This chapter compiles various auxiliary results; including variants of van der Corput-type estimates for one-dimensional oscillatory integrals and related sublevel estimates through “integrals of sublevel type.” It also derives a straightforward variant of a beautiful real interpolation method that has been devised by Bak and Seeger and that will allow in some cases the replacement of the more classical complex interpolation methods in the proof of Stein–Tomas-type Fourier restriction estimates by substantially shorter arguments. Last, this chapter derives normal forms for phase functions φ of linear height < 2 for which no linear coordinate system adapted to φ does exist.
Keywords: auxiliary results, van der Corput-type estimates, one-dimensional oscillatory integrals, sublevel type, real interpolation, Stein–Tomas-type Fourier restriction, phase functions, linear coordinates
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