Multidimensional Van Der Corput and Sublevel Set Estimates

If a function has a large derivative, then it changes rapidly, and so spends little time near any particular value. This paper is devoted to quantifying that principle for functions of several variables, particularly as it pertains to two problems in harmonic analysis. Along the way we shall encounter diverse problems and techniques, and shall be led to issues distinctly combinatorial in nature. We begin by reviewing two well-known questions in one-dimensional analysis. Suppose that u is a (smooth) real-valued function on the real line R such that for some k ∈ N, u(t) ≥ 1 for all t ∈ R. Here, and in what follows, u denotes the k’th derivative of u. a) How small are the sublevel sets {t ∈ R : |u(t)| ≤ α} for small α? In particular, at what rate does the Lebesgue measure |{t ∈ R : |u(t)| ≤ α}| tend to zero as α→ 0? b) How quickly does the oscillatory integral I(λ) = ∫ b a edt tend to zero as λ→∞? Answers are given by the following two results. The first is known as van der Corput’s lemma; see, for example, [S]. Lemma 1.1. a) Suppose k ≥ 2. If u ≥ 1, then |I(λ)| ≤ Ck/ |λ| 1 k where Ck is an absolute constant depending only upon k. b) There is no constant C such that if u′ ≥ 1, then |I(λ)| ≤ C/ |λ|. c) If u′ ≥ 1 and if in addition u′ is monotonic, then |I(λ)| ≤ C1/ |λ| where C1 is an absolute constant. Lemma 1.2. For each k ≥ 1, there exists a finite absolute constant Ck such that for any function satisfying u(x) ≥ 1 for all x, |{t : |u(t)| ≤ α}| ≤ Ckα . The estimates in Lemmas 1.1 and 1.2 are uniform over classes of functions u whose k’th derivatives are bounded below; they are independent of any upper bounds for higher order derivatives of u. Our chief goal is to establish analogous uniform estimates in higher dimensions.