A theory of complex oscillatory integrals: A case study

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$z = (x,y) \in {\mathbb R}^2$)andconsideroscillatoryintegralsoftheform$$ I \ =
\int_{{\mathbb C}^n} {\rm e}(f({\underline{z}})) , \phi({\underline{z}}) , d{\underline{z}} $$where$\phi \in C^{\infty}_c({\mathbb C}^n)$. 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.