Oscillatory integrals and weighted gradient flows
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913532875898880 |
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| author | Greenblatt, Michael |
| author_facet | Greenblatt, Michael |
| contents | We investigate estimating scalar oscillatory integrals by integrating by parts in directions based on $(x_1 \partial_{x_1} f(x) ,..., x_n \partial_{x_n}f(x))$, where $f(x)$ is the phase function. We prove a theorem which provides estimates that are uniform with respect to linear perturbations of the phase and investigate some consequences. When the phase function is quasi-homogeneous the theorem gives estimates for the associated surface measure Fourier transforms that are generally not too far off from being sharp. In addition, the theorem provides a new proof, up to endpoints, that the well-known oscillatory integral estimates of Varchenko [V] when the Newton polyhedron of the phase function is nondegenerate extend to corresponding bounds for surface measure Fourier transforms when the index is less than $\frac{1}{2}$. A sharp version of this was originally proven in [G2]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_12751 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Oscillatory integrals and weighted gradient flows Greenblatt, Michael Classical Analysis and ODEs 41A60, 42A38, 42B99 We investigate estimating scalar oscillatory integrals by integrating by parts in directions based on $(x_1 \partial_{x_1} f(x) ,..., x_n \partial_{x_n}f(x))$, where $f(x)$ is the phase function. We prove a theorem which provides estimates that are uniform with respect to linear perturbations of the phase and investigate some consequences. When the phase function is quasi-homogeneous the theorem gives estimates for the associated surface measure Fourier transforms that are generally not too far off from being sharp. In addition, the theorem provides a new proof, up to endpoints, that the well-known oscillatory integral estimates of Varchenko [V] when the Newton polyhedron of the phase function is nondegenerate extend to corresponding bounds for surface measure Fourier transforms when the index is less than $\frac{1}{2}$. A sharp version of this was originally proven in [G2]. |
| title | Oscillatory integrals and weighted gradient flows |
| topic | Classical Analysis and ODEs 41A60, 42A38, 42B99 |
| url | https://arxiv.org/abs/2403.12751 |