ABSTRACT
In this chapter we present a Fourier transform approach to the proof of the CR extension theorem. This technique has the advantage in that it can be more easily adapted to the holomorphic extension of CR distributions, which will be discussed at the end of Part III. However, the goal of this chapter is to introduce the technique rather than to prove the most general theorem. Therefore, to avoid some of the cumbersome technicalities of more general results, we shall assume the given CR function is sufficiently smooth (class C d+2 where d = codimR M will suffice). In addition, we shall assume that the submanifold M is rigid, which means that near a given point p ∈ M there is a local biholomorphic change of coordinates so that p is the origin and M = { ( z = x + i y , w ) ∈ ℂ d × ℂ n − d ; y = h ( w ) } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315140445/fbe50be4-2923-45ad-961d-51ed68ffe025/content/equation851.tif"/> where h: ℂ n–d → ℝ d is smooth (say class C d+2) with h(0) = 0 and Dh(0) = 0. The point is that the graphing function, h, for a rigid submanifold is independent of the variable x ∈ ℝ d . The modifications required to handle the more general case where h depends on both x and w will be mentioned at the end of Part III.
