A theory of complex oscillatory integrals: A case study

Abstract

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 $\mathrm{e}(z):=e^{2\pi ix}{e}^{2\pi iy}$ 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 $$I={\int }_{{\mathbb{C}}^n}\mathrm{e}(f(\underline{z}))\varphi (\underline{z})d\underline{z}$$ where $\varphi \in C_c^{\infty }({\mathbb{C}}^n)$. 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.