论文标题
复杂振荡积分理论：案例研究
A theory of complex oscillatory integrals: A case study
论文作者
论文摘要
在本文中，我们开发了具有复杂阶段的振荡积分理论。当$ f：{\ mathbb c}^n \ to {\ mathbb c} $，我们在基本字符$ {\ rm e}（z）上评估此阶段函数：= e^e^{2πix} e^{2πiy} $ \在{\ Mathbb c} $或$ z =（x，y）\ in {\ Mathbb r}^2 $）中，并考虑表单$ i \ = \ = \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ f {{\ mathbb c}^n}^n}^n} {\ rm e} {\ rm e} {z {z {z {Z { ϕ（{\ usewissline {z}}）\，d {\ usewissline {z}} $$其中$ ϕ \ in c^{\ infty} _C（{\ mathbb c}^n）$。 不幸的是，振荡积分$ i $的基本规模不变界限并不符合它们在实际环境中所做的一般性。我们的主要努力是开发一个观点和论据，以在（必然）的一般性中找到比例不变的界限，而不是我们在真实环境中习惯的范围。
In this paper we develop a theory for oscillatory integrals with complex phases. When $f:{\mathbb C}^n \to {\mathbb C}$, we evaluate this phase function on the basic character ${\rm e}(z) := e^{2πi x} e^{2πi y}$ of ${\mathbb C} \simeq {\mathbb R}^2$ (here $z = x+iy \in {\mathbb C}$ or $z = (x,y) \in {\mathbb R}^2$) and consider oscillatory integrals of the form $$ I \ = \ \int_{{\mathbb C}^n} {\rm e}(f({\underline{z}})) \, ϕ({\underline{z}}) \, d{\underline{z}} $$ where $ϕ\in C^{\infty}_c({\mathbb C}^n)$. Unfortunately basic scale-invariant bounds for the oscillatory integrals $I$ do not hold in the generality that they do in the real setting. Our main effort is to develop a perspective and arguments to locate scale-invariant bounds in (necessarily) less generality than we are accustomed to in the real setting.
