On an oscillatory integral involving a homogeneous form

Abstract

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

32 pages