What Is Van Der Corput’s Lemma in Higher Dimensions?

We consider variants of van der Corput’s lemma in higher dimensions. 1. The very well-known and extremely useful van der Corput lemma is the following: Van der Corput’s lemma. Let I ⊆ R be an interval and suppose φ : I → R satisfies φ(k) ≥ 1 on I (where k ∈ N). Then, for λ ∈ R, ∣∣∣∣∣ ∫ I e ∣∣∣∣∣ ≤ Ck|λ| 1 k , provided, in addition when k = 1, that φ′ is monotonic on I. An extensive discussion of van der Corput’s lemma, its proof and its applications is given in Stein’s book [S]. Amongst the features of van der Corput’s lemma emphasised in [S] are • The sharpness of the decay rate (seen by taking I = [0, 1] and φ(t) = tk/k!). • The fact that the constants Ck are absolute —that is independent of I, λ and φ. This can be useful even if we do not have the sharp decay rate. • The scaling property of the inequality: knowing the inequality for λ = 1 and arbitrary I automatically gives the inequality for general λ; or knowing the inequality for I = [0, 1] and arbitrary λ automatically gives the inequality for general I. (Note that scaling can only occur because we have the sharp decay rate.) 2000 Mathematics Subject Classification. 42B99.