This is the second part of a two-part post on the theory of oscillatory integrals. In the first part we studied the theory of oscillatory integrals whose phases are functions of a single variable. In this post we will instead study the case in which the phase is a function of several variables (and we integrate in all of them). Here the theory becomes weaker because these objects can indeed have a worse behaviour. We will proceed by analogy following the same footsteps as in the single-variable case.

Part I

**3. Oscillatory integrals in several variables **

In the previous section we have analysed the situation for single variable phases, that is for integrals over (intervals of) . In this section, we will instead study the higher dimensional situation, when the phase is a function of several variables. Things are unfortunately generally not as nice as in the single variable case, as you will see.

In order to avoid having to worry about connected open sets of (see Exercise 18 for the sort of issues that arise in trying to deal with general open sets of ), in this section we will study mainly objects of the form

where has compact support. We have switched to for the phase to remind the reader of the fact that it is a function of several variables now.

** 3.1. Principle of non-stationary phase – several variables **

The principle of non-stationary phase we saw in Section 2 of part I continues to hold in the several variables case.

Given a phase , we say that is a *critical point* of if

Proposition 8 (Principle of non-stationary phase – several variables)Let (that is, smooth and compactly supported) and let the phase be such that does not have critical points in the support of . Then for any we have