In short, the method of Stationary Phase concerns itself with obtaining bounds for oscillatory integrals of the following form

I(f;R) = Z Rn eiRf(x) (x) dx

where (x) is a bump function. Dierent bounds are obtained depending on the conditions imposed on f(x). We’ll explore the Van Der Corput Lemmas and a possible generalization of them and then aim to give estimates for the Fourier transform of the surface measure on the sphere.