A sharp bound for the Stein-Wainger oscillatory integral

Let Pd denote the space of all real polynomials of degree at most d. It is an old result of Stein and Wainger that for every polynomial P in Pd: |p.v.\int_R {e^{iP(t)} dt/t} |<C(d) for some constant C(d) depending only on d. On the other hand, Carbery, Wainger and Wright claim that the true order of magnitude of the above principal value integral is log d. We prove this conjecture.

claim that the true order of magnitude of the above principal value integral is log d. We prove that sup P ∈P d˛p .v.
Z R e iP (t) dt t˛∼ log d.

Introduction
Let P d be the vector space of all real polynomials of degree at most d in R. For P ∈ P d we consider the principal value integral We wish to estimate the quantity I(P ) by a constant C(d) depending only on the degree of the polynomial d. This amounts to estimating the integral I (ǫ,R) (P ) = ǫ≤|t|≤R e iP (t) dt t by some constant C(d) independent of ǫ, R and P . This problem is quite old and in fact has been answered some thirty years ago by Stein and Wainger in [4] and [6]. They showed that the quantity I(P ) is bounded by a constant C d depending only on d. Their proof is very simple and uses a combination of induction and Van der Corput's lemma. Let us recall the latter since we'll also be using it in what follows.
2000 Mathematics Subject Classification. Primary 42A50; Secondary 42A45. 1 Proposition 1.1 (van der Corput). Let φ : [a, b] → R be a C k function and suppose that |φ (k) (t)| ≥ 1 for some k ≥ 1 and all t ∈ [a, b]. If k = 1 suppose in addition that φ ′ is monotonic. Then, for every where C is an absolute constant independent of a,b,k and φ.
For a proof of this very well known result with Ck replaced by C k see for example [3]. A proof that the constant C k can be taken to be linear in k can be found in [1].
On the other hand, Carbery, Wainger and Wright have conjectured in [2] that the true order of magnitude of the principal value integral is log d. The main result of this paper is the proof of this conjecture. This is the content of: Theorem. There exist two absolute positive constants c 1 and c 2 such that Remark 1.2. Suppose that K is a −n homogeneous function on R n , odd and integrable on the unit sphere. Then, by the one-dimensional result, we trivially get that there is an absolute positive constant c, such that: for every polynomial P on R n , of degree at most d.
Notation. We will use the letter c to denote an absolute positive constant which might change even in the same line of text. Also, the notation A ∼ B means that there exist absolute positive constants c 1 and c 2 such that c 1 B ≤ A ≤ c 2 B.

Aknowledgements
I would like to thank James Wright for bringing this problem to our attention and for many helpful discussions. I would also like to thank Mihalis Papadimitrakis from the University of Crete, my thesis supervisor, for his constant support.

The lower bound in the Theorem
In this section we will construct a real polynomial P of degree at most d such that the inequality The general plan of the construction is as follows. We will first construct a function f (which will not be a polynomial) such that I(f ) ≥ c log n. We will then construct a polynomial P of degree d = 2n 2 − 1 that approximates the function f in a way that |I(f ) − I(P )| is small (small means o(log n) here). Since log n ∼ log d this will yield our result.
Proof. The proof is more or less straightforward.
We now want to construct a polynomial which approximates the function f . We will do so by convolving the function f with a "polynomial approximation to the identity". To be more specific, for k ∈ N and x ∈ R define the function where the constant c k is defined by means of the normalization Observe that where B(·, ·) is the beta function. Using standard estimates for the beta function we see that c k ∼ k. Define, next, the functions P k in R as where f is the function of Lemma 3.1. It is clear that the functions P k are polynomials of degree at most 2k 2 . The following lemma deals with some technical issues concerning the polynomials P k .
Lemma 3.2. Let P k be defined as in (3.5) above.
(i) P k is an odd polynomial of degree 2k 2 − 1 with leading coefficient That is (ii) As a consequence of (i) we have for all t Next, from (3.5) we have that It is now easy to see that the two highest order terms come from the first summand in the above formula. Therefore, (ii) We just use the result of (i) and that c k ∼ k.
However, since φ k is even, We are now ready to prove the lower bound for I(P ).
Proposition 3.3. Let P n be the polynomial defined in (3.5) where n is the large positive integer used to define the function f in Lemma 3.1. Then P n is a polynomial of degree d = 2n 2 − 1 and Proof. Since P n is odd, and it suffices to show that for all R ≥ 1 By part (ii) of Lemma 3.2 and a standard application of Proposition 1.1 (Van der Corput) we see that for all R ≥ 1. As a result, the proof will be complete if we show that Using Lemma 3.1 and the triangle inequality we get We have that Using part (iii) of Lemma 3.2 and (3.4), we get Now, the desired result, condition (3.9), is the content of the following lemma.

Proof. Firstly, it is not difficult to establish that
A(x, t) ≤ 4 min(nx, nt, 1) (3.10) On the other hand, Inequality (3.10) now follows by the fact that |f | is bounded by 1 and (3.11) is trivial to prove. We split the integral 2 0 1 0 · · · dtdx into seven integrals: We estimate each of the seven integrals separately. A(x, t) t dtφ n (x)dx ≤ 4 log 2 2 0 φ n (x)dx = 2 log 2.

The upper bound in the Theorem
We set We take any polynomial P , of degree at most d, which we can assume has no constant term, that is, P (0) = 0. We set k = [ d 2 ] and we write where Q(t) = a 1 t + a 2 t 2 + · · · + a k t k and R(t) = a k+1 t k+1 + · · · + a d t d . Let |a l | = max k+1≤j≤d |a j | for some k + 1 ≤ l ≤ d. By a change of variables in the integral in (4.1) we can assume that |a l | = 1 and thus that |a j | ≤ 1 for every k + 1 ≤ j ≤ d. Now split the integral in (4.1) in two parts as follows e iP (t) dt t (4.2) = I 1 + I 2 .
For I 1 we have that For the second integral in (4.2) we have that For some α > 0 to be defined later split I + 2 into two parts as follows: For the logarithmic measure of the set {t ∈ [1, +∞) : |P ′ (t)| ≤ α}, observe that We have thus showed that In order to finish the proof we need a suitable estimate for the sublevel set of a polynomial. This is the content of the following lemma. This Lemma is due to Vinogradov [5]. We postpone the proof of Lemma 4.1 until after the end of the proof of the upper bound.
. Using Lemma 4 and (4.3), we get Obviously, a similar estimate holds for I − 2 . Summing up the estimates we get Optimizing in α we get that (4.4) and hence In particular we have K 2 n ≤ c + K 2 n−1 . Using induction on n we get that K 2 n ≤ cn. It is now trivial to show the inequality for general d. Indeed, if 2 n−1 < d ≤ 2 n then K d ≤ K 2 n ≤ cn ≤ c log d.
For the sake of completeness we give the proof of Lemma 4.1.
Proof of Lemma 4.1. The set E α = {t ∈ [1,2] : |h(t)| ≤ α} is a union of intervals. We slide them together to form a single interval I of length |E α | and pick n + 1 equally spaced points in I. If we slide the intervals back to their original position we end up with n + 1 points x 0 , x 1 , x 2 , . . . , x n ∈ E α which satisfy |x j − x k | ≥ |E α | |j − k| n .