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March 2020 On an oscillatory integral involving a homogeneous form
Shuntaro Yamagishi
Funct. Approx. Comment. Math. 62(1): 21-58 (March 2020). DOI: 10.7169/facm/1775

## Abstract

Let $F \in \mathbb{R}[x_1, \ldots, x_n]$ be a homogeneous form of degree $d > 1$ satisfying $(n - \dim V_{F}^*) > 4$, where $V_F^*$ is the singular locus of $V(F) = \{ \mathbf{z} \in {\mathbb{C}}^n: F(\mathbf{z}) = 0 \}$. Suppose there exists $\mathbf{x}_0 \in (0,1)^n \cap (V(F) \backslash V_F^*)$. Let $\mathbf{t} = (t_1, \ldots, t_n) \in \mathbb{R}^n$. Then for a smooth function $\varpi:\mathbb{R}^n \rightarrow \mathbb{R}$ with its support contained in a small neighbourhood of $\mathbf{x}_0$, we prove $$\Big{|} \int_{0}^{\infty} \cdots \int_{0}^{\infty} \varpi(\mathbf{x}) x_1^{i t_1} \cdots x_n^{i t_n} e^{2 \pi i \tau F(\mathbf{x})} d \mathbf{x} \Big{|} \ll \min \{ 1, |\tau|^{-1} \},$$ where the implicit constant is independent of $\tau$ and $\mathbf{t}$.

## Citation

Shuntaro Yamagishi. "On an oscillatory integral involving a homogeneous form." Funct. Approx. Comment. Math. 62 (1) 21 - 58, March 2020. https://doi.org/10.7169/facm/1775

## Information

Published: March 2020
First available in Project Euclid: 26 October 2019

zbMATH: 07225497
MathSciNet: MR4074385
Digital Object Identifier: 10.7169/facm/1775

Subjects:
Primary: 42B20