$ denote the Gaussian curvature on $H$. We consider the oscillatory integral $(κ^{1/2} σ)^\wedge$ with the damping factor $κ^{1/2}$ and prove the optimal decay estimate \[ |(κ^{1/2} σ)^\wedge(ξ)|\le C|ξ|^{-d/2}\] for $d=2,3,$ and with an extra logarithmic factor for $d=4$. Our result provides an essentially complete answer, since such decay estimates generally fail for $d \ge 5$, even for convex analytic hypersurfaces, as shown by Cowling--Disney--Mauceri--Müller. Furthermore, we prove the same estimates for $(κ^{1/2+it} σ)^\wedge$ with $C$ growing polynomially in $|t|$. As consequences, we obtain the best possible estimates for the convolution, maximal, and adjoint restriction operators associated with $H$, incorporating the mitigating factors of optimal orders. In particular, for $d=2, 3$, we prove the $L^2$--$L^{2(d+2)/(d+4)}$ restriction estimate with respect to the affine surface measure $κ^{1/(d+2)} σ$. This work was inspired by the stationary set method due to Basu--Guo--Zhang--Zorin-Kranich. |
| title | Damping oscillatory Integrals of convex analytic functions |
| topic | Classical Analysis and ODEs Primary 42B20, Secondary 42B10 |
| url | https://arxiv.org/abs/2505.15492 |