Some remarks on the $L^p-L^q$ boundedness of trigonometric sums and oscillatory integrals

We discuss the asymptotic behaviour for the best constant in L^p-L^q estimates for trigonometric polinomials and for an integral operator which is related to the solution of inhomogeneous Schrodinger equations. This gives us an opportunity to review some basic facts about oscillatory integrals and the method of stationary phase, and also to make some remarks in connection with Strichartz estimates.


Introduction
Let u(t, x) be the solution of the homogeneous Schrödinger equation i∂ t u − ∆u = 0, with initial data u(0, x) = f (x). Let v(t, x) be the solution of the inhomogeneous Schrödinger equation i∂ t v − ∆v = F (t, x), with zero initial data. It is known [6,3,4] that inhomogeneous Strichartz estimates of the form v L q (R;L r (R n )) F L q ′ (R;L r (R n )) are valid even for some pairs of exponents (q, r), ( q, r) which are not admissible for the homogeneous Strichartz estimate (1) u L q (R;L r (R n )) f L 2 (R n ) .
While searching for counterexamples which could help us understand what the optimal range for the exponents q, r, q, r in (1) could be, a simplification of the problem led us to consider the integral operator T : L p (0, 1) → L q (0, 1) defined by (2) T f (t) = 1 0 e iN/(1+t+s) f (s) ds.
A further simplified discrete version of this integral operator is represented by the operator which assigns to N complex numbers a 0 , . . . , a N −1 the trigonometric polynomial N −1 n=0 a n e int , acting from ℓ p (C N ) to L q (−π, π). We are interested in the asymptotic behaviour of its operator norm as N → ∞. This becomes an interesting exercise in elementary harmonic analysis whose solution (theorem 2.3) is discussed in sections 2, 3, 4. In section 5 we obtain estimates for integral operators like (2). We then use them in section 6 to find the optimal range of exponent for a weaker local version of Strichartz estimates (theorem 6.1). The details of the proofs of the various lemmata about oscillatory integrals which are needed throughout the paper are collected in section 7.

Trigonometric sums
Given N complex numbers a 0 , a 2 , . . . , a N −1 , the trigonometric sum (3) f (t) = N −1 n=0 a n e int defines a smooth 2π-periodic function. Let T N be the linear operator from C N to C ∞ ([−π, π]; C) which maps the vector a = (a 0 , a 1 , . . . , a N −1 ) to the function f . For any p, q ∈ [1, ∞], let us denote by C N (p → q) the best constant for the estimate where the norms are defined by Problem: Is it possible to compute C N (p → q), or at least to describe its asymptotic behaviour as N → ∞? Remark 2.1. Since the operator T N is defined on a finite dimensional vector space, we know that the constant C N (p → q) is always finite and that for any choice of N, p, q there esists some maximizer a ∈ C N for which we have T N (a) L q = C N (p → q) · a ℓ p .
Remark 2.2. In order to facilitate the visualization of relations among the various estimates, it will be convenient to use the notation (p → q) to indicate the point (1/p, 1/q) in the unit square Q = [0, 1] 2 and view C N as a function defined on Q.
Let's decompose the square Q into the three regions (see figure 1) In sections 3 and 4 we calculate upper and lower bounds for C N which are summarized in the following theorem.
Theorem 2.3. There exists positive absolute constants c B and c C such that:
Remark 3.2 (Energy estimate). We can exploit L 2 orthogonality of the oscillating terms in (3) and obtain the (2 → 2) estimate, This implies that Remark 3.3 (Interpolation). The Riesz-Thorin interpolation theorem ( [1]) applied to our operator T N tells us that This amounts to saying that log C N is a convex function on Q. In particular, if we have bounds for C N at any two points X and Y of Q, then interpolation gives us bounds for C N on the whole segment in Q connecting X with Y .
Remark 3.4 (Hölder inclusions). If 1 ≤ q ≤ q ≤ ∞ we can apply Hölder's inequality to the norm of f , f L q ≤ f L q , and obtain the following condition for C N : If 1 ≤ p ≤ p ≤ ∞ we can apply Hölder's inequality to the norm of a, a ℓ p ≤ N 1/p−1/ p a ℓ p , and obtain the following condition for C N : Looking at the Q square, these conditions mean that an upper bound at one point in Q implies upper bounds at any point which can be reached by moving upward or leftward.
Remark 3.6. It is interesting to note that we had to look at the structure of the operator T N only for the dispersive (1 → ∞) estimate and the energy (2 → 2) estimate. All other estimates followed from these two cases using only the structure and interpolation properties of L p spaces, without having to look at the structure of the operator T N . A similar situation happens when we want to prove Strichartz estimates for dispersive evolution operators [5].

Lower bounds
We can obtain lower bounds for C N by computing the norms of f and a for specific examples.
Together with the upper bound (6), this proves that C N (p → q) = 1 in the region A. Moreover, in this region any choice of a ∈ C N whose components are all vanishing except for one is a maximizer.

Lower bounds for region B.
Example 4.2 (Dirichlet kernels). Let us choose a = (1, 1, . . . , 1). Its norm is a ℓ p = N 1/p . We can compute f explicitly, where D N is the Dirichlet kernel To estimate D N from below, we use the fact that It follows that from which we obtain In particular, this example shows that in region B the exponent 1 − 1/q − 1/p which appears in the upper bound (6) is sharp and that we can take c B = 2/π in theorem 2.3.
The following lemma, which we prove in section 7, improves the estimate (8).
Lemma 4.3. Let D N be the Dirichlet kernel (7). When q > 1, the limit exists, is finite, and its value is Moreover, if q > 2 then γ(q) < 1.
Remark 4.4. When q = 2m is an even integer and p = ∞, it is easy to see that example 4.2 provides a maximizer and hence Indeed, this is an immediate consequence of the following monotonicity property. Let us assume that |a n | ≤ b n for all n = 0, . . . , N − 1, and let f = T N (a) and g = T N (b), then, because of the positivity of the delta function, we have Remark 4.5. Using interpolation, from lemma 4.3 and remark 4.4 it follows that we must have strict inequality, C N (p → q) < N 1−1/q−1/p , and also that lim sup in the interior of region B where 2 < q < ∞ and 1/p < 1 − 1/q.

4.3.
Lower bounds for region C. We now need an example for region C which could give us a lower bound of the type A good candidate would be a choice of a ∈ C N with |a n | ≈ 1 for most of the n's, and such that |f (t)| N 1/2 for most of the t's. The idea is to set a n = e iϕ(n) , for some real valued function ϕ, and compare the sum N −1 n=0 e iϕ(n) e int with the integral N 0 e iϕ(x) e ixt dx, with the help of the following lemma taken from Zygmund's "Trigonometric series" [8]. For the sake of completeness, we present the interesting proof of the lemma in section 7.
We also need another lemma whose proof is given in section 7.
We are now ready to construct our example for region C.
We have a ℓ p = N 1/p . We fix t ∈ [−π, π] and set Φ(x) = ϕ(x) + xt. We have The phase function Φ satisfies the hypotheses of lemma 4.6: It follows that the difference between the sum (10) and the integral is bounded by an absolute constant (independent of N and t), We have When 0 < t < 2 we apply lemma 4.7 and obtain .

An integral operator with oscillating kernel
We turn our attention to the linear integral operator defined by for some fixed γ ≥ 0. Let us denote now by C N (p → q) the best constant which can appear in the estimate where this time We ask the same question as before: what can we say about the behaviour of C N as N → ∞?
Theorem 5.1. There exists positive absolute constants c A , c B and c C such that: The following proof is similar to the proof of theorem 2.3.

Upper bounds.
In order to prove the upper bounds in theorem 5.1, it is enough to observe that we have the (1 → ∞) dispersive estimate Interpolation yields the (p → p ′ ) estimate when 1 ≤ p ≤ 2, and Hölder's inequality does the rest. Estimate (14) is trivial and simply follows by taking absolute values inside the integral.
To get estimate (15), let χ ∈ C ∞ 0 (R) be a non-negative cut-off function such that where the kernel K N is given by the oscillatory integral and amplitude To estimate K N we apply the principle of non-stationary phase as illustrated by the following lemma.

Local estimates for inhomogeneous Schrödinger equations
Now we come to the problem which motivated the above study, namely the problem of determining the optimal range of exponents for local inhomogeneous Strichartz estimates.
Let n ≥ 3. Let u(t, x) be the solution of the inhomogeneous Schrödinger equation with zero initial data u(0, x) = 0. We assume that the support of F is contained in the region where 0 ≤ t ≤ 1 and we look at the solution u(t, x) in the region where 2 ≤ t ≤ 3. Using the fundamental solution of the Schrödinger equation we can write an explicit formula for u in terms of F : Local Strichartz estimates of the type are known to hold when the pairs of exponents (q, r) and ( q, r) satisfy the conditions (see [4] and [5] for details and references). The norms which appear in (18) have the integration with respect to space variables computed before doing the integration with respect to time. Interpolation with the easy dispersive estimate proves that when exponents r and r satisfy the conditions then there exist some exponents q, q ∈ [1, ∞] for which estimate (18) holds ( [2]). The dark shaded area in figure 2 shows the region in the (r, r) plane where (19) is satisfied.
Here, we use the result of section 5 to obtain estimates similar to (18) but with norms which have the integration with respect to time computed before doing the integration with respect to the space variables. Theorem 6.1. Let us suppose that exponents r and r satisfy the conditions then there exist some exponents q, q ∈ [1, ∞] for which we have the estimate (21) u L r (R n ;L q ([2,3])) F L r ′ (R n ;L q ′ ([0,1])) .
The light shaded area in figure 2 shows the region in the (r, r) plane where (20) is satisfied.  (18) and (21) in the (1/r, 1/ r) plane.
Remark 6.2. Estimate (21) is weaker than (18) in the sense that when (18) holds for the pairs of exponents (q, r) and ( q, r) then (21) holds if we replace q with min{q, r} and q with min{ q, r}. Indeed, if Q = min{q, r}, then we have G L r (R n ;L Q (I)) ≤ G L Q (I;L r (R n )) ≤ G L q (I;L r (R n )) , and G L q ′ (I;L r ′ (R n )) ≤ G L Q ′ (I;L r ′ (R n )) ≤ G L r ′ (R n ;L Q ′ (I)) , for any function G(t, x) and time interval I of length 1.
Proof of theorem 6.1. Using the change of variables we have t − s = 1 + τ + σ and we see that The inner integral is exacly the operator T N described in section 5, with N = |x − y| 2 /4 and γ = n/2, acting on the function σ → F (1 − σ, y). We can apply theorem 5.1 and use (13) to obtain Estimate (21) then follows from (22) and Young's inequality when 1/r + 1/ r < α/n, or the Hardy-Littlewood-Sobolev inequality ([7, chapter VIII, section 4.2]) when 1/r + 1/ r = α/n and r, r < ∞. This means that we proved estimate (21) when with strict inequalities if r = ∞ or r = ∞. On the other hand, it follows from remark 6.2 that estimate (21) holds for some q and q when r and r satisfy conditions (19). Using interpolation we obtain estimate (21) for some q and q whenever r and r are in the convex hull of the two regions in the (1/r, 1/ r) plane described by (19) and (23). This convex hull is precisely the region described by (20) plus the two endpoints r = 2, r = 2n/(n − 2) and r = 2n/(n − 2), r = 2.
By the above computations, when t ∈ [2, 3] and |z| < 1 we have ). In particular this shows that the oscillatory factor does not oscillates too much when |y| ≤ η/R, R ≤ |x − y| ≤ 2R and η is sufficiently small. It follows that which inserted in (24) and using (25) proves that cannot be bounded as R → ∞ unless the necessary condition

Proof of the lemmata
Proof of lemma 4.3. We want to compute the leading term in the asymptotic espansion of the L q norm of the Dirichlet kernel D N as N → ∞. By a simple change of variable we have When |α| < π/2 we have It follows that When q > 1, we have
Proof of lemma 4.6. Integration by parts gives We write e iΦ(n+1) − e iΦ(n) = n+1 n ∂ x e iΦ(x) dx and obtain where [x] denotes the largest integer smaller or equal to x. The function x − [x] is a piecewise-continuous periodic function with period 1 and mean value 1/2. When x / ∈ Z, it coincides almost everywhere with its Fourier series given by Thus, we have It follows that Ψ = Γ • Φ ′ is also smooth and monotone, being the composition of two smooth and monotone functions. In particular ∂ x Ψ does not change sign and we have Integrating (31) by parts we obtain We now take absolute values and use (33) and (32), Substituting this in (30) we obtain and integrate by parts, It follows that If 0 < t < 1 the result follows from (36) and a rescaling of the Fresnel integral Proof of lemma 5.2. The method is standard. We begin by noticing that thus e iλψ is an eigenfunction for the differential operator relative to the eigenvalue λ, and in particular Let L * be the transpose of L, Then, integrating by parts k times we obtain thus, for k = 0, . . . , K we have The quantity (L * ) k (χ) is a linear combination of terms which are the ratio of products of χ, its derivatives and derivatives of ψ ′ , with powers of ψ ′ ; hence, its integral can be bounded by a constant which depends on k, M , and the measure of the support of χ.
Proof of lemma 5.6. As a first step we prove the lemma in the special case of quadratic phase ϕ(s) = k(s − s * ) 2 , for some k ≥ 1/2. We write Integration by parts in the second integral in the right hand side of (37) gives From the explicit formulas and the hypotheses on χ, it follows that max |Ψ| ≤ max |χ ′ | ≤ 1, max |Ψ ′ | ≤ max |χ ′′ | ≤ 1. Hence, which concludes the proof in this special case.
In the second step, we prove the lemma for a general phase function ϕ but we assume the amplitude χ to have compact support in ]a, b[. Let In order to control I(N ; 1), it will be enough to provide uniform bounds of order O(1/N ) for the derivative ∂ λ I(N ; λ). We derive (38) with respect to λ and then integrate by parts twice, There is no contribution coming from the boundary points, for we have chosen χ smooth and compactly supported in the interval ]a, b[. Here we have used subscript notation for derivatives of Φ, ϕ ′′ (s * + θ(s − s * )) dθ.