Sublevel Set Estimates and the Van der Corput Lemma

2.6 Oscillatory Integrals

2.6.2 Sublevel Set Estimates and the Van der Corput Lemma

We discuss a sharp decay estimate for one-dimensional oscillatory integrals. This estimate is obtained as a consequence of delicate size estimates for the Lebesgue measures of the sublevel sets{|u| ≤α}for a functionu. In what follows,u^(k)de-

notes thekth derivative of a function u(t)defined on R, andC^k the space of all functions whosekth derivative exists and is continuous.

Lemma 2.6.5.Let k≥1and suppose that a₀, . . . ,a_kare distinct real numbers. Let a=min(a_j)and b=max(a_j)and let f be a real-valuedC^k−1function on[a,b]that isC^kon(a,b). Then there exists a point y in(a,b)such that

k m=0

∑

cmf(a_m) =f^(k)(y),

where c_m= (−1)^kk! ∏^k

`6=m`=0

(a_`−a_m)⁻¹.

Proof. Suppose we could find a polynomialp_k(x) =∑^k_j=0b_jx^jsuch that the function ϕ(x) = f(x)−p_k(x) (2.6.8) satisfiesϕ(a_m) =0 for all 0≤m≤k. Since thea_j are distinct, we apply Rolle’s theoremktimes to find a pointyin(a,b)such that f^(k)(y) =k!b_k.

The existence of a polynomialp_ksuch that (2.6.8) is satisfied is equivalent to the existence of a solution to the matrix equation















a^k₀ a^k−1₀ . . . a₀ 1 a^k₁ a^k−1₁ . . . a₁ 1 ... ... ... ... ... a^k_k−1a^k−1_k−1. . . ak−11 a^k_k a^k−1_k . . . a_k 1



























 b_k b_k−1

... b₁ b₀















=













 f(a₀) f(a₁)

... f(ak−1)

f(a_k)













 .

The determinant of the square matrix on the left is called theVandermonde determi- nantand is equal to

k−1

∏

`=0 k j=`+1

∏

(a_`−a_j)6=0.

Since thea_j are distinct, it follows that the system has a unique solution. Using Cramer’s rule, we solve this system to obtain

b_k =

k m=0

∑

(−1)^mf(a_m)

k−1

∏

`6=m`=0 k

∏

j=`+1 j6=m

(a_`−a_j)

k−1

∏

`=0 k

∏

j=`+1

(a_`−a_j)

=

k

∑

m=0

(−1)^mf(a_m)

k

∏

`6=m`=0

(a_`−a_m)⁻¹(−1)^k−m.

The required conclusion now follows withc_mas claimed.

Lemma 2.6.6.Let E be a measurable subset ofRwith finite nonzero Lebesgue mea- sure and let k∈Z⁺. Then there exist a0, . . . ,a_kin E such that for all`=0,1, . . . ,k we have

k

∏

j=0 j6=`

|a_j−a_`| ≥(|E|/2e)^k. (2.6.9)

Proof. Given a measurable setE with finite measure, pick a compact subsetE⁰of E such that|E\E⁰|<δ, for someδ >0. Forx∈RdefineT(x) =|(−∞,x)∩E⁰|.

ThenT enjoys the distance-decreasing property

|T(x)−T(y)| ≤ |x−y|

for allx,y∈E⁰; consequently, by the intermediate value theorem,T is a surjective map from E⁰ to[0,|E⁰|]. Leta_j be points in E⁰ such that T(a_j) = _k^j|E⁰| for j= 0,1, . . . ,k. Forkan even integer, we have

k

∏

j=0 j6=`

|a_j−a_`| ≥

k

∏

j=0 j6=`

j k|E⁰| −`

k|E⁰| ≥

k

∏

j=0 j6=^k₂

j k−1

2 |E⁰|^k=

k 2−1 r=0

∏

r−^k₂ k

2

|E⁰|^k,

and it is easily shown that (k/2)!2

k^−k≥(2e)^−k. Forkan odd integer we have

k

∏

j=0 j6=`

|a_j−a_`| ≥

k

∏

j=0 j6=`

j k|E⁰| −`

k|E⁰| ≥

k

∏

j=0 j6=^k+1₂

j k−k+1

2k |E⁰|^k,

while the last product is at least n1

k·2 k· · ·

k−1 2

k

o2k+1

2k ≥(2e)^−k.

We have therefore proved (2.6.9) withE⁰replacingE. Since|E\E⁰|<δ andδ>0 is arbitrarily small, the required conclusion follows.

The following is the main result of this section.

Proposition 2.6.7.(a) Let u be a real-valued C^k function, k∈Z⁺, that satisfies u^(k)(t)≥1for all t∈R. Then the following estimate is valid for allα>0:

t∈R: |u(t)| ≤α ≤(2e)((k+1)!)^1kα

1

k. (2.6.10)

(b) For all k≥2, for every real-valued C^k function u on the line that satisfies u^(k)(t)≥1, for any−∞<a<b<∞, and everyλ>0, the following is valid:

Z b a

e^iλu(t)dt

≤12k|λ|^−1k. (2.6.11) (c) If k=1, u⁰(t)is monotonic on(a,b), and u⁰(t)≥1for all t∈(a,b), then

Z _b a

e^iλu(t)dt

≤3|λ|⁻¹. (2.6.12)

Proof. Part (a): LetE ={t ∈R: |u(t)| ≤α}. If|E| is nonzero, then by Lemma 2.6.6 there exista₀,a₁, . . . ,a_kinEsuch that for all`we have

|E|^k≤(2e)^k

k

∏

j=0 j6=`

|a_j−a_`|. (2.6.13)

Lemma 2.6.5 implies that there existsy∈ mina_j,maxa_j

such that u^(k)(y) = (−1)^kk!

k

∑

m=0

u(a_m)

k

∏

`6=m`=0

(a_`−a_m)⁻¹. (2.6.14)

Using (2.6.13), we obtain that the expression on the right in (2.6.14) is in absolute value at most

(k+1)! max

0≤j≤k|u(a_j)|(2e)^k|E|^−k≤(k+1)!α(2e)^k|E|^−k, sincea_j∈E. The boundu^(k)(t)≥1 now implies

|E|^k≤(k+1)!(2e)^kα as claimed. This proves (2.6.10).

Part (b): We now takek≥2 and we split the interval(a,b)in (2.6.11) into the sets

R₁ = {t∈(a,b): |u⁰(t)| ≤β}, R₂ = {t∈(a,b): |u⁰(t)|>β},

for some parameter β to be chosen momentarily. The function v=u⁰ satisfies v^(k−1)≥1 andk−1≥1. It follows from part (a) that

Z

R₁

e^iλu(t)dt

≤ |R₁| ≤2e(k!)^k−11 β

1

k−1 ≤6kβ^k−11 .

To obtain the corresponding estimate overR₂, we note that ifu^(k)≥1, then the set {|u⁰|>β}is the union of at most 2k−2 intervals on each of whichu⁰is monotone.

Let(c,d)be one of these intervals on whichu⁰is monotone. Thenu⁰has a fixed sign

on(c,d)and we have

Z _d c

e^iλu(t)dt

=

Z _d c

e^iλu(t)0 1 λu⁰(t)dt

≤

Z _d c

e^iλu(t) 1 λu⁰(t)

0

dt

+ 1

|λ|

e^iλu(d)

u⁰(d) −e^iλu(c) u⁰(c)

≤ 1

|λ| Z _d

c

1 u⁰(t)

0 dt+ 2

|λ|β

= 1

|λ|

Z _d c

1 u⁰(t)

0

dt

+ 2

|λ|β

≤ 1

|λ|

1 u⁰(d)− 1

u⁰(c) + 2

|λ|β ≤ 3

|λ|β,

where we use the monotonicity of 1/u⁰(t)in moving the absolute value from inside the integral to outside. It follows that

Z

R₂

e^iλu(t)dt

≤ 6k

|λ|β.

Choosingβ =|λ|^−(k−1)/k to optimize and adding the corresponding estimates for R₁andR₂, we deduce the claimed estimate (2.6.11).

Part (c): Repeat the argument in part (b) settingβ =1 and replacing the interval

(c,d)by(a,b).

Corollary 2.6.8.Let(a,b), u(t),λ >0, and k be as in Proposition 2.6.7. Then for any functionψon(a,b)with an integrable derivative and k≥2, we have

Z _b a

e^iλu(t)ψ(t)dt

≤12kλ^−1/k

|ψ(b)|+ Z _b

a

|ψ⁰(s)|ds

.

We also have

Z b a

e^iλu(t)ψ(t)dt

≤3λ⁻¹

|ψ(b)|+ Z b

a

|ψ⁰(s)|ds

,

when k=1and u⁰is monotonic on(a,b).

Proof. Set

F(x) = Z _x

a

e^iλu(t)dt and use integration by parts to write

Z b a

e^iλu(t)ψ(t)dt=F(b)ψ(b)− Z b

a

F(t)ψ⁰(t)dt.

The conclusion easily follows.

Example 2.6.9.TheBessel functionof ordermis defined as J_m(r) = 1

2π Z _2π

0

e^irsinθe^−imθdθ.

Here we take bothrandmto be real numbers, and we suppose thatm>−¹₂; we refer to Appendix B for an introduction to Bessel functions and their basic properties.

We use Corollary 2.6.8 to calculate the decay of the Bessel functionJ_m(r)as r→∞. Set

ϕ(θ) =sin(θ)

and note thatϕ⁰(θ)vanishes only atθ=π/2 and 3π/2 inside the interval[0,2π]and

thatϕ⁰⁰(π/2) =−1, whileϕ⁰⁰(3π/2) =1.We now write 1=ψ1+ψ2+ψ3, where

ψ1is smooth and compactly supported in a small neighborhood ofπ/2, andψ2is smooth and compactly supported in a small neighborhood of 3π/2. For j=1,2, Corollary 2.6.8 yields

Z 2π 0

e^irsin(θ) ψ_j(θ)e^−imθ dθ

≤C m r^−1/2

for some constantC, while the corresponding integral containingψ₃has arbitrary decay inrin view of estimate (2.6.3) (or Proposition 2.6.4 whenn=1).

Exercises

2.6.1.Suppose thatuis aC^kfunction on the line that satisfies|u^(k)(t)| ≥c₀>0 for somek≥2 and allt∈(a,b). Prove that forλ >0 we have

Z _b a

e^iλu(t)dt

≤12k(λc₀)^−1/k

and that the same conclusion is valid whenk=1, providedu⁰is monotonic.

2.6.2.Show that ifu⁰ is not monotonic in part (c) of Proposition 2.6.7, then the conclusion may fail.

Hint:Letϕ(t)be a smooth function on the real line that is equal to 10ton intervals [2πk+ε,2π(k+¹

2)−ε]and equal tot on intervals[2π(k+¹

2) +ε,2π(k+1)−ε].

Show that the imaginary part of the oscillatory integral in question may tend to infinity over the union of several such intervals.

2.6.3.Prove that the dependence onkof the constant in part (b) of Proposition 2.6.7 is indeed linear.

Hint:Takeu(t) =t^k/k! over the interval(0,k!).

2.6.4.Follow the steps below to give an alternative proof of part (b) of Proposition 2.6.7. Assume that the statement is known for somek≥2 and some constantC(k)

for all intervals[a,b]and allC^kfunctions satisfyingu^(k)≥1 on[a,b]. Letcbe the unique point at which the functionu^(k)attains its minimum in[a,b].

(a) Ifu^(k)(c) =0, then for allδ >0 we haveu^(k)(t)≥δ in the complement of the interval(c−δ,c+δ)and derive the bound

Z b a

e^iλu(t)dt

≤2C(k)(λ δ)^−1/k+2δ. (b) Ifu^(k)(c)6=0, then we must havec∈ {a,b}. Obtain the bound

Z _b a

e^iλu(t)dt

≤C(k)(λ δ)^−1/k+δ.

(c) Choose a suitableδ to optimize and deduce the validity of the statement fork+1 withC(k+1) =2C(k) +2=5·2^k−2. (Note thatC(1) =3.)

2.6.5.(a) Prove that for some constantCand allλ∈Randε∈(0,1)we have

Z

ε≤|t|≤1e^iλtdt t

≤C.

(b) Prove that for someC⁰<∞, allλ∈R,k>0, andε∈(0,1)we have

Z

ε≤|t|≤1e^iλt±tkdt t

≤C⁰.

(c) Show that there is a constantC⁰⁰such that for any 0<ε<N<∞, for allξ1,ξ2

inR, and for all integersk≥2, we have

Z

ε≤|s|≤Ne^{i(ξ1s+ξ2sk)}ds s

≤C⁰⁰.

Hint:Part (a): For |λ| small use the inequality |e^iλt−1| ≤ |λt|. If |λ| is large, split the domains of integration into the regions|t| ≤ |λ|⁻¹and|t| ≥ |λ|⁻¹and use integration by parts in the second case. Part (b): Write

e^i(λt+±tk)−1

t =e^iλte^±itk−1 t +e^iλt

t

and use part (a). Part (c): Whenξ1=ξ2=0 it is trivial. Ifξ2=0,ξ16=0, change variablest=ξ1sand then split the domain of integration into the sets|t| ≤1 and

|t| ≥1. In the interval over the set|t| ≤1 apply part (b) and over the set|t| ≥1 use integration by parts. In the caseξ26=0, change variablest=|ξ₂|^1/ksand split the domain of integration into the sets|t| ≥1 and|t| ≤1. When|t| ≤1 use part (b) and in the case|t| ≥1 use Corollary 2.6.8, noting that^dk(ξ1|ξ2|_dt^−1/kt±tk) =k!≥1.

2.6.6.(a) Show that for alla≥1 andλ>0 the following is valid:

Z

|t|≤aλe^iλlogtdt

≤6a.

(b) Prove that there is a constantc>0 such that for allb>λ>10 we have

Z b 0

e^iλtlogtdt

≤ c

λlogλ.

Hint:Part (b): Consider the intervals(0,δ)and[δ,b)for someδ. Apply Propo- sition 2.6.7 withk=1 on one of these intervals and withk=2 on the other. Then optimize overδ.

2.6.7.Show that there is a constantC<∞such that for all nonintegersγ>1 and allλ,b>1 we have

Z b 0

e^iλtγdt

≤ C λ^γ.

Hint:On the interval (0,δ)apply Proposition 2.6.7 with k= [γ] +1 and on the interval(δ,b)withk= [γ]. Then optimize by choosingδ =λ^−1/γ.

HISTORICAL NOTES

The one-dimensional maximal function originated in the work of Hardy and Littlewood [123].

Itsn-dimensional analogue was introduced by Wiener [291], who used Lemma 2.1.5, a variant of the Vitali covering lemma, to derive itsL^pboundedness. One may consult the books of de Guzm´an [72], [73] for extensions and other variants of such covering lemmas. The actual covering lemma proved by Vitali [285] says that if a family of closed cubes inRⁿhas the property that for every pointx∈A⊆Rⁿthere exists a sequence of cubes in the family that tends tox, then it is always possible to extract a sequence of pairwise disjoint cubesEjfrom the family such that

|A\^S_jEj|=0. We refer to Saks [233] for details and extensions of this theorem.

The classLlogLwas introduced by Zygmund to give a sufficient condition on the local integra- bility of the Hardy–Littlewood maximal operator. The necessity of this condition was observed by Stein [255]. Stein [259] also showed that theL^p(Rⁿ)norm of the centered Hardy–Littlewood maxi- mal operatorMis bounded above by some dimension-free constant; see also Stein and Str¨omberg [262]. Analogous results for maximal operators associated with convex bodies are contained in Bourgain [29], Carbery [42], and M¨uller [204]. The situation for the uncentered maximal operator Mis different, since given any 1<p<∞there existsCp>1 such thatkMk_Lp(Rⁿ)→L^p(Rⁿ)≥Cⁿ_p(see Exercise 2.1.8 for a value of such a constantC_pand also the article of Grafakos and Montgomery- Smith [109] for a larger value). The centered maximal functionMµwith respect to a general inner regular locally finite positive measureµ onRⁿis bounded onL^p(Rⁿ,µ)without the additional hypothesis that the measure is doubling; see Fefferman [93]. The proof of this result requires the following covering lemma, obtained by Besicovitch [23]: Given any family of closed balls whose centers form a bounded subset ofRⁿ, there exists an at most countable subfamily of balls that covers the set of centers and has bounded overlap, i.e., no point inRⁿbelongs to more than a finite number (depending on the dimension) of the balls in the subfamily. A similar version of this lemma was obtained independently by Morse [202]. See also Ziemer [300] for an alternative formulation.

The uncentered maximal operatorM_µof Exercise 2.1.1 may not be weak type(1,1)if the mea- sureµ is nondoubling, as shown by Sj¨ogren [243]; related positive weak type(1,1)results are

contained in the article of Vargas [283]. The precise value of the operator norm of the uncentered Hardy–Littlewood maximal function onL^p(R)was shown by Grafakos and Montgomery-Smith [109] to be the unique positive solution of the equation(p−1)x^p−px^p−1−1=0. This con- stant raised to the powernis the operator norm of the strong maximal functionM_sonL^p(Rⁿ)for 1<p≤∞. The best weak type(1,1)constant for the centered Hardy–Littlewood maximal opera- tor was shown by Melas [193] to be the largest root of the quadratic equation 12x²−22x+5=0.

The strong maximal operatorMsis not weak type(1,1), but it satisfies the substitute inequality d_Ms(f)(α)≤C^R_Rn

|f(x)|

α (1+log^+|f(x)|_α )ⁿ⁻¹dx. This result is due to Jessen, Marcinkiewicz, and Zygmund [140], but a geometric proof of it was obtained by C´ordoba and Fefferman [58].

The basic facts about the Fourier transform go back to Fourier [95]. The definition of distribu- tions used here is due to Schwartz [235]. For a concise introduction to the theory of distributions we refer to H¨ormander [130] and Yosida [296]. Homogeneous distributions were considered by Riesz [222] in the study of the Cauchy problem in partial differential equations, although some earlier accounts are found in the work of Hadamard. They were later systematically studied by Gelfand and ˇSilov [100], [101]. References on the uncertainty principle include the articles of Fefferman [90] and Folland and Sitaram [94]. The best possible constantBpin the Hausdorff–

Young inequality bf

L^p0(Rⁿ)≤B_p f

L^p(Rⁿ)when 1≤p≤2 was shown by Beckner [16] to be B_p= (p^1/p(p⁰)^−1/p0)^n/2. This best constant was previously obtained by Babenko [13] in the case whenp⁰is an even integer.

A nice treatise of the spacesM^p,qis found in H¨ormander [129]. This reference also contains Theorem 2.5.6, which is due to him. Theorem 2.5.16 is due to de Leeuw [74], but the proof pre- sented here is taken from Jodeit [142]. De Leeuw’s result in Exercise 2.5.9 says that periodic elements ofM^p(Rⁿ)can be isometrically identified with elements ofM(Tⁿ), the latter being the space of all multipliers on`^p(Zⁿ).

Parts (b) and (c) of Proposition 2.6.7 are due to van der Corput [282] and are referred to in the literature as van der Corput’s lemma. The refinenment in part (a) was subsequently obtained by Arhipov, Karachuba, and ˇCubarikov [6]. The treatment of these results in the text is based on the article of Carbery, Christ, and Wright [44], which also investigates higher-dimensional analogues of the theory. Precise asymptotics can be obtained for a variety of oscillatory integrals via the method of stationary phase; see H¨ormander [130]. References on oscillatory integrals include the books of Titchmarsh [280], Erd´elyi [83], Zygmund [303], [304], Stein [261], and Sogge [248]. The latter provides a treatment of Fourier integral operators.

Chapter 3

Fourier Analysis on the Torus

Principles of Fourier series go back to ancient times. The attempts of the Pythagorean school to explain musical harmony in terms of whole numbers embrace early ele- ments of a trigonometric nature. The theory of epicycles in theAlmagestof Ptolemy, based on work related to the circles of Appolonius, contains ideas of astronomical periodicities that we would interpret today as harmonic analysis. Early studies of acoustical and optical phenomena, as well as periodic astronomical and geophysical occurrences, provided a stimulus of the physical sciences to the rigorous study of expansions of periodic functions. This study is carefully pursued in this chapter.

The modern theory of Fourier series begins with attempts to solve boundary value problems using trigonometric functions. The work of d’Alembert, Bernoulli, Euler, and Clairaut on the vibrating string led to the belief that it might be possible to rep- resent arbitrary periodic functions as sums of sines and cosines. Fourier announced belief in this possibility in his solution of the problem of heat distribution in spatial bodies (in particular, for the cubeT³) by expanding an arbitrary function of three variables as a triple sine series. Fourier’s approach, although heuristic, was appeal- ing and eventually attracted attention. It was carefully studied and further developed by many scientists, but most notably by Laplace and Dirichlet, who were the first to investigate the validity of the representation of a function in terms of its Fourier series. This is the main topic of study in this chapter.