Restriction of the Fourier transform to some oscillating curves

0.44 MB·

by  Xianghong Chen

RESTRICTION OF THE FOURIER TRANSFORM TO
SOME OSCILLATING CURVES
XIANGHONG CHEN, DASHAN FAN, AND LIFENG WANG
7
1
0 Abstract. Let φ be a smooth function on a compact interval I. Let
2 γ(t)=(cid:0)t,t2,··· ,tn−1,φ(t)(cid:1).

n In this paper, we show that
a
(cid:18)(cid:90) (cid:19)1/q
J (cid:12)(cid:12)fˆ(γ(t))(cid:12)(cid:12)q(cid:12)(cid:12)φ(n)(t)(cid:12)(cid:12)n(n2+1)dt ≤C(cid:107)f(cid:107)Lp(Rn)
2 I
holds in the range

]
A n2+n+2 2
1≤p< , 1≤q< p(cid:48).
n2+n n2+n
C
This generalizes an affine restriction theorem of Sjo¨lin [22] for n = 2.
.
h OurproofreliesonideasofSjo¨lin[22]andDrury[11],andmorerecently
t Bak-Oberlin-Seeger[3]andStovall[24],aswellasavariationboundfor
a
m smooth functions.
[

1
v
7 1. Introduction
7
4 Let γ : I → Rn be a smooth curve and let f be a Schwartz function on
0 Rn. We are interested in understanding restriction bounds of the form
0
. (cid:18)(cid:90) (cid:19)1/q
01 (1) (cid:12)(cid:12)fˆ(γ(t))(cid:12)(cid:12)qw(t)dt ≤ C(cid:107)f(cid:107)Lp(Rn)
I
7
1 where fˆis the Fourier transform of f, C is a constant independent of f, and
:
v p,q ≥ 1. In this context, it is natural to take
i
X 2
(2) w(t) = |τ(t)|n(n+1)
r
a
where
τ(t) = det[γ(cid:48)(t),··· ,γ(n)(t)]
2
is the torsion of γ. The image of the measure |τ(t)|n(n+1)dt under γ is
called the affine arclength measure of γ, which possesses several invariance
properties. We refer the reader to Guggenheimer [16] for more background
on this notion. In what follows, unless otherwise stated, we will always
2
assume that w(t) = |τ(t)|n(n+1).
2010 Mathematics Subject Classification. 42B10, 42B99.
Keywordsandphrases. Fourierrestriction,affinearclengthmeasure,oscillatingcurve.
1
2 XIANGHONGCHEN,DASHANFAN,ANDLIFENGWANG
Figure 1. The shaded region corresponds to the range (3).
When γ is nondegenerate, that is when τ(t) is nonvanishing on I, it has
been shown by Sj¨olin [22] (for n = 2) and Drury [11] (for n ≥ 3) that the
restriction bound (1) holds for
n2+n+2 2
(3) 1 ≤ p < , 1 ≤ q ≤ p(cid:48)
n2+n n2+n
provided that I is a compact interval. Here p(cid:48) = p is the conjugate
p−1
exponent of p. The range of p is sharp, for example when γ is the moment
curve
γ(t) = (t,t2,··· ,tn).
See Arkhipov, Chubarikov and Karatsuba [1]. Sharpness of the range of q
follows from Knapp’s homogeneity argument.
Formoregeneralcurvesγ,itishardertoobtain(1)inthewholerange(3).
Partial results were obtained by Ruiz [21], Christ [7], Drury and Marshall
[13], [14]andDrury[12]forsomeclassesoffinite-typecurves. Formonomial
curves of the form
γ(t) = (ta1,··· ,tan)
(where a ,··· ,a are distinct nonzero real numbers), sharp global results
1 n
are due to Drury [11] for (a ,··· ,a ) = (1,··· ,n) and Bak, Oberlin and
1 n
Seeger [4], [5] for general exponents. In [5] a restricted strong type bound at
the endpoint p = n2+n+2 is also obtained. For curves that are perturbations
n2+n
of the monomial curves, sharp local results are due to Dendrinos and Mu¨ller
[8]. Improving upon earlier work of Dendrinos and Wright [10], Stovall [24]
obtainedsharpglobalresultsforallpolynomialcurves, withboundsuniform
over polynomials of given degree.
FOURIER RESTRICTION TO OSCILLATING CURVES 3
In this paper, we focus on smooth curves of the form
(4) γ(t) = (cid:0)t,t2,··· ,tn−1,φ(t)(cid:1).
For such curves one has τ(t) = Cφ(n)(t) where C > 0 is a dimensional
constant. Following [13] and [3], we will call such curves simple curves.
Note that if φ is so that γ is of finite type or is a perturbation of a monomial
curve, sharp restriction bounds for γ follow from the work of Dendrinos and
Mu¨ller [8] mentioned above. If φ is a polynomial, a uniform bound at the
endpoint p = n2+n+2 is obtained by Bak, Oberlin and Seeger [5].
n2+n
Inthecasen = 2, Sj¨olin[22]obtainedsharpuniformboundsforallsimple
curveswithconvexφ. Forn ≥ 3,Bak,OberlinandSeeger[3]identifiedsome
general conditions on φ that imply sharp uniform bounds. Their conditions
in particular require φ(n)(t) to be nondecreasing and to satisfy a geometric
inequality. As a consequence, one obtains sharp restriction bounds for the
simple curves with
φ(t) = e−t−α, α > 0.
See [3, Section 4] for details.
For general simple curves in n = 2 defined on compact I, Sj¨olin [22]
showed that (1) holds in the whole range (3) if one instead takes w(t) =
w (t), where
ε
2 +ε
wε(t) := |τ(t)|n(n+1) , ε > 0.
Moreover, he showed that (1) fails with w(t) = w (t) if q = 2 p(cid:48) < ∞ and
0 n2+n
(5) φ(t) = e−t−αsin(t−β)
where α,β > 0 and β is sufficiently large.
Our main result extends Sj¨olin’s result to n ≥ 3.1
Theorem 1. Let n ≥ 3 and let γ be a simple curve as in (4) with φ de-
fined on a compact interval I. Then the restriction bound (1) holds in the
following cases:
(i) w(t) = w (t), ε > 0 and
ε
n2+n+2 2
(6) 1 ≤ p < , 1 ≤ q ≤ p(cid:48);
n2+n n2+n
(ii) w(t) = w (t) and
0
n2+n+2 2
(7) 1 ≤ p < , 1 ≤ q < p(cid:48).
n2+n n2+n
Case (ii) is not explicitly formulated in Sj¨olin [22], but follows easily from
the treatment for case (i). See also Drury and Marshall [13, Theorem 1] for
a similar formulation.
The sharpness of Theorem 1 can be justified by Sj¨olin’s oscillating curves.
1The same statement holds true for n=2 and n=1.
4 XIANGHONGCHEN,DASHANFAN,ANDLIFENGWANG
Proposition 1. Let n ≥ 2 and α,β > 0. Let γ be the simple curve as in (4)
with φ given by (5) and defined on I = [0,1]. Then the restriction bound (1)
holds in cases (i) and (ii) of Theorem 1. Moreover, (1) fails if w(t) = w (t),
0
q = 2 p(cid:48) < ∞, and β > (n+1)α.
n2+n 2
Sj¨olin’s result in n = 2 relies on uniform restriction bounds for simple
curves with convex φ. In n ≥ 3, we deduce Theorem 1 based on uniform
restriction bounds for simple curves with essentially constant torsion.
Lemma 1. Let n ≥ 3 and let γ be a simple curve as in (4) with φ satisfying
(8) 1/2 ≤ |φ(n)(t)| ≤ 1, t ∈ I.
Then the restriction bound (1) holds for
n2+n+2 2
(9) 1 ≤ p < , q = p(cid:48)
n2+n n2+n
and for a constant C depending only on p and n.
It is important to note that the constant C is independent of φ. Drury
[11, Theorem 2] has obtained a similar result for all smooth curves defined
on compact I, but with the constant C depending on other information of
the curve (besides the assumption (8)). Such dependence cannot be avoided
in Drury’s result as can be seen by taking γ to be a nondegenerate closed
curve (for the existence such curves, see Costa [20]). Note also that Lemma
1 follows from Theorem 1 in Bak, Oberlin and Seeger [4] if φ satisfies some
additional mild assumptions.
With the uniform restriction bounds for convex simple curves in n = 2,
Sj¨olindeducedhisresultforgeneralsimplecurvesusingthefollowinglemma.
Lemma 2. [22, Lemma1] Let ϕ be a smooth function on a compact interval
I. Let E = {t ∈ I : ϕ(t) = 0} and let {I }∞ be the connected components
k k=1
of I\E. Then
∞
(cid:88)(cid:0) (cid:1)δ
sup|ϕ| < ∞, ∀δ > 0.
I
k=1 k
Inn ≥ 3,wededuceTheorem1fromLemma1usingthefollowinglemma.
Lemma 3. Let ϕ be a smooth function on a compact interval I. For k ∈ Z,
let E = {t ∈ I : 2−k−1 ≤ |ϕ(t)| ≤ 2−k}. Then there exist intervals {I ⊂
k k,j
I}Nk such that
j=1
N
(cid:91)k
E ⊂ I
k k,j
j=1
and such that
2−k−2 ≤ |ϕ(t)| ≤ 2−k+1, t ∈ I ;
k,j
moreover, N satisfies
k
N ≤ C 2δk, ∀δ > 0.
k δ
FOURIER RESTRICTION TO OSCILLATING CURVES 5
Note that Lemma 2 follows from Lemma 3, but not vice versa. More
general versions of Lemma 3 and Theorem 1 under weaker assumptions are
stated in Section 3 and Section 4 respectively.
The proof of Lemma 1 follows Drury’s argument in [11] using offspring
curves. We are able to obtain uniform bounds here because the special form
of the simple curves allows several technical steps in the iteration process
to go through uniformly. The proof of Theorem 1 proceeds by decomposing
the curve into segments according to the size of φ(n)(t) and then summing
up the pieces directly using Lemma 1 and rescaling. Such an approach has
also been applied to polynomial curves in Stovall [24], where it is coupled
with a square function estimate so that the pieces are summed up in a more
efficient way.
The paper is organized as follows. In Section 2 we give a detailed proof of
Lemma 1. In Section 3 we prove a more general version of Lemma 3 under
weaker smoothness assumptions. In Section 4 we prove a more general
version of Theorem 1 based on Sections 2 and 3. In Section 5 we prove
Proposition 1 by examining Knapp type examples. A technical calculation
needed in Section 5 is postponed to Section 6. Throughout the paper, C
denotes a constant whose value may change from line to line.
Acknowledgment. We would like to thank Andreas Seeger for bringing
this subject to our attention and for many useful suggestions.
2. Proof of Lemma 1
In this section, we give a detailed proof of Lemma 1 following Drury’s
argument in [11]. We will assume that I = [a,b] is a compact interval and
φ ∈ Cn(I) satisfies (8).
Associated with the given curve
γ(t) = (cid:0)t,t2,··· ,tn−1,φ(t)(cid:1), t ∈ I,
define a family of curves
(cid:40) N (cid:41)
1 (cid:88)
Υ = γ : γ (t) = γ(t+α )
α α k
N
k=1
where N ∈ N and α = (α ,··· ,α ) ∈ RN satisfies 0 ≤ α ≤ ··· ≤ α . The
1 N 1 N
domain of γ is I = [a−α ,b−α ], so that each γ(t+α ) is well defined.
α α 1 N k
Note that γ ∈ Υ. A curve γ ∈ Υ takes the form
α
(cid:32) N N (cid:33)
1 (cid:88) 1 (cid:88)
γ (t) = t+ α ,··· , (t+α )n−1,Φ (t)
α k k α
N N
k=1 k=1
where
N
1 (cid:88)
Φ (t) = φ(t+α )
α k
N
k=1
6 XIANGHONGCHEN,DASHANFAN,ANDLIFENGWANG
satisfies
(10) 1/2 ≤ |Φ(n)(t)| ≤ 1, t ∈ I .
α α
To prove Lemma 1, we will show that
(cid:18)(cid:90) (cid:12) (cid:12)q (cid:19)1/q
(11) (cid:12)fˆ(γ (t))(cid:12) dt ≤ C (cid:107)f(cid:107)
(cid:12) α (cid:12) p,n Lp(Rn)
Iα
holds uniformly for all γ ∈ Υ, where p,q satisfy (9) and C depends only
α p,n
on p and n. For simplicity, we will denote γ = γ , I = I .
α α
By duality, it suffices to show that the operator
(cid:90)
E(g)(x) := eiγ(t)·xg(t)dt
I
satisfies
(12) (cid:107)E(g)(cid:107) ≤ C (cid:107)g(cid:107)
Lp(cid:48)(Rn) p,n Lq(cid:48)(I)
for
n2+n n2+n 1 1
(13) < p(cid:48) ≤ ∞, + = 1.
2 2 p(cid:48) q(cid:48)
The proof is by induction on p(cid:48). The induction hypothesis is that for some
p(cid:48) and q(cid:48) with
0 0
n2+n 1 1
+ = 1,
2 p(cid:48) q(cid:48)
0 0
we have
(14) (cid:107)E(g)(cid:107) ≤ C (cid:107)g(cid:107) ,
Lp(cid:48)0(Rn) p0,n Lq0(cid:48)(I)
that is,
(cid:32)(cid:90) (cid:12)(cid:90) (cid:12)p(cid:48) (cid:33)1/p(cid:48)0 (cid:18)(cid:90) (cid:19)1/q(cid:48)
(15) Rn(cid:12)(cid:12)(cid:12) Iαeiγα(t)·xg(t)dt(cid:12)(cid:12)(cid:12) 0dx ≤ Cp0,n Iα|g(t)|q0(cid:48)dt 0
holds uniformly for γ ∈ Υ. The base case is that (14) holds for p(cid:48) = ∞
α 0
and q(cid:48) = 1, with C = 1.
0
By Fubini’s theorem, we can write
(cid:18)(cid:90) (cid:19)n
(cid:0)E(g)(x)(cid:1)n = eiγ(t)·xg(t)dt
I
= (cid:90) ei(cid:80)nk=1nγ(tk)·nx (cid:89)n g(tk)dt1···dtn.
In
k=1
Since the last integral is symmetric in t ,··· ,t , we have
1 n
(16) (cid:0)E(g)(x)(cid:1)n = n!(cid:90) ei(cid:80)nk=1nγ(tk)·nx (cid:89)n g(tk)dt1···dtn
A
k=1
where
A = {(t ,··· ,t ) ∈ In : t < t < ··· < t }.
1 n 1 2 n
FOURIER RESTRICTION TO OSCILLATING CURVES 7
Apply the change of variables
(17) t = t , h = t −t , k = 2,··· ,n.
1 k k 1
From (16) we can write
(18) (cid:0)E(g)(x)(cid:1)n = n!(cid:90) ei(cid:80)nk=1γn(t+hk)·nx (cid:89)n g(t+hk)dtdh2···dhn
B
k=1
where h ≡ 0 and B is the image of A under the change of variables (17).
1
Note that the curves
n
1 (cid:88)
t (cid:55)→ γ(t+h ),
k
n
k=1
belong to the family Υ.
Write
(cid:89)
v(h) = h ···h (h −h )
2 n j i
2≤i<j≤n
and define
(19) T(F)(x) = (cid:90) ei(cid:80)nk=1γn(t+hk)·xF(t,h)v(h)dtdh.
B
By Minkowski’s inequality and the induction hypothesis (15),
(20) (cid:107)T(F)(cid:107)Lp(cid:48)0(dx) ≤ (cid:90) (cid:13)(cid:13)(cid:13)(cid:13)(cid:90) ei(cid:80)nk=1γn(t+hk)·xF(t,h)dt(cid:13)(cid:13)(cid:13)(cid:13)Lp(cid:48)0(dx)v(h)dh
(cid:90)
≤ C (cid:107)F(·,h)(cid:107) v(h)dh.
p0,n Lq0(cid:48)(Ih,dt)
≤ C (cid:107)F(cid:107) .
p0,n L1(vdh,Lq0(cid:48)(dt))
On the other hand, consider the change of variables
(cid:80)n γ(t+h )
(21) y = k=1 k .
n
The corresponding Jacobian is
J(t,h) = n1n (cid:12)(cid:12)det[γ(cid:48)(t),γ(cid:48)(t+h2)··· ,γ(cid:48)(t+hn)](cid:12)(cid:12).
ByageneralizedRolle’stheorem(seeExercises95and96inPartV,Chapter
1 of [18]), we have
(n)
det[γ(cid:48)(t),γ(cid:48)(t+h )··· ,γ(cid:48)(t+h )] = Φγ (ξ)h ···h (cid:89) (h −h )
2 n 2 n j i
n!
2≤i<j≤n
for some ξ ∈ (t+h ,t+h ). Therefore, by (10),
n−1 n
(22) J(t,h) ≥ C v(h).
n
8 XIANGHONGCHEN,DASHANFAN,ANDLIFENGWANG
In particular, J is nonvanishing on B. It follows that (see [9, Prop. 3.9]) the
change of variables (21) is injective. So we can write (19) as
(cid:90)
v(h)
T(F)(x) = n! eiy·xF(t,h) dy
J(t,h)
A(cid:101)
where A(cid:101) is the image of B under the change of variables (21). By the
Plancherel theorem, we have
(cid:32)(cid:90) (cid:20) v(h) (cid:21)2 (cid:33)1/2
(cid:107)T(F)(cid:107) = C |F(t,h)|2 dy .
L2(dx) n J(t,h)
A(cid:101)
Changing the variables back and using (22), we get
(cid:18)(cid:90) (cid:20) v(h) (cid:21) (cid:19)1/2
(23) (cid:107)T(F)(cid:107) = C |F(t,h)|2 v(h)dtdh
L2(dx) n J(t,h)
B
(cid:18)(cid:90) (cid:19)1/2
≤ C |F(t,h)|2v(h)dtdh
n
B
≤ C (cid:107)F(cid:107) .
n L2(vdh,L2(dt))
By an interpolation argument (see [6]), from (20) and (23) we obtain
(24) (cid:107)T(F)(cid:107) ≤ C (cid:107)F(cid:107)
Lu(dx) p0,n Lr(vdh,Ls(dt))
for any
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
1 1 1 1 1 1 1 1
(25) , , = (1−θ) 1, , +θ , ,
r s u q(cid:48) p(cid:48) 2 2 2
0 0
with θ ∈ (0,1), and for a constant C depending only on p and n. In
p0,n 0
particular, taking
n
1 (cid:89)
F(t,h) = g(t+h )
k
v(h)
k=1
in (24), we obtain, by (18),
(cid:18)(cid:90) (cid:19)1/r
(26) (cid:107)E(g)(cid:107)n ≤ C G(h)r/sv1−r(h)dh .
Lnu(dx) p0,n
where
(cid:90) (cid:12)(cid:89)n (cid:12)s
G(h) = (cid:12) g(t+h )(cid:12) dt.
(cid:12) k (cid:12)
k=1
By [13, Lemma 1], we have
v−n2 ∈ L1,∞(dh).
Therefore, with
2
(27) 1 < r < 1+
n
FOURIER RESTRICTION TO OSCILLATING CURVES 9
and
2
ρ = > 1,
n(r−1)
we can bound
(cid:90)
(28) G(h)r/sv1−r(h)dh ≤ C (cid:107)Gr/s(cid:107) (cid:107)v1−r(cid:107)
ρ Lρ(cid:48),1(dh) Lρ,∞(dh)
= Cρ(cid:107)Gr/s(cid:107)Lρ(cid:48),1(dh)(cid:107)v−n2(cid:107)1L/1ρ,∞(dh)
≤ C (cid:107)Gr/s(cid:107) .
r,n Lρ(cid:48),1(dh)
If for some E ⊂ I,
(29) |g(t)| ≤ 1 (t), t ∈ I.
E
Then
(cid:90)
G(h) ≤ |E|, G(h)dh ≤ |E|n.
It follows that
(30) (cid:107)Gr/s(cid:107)Lρ(cid:48),1(dh) ≤ Cr,n|E|rs+nρ−(cid:48)1
provided
r
(31) ρ(cid:48) > 1.
s
Combining (26), (28) and (30), we obtain
1 +n−1
(cid:107)E(g)(cid:107)Lnu(dx) ≤ Cp0,r,n|E|ns nrρ(cid:48)
as long as (25), (27), (31) and (29) are satisfied. By interpolation, this
produces a new bound (12) for any p(cid:48),q(cid:48) satisfying (13) with
p(cid:48) > p(cid:48), q(cid:48) < q(cid:48),
1 1
where
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
1 1 n−2 1 1 2 2
, = , + , .
p(cid:48) q(cid:48) n(n+2) p(cid:48) q(cid:48) n(n+2) n(n+2)
1 1 0 0
Iterating this process, we see that (12) holds for p(cid:48),q(cid:48) in the full range
(13). This completes the proof of Lemma 1.
3. Proof of Lemma 3
We now prove Lemma 3. Let ϕ be a continuous (real-valued) function
defined on a compact interval [a,b]. Let r > 0. Suppose the set
E = {t ∈ [a,b] : r/2 ≤ |ϕ(t)| ≤ r} =(cid:54) ∅.
r
For t ∈ E , define
r
a = sup{s ∈ [a,t] : |ϕ(s)| ≤ r/4 or |ϕ(s)| ≥ 2r}
t
10 XIANGHONGCHEN,DASHANFAN,ANDLIFENGWANG
if the set on the right-hand side is not empty; otherwise define a = a.
t
Similarly, define
b = inf{s ∈ [t,b] : |ϕ(s)| ≤ r/4 or |ϕ(s)| ≥ 2r}
t
if the set on the right-hand side is not empty; otherwise define b = b. Let
t
I = (a ,b ) if a > a and b < b,
t t t t t
I = [a ,b ) if a = a and b < b,
t t t t t
I = (a ,b ] if a > a and b = b,
t t t t t
I = [a ,b ] if a = a and b = b.
t t t t t
Note that ϕ has a definite sign on I .
t
Lemma 4. For any t ,t ∈ E , we have either I = I or I ∩I = ∅.
1 2 r t1 t2 t1 t2
Proof. If t ∈ I , then by the definitions of I and I , we must have
2 t1 t1 t2
I = I . If t ∈/ I , then I ∩I = ∅. (cid:3)
t1 t2 2 t1 t1 t2
Notice that {I } forms an open cover of the compact set E in [a,b].
t t∈Er r
So there must be a finite subcover. Combining with Lemma 4, we get the
following.
Lemma 5. {I } is a finite set of disjoint intervals.
t t∈Er
With Lemma 5, we can make the following.
Definition 1. Let ϕ be a continuous function defined on a compact interval
[a,b]. For r > 0, define
N(r;ϕ) = #{I }
t t∈Er
where {I } is as described above.
t t∈Er
We now prove the main technical lemma. Denote by C1[a,b] the space of
functions on [a,b] whose derivative is continuous on [a,b].
Lemma 6. Suppose ϕ ∈ C1[0,1] and N(r;ϕ) ≥ N ≥ 20. Then
(cid:18) (cid:19)
N N
N r;ϕ(cid:48) ≥ .
8 16
Proof. Since the intervals I corresponding to ϕ and r are disjoint, there
t
must be at least N/2 many of them that satisfy
2
|I | ≤ .
t
N
After perhaps excluding one such interval (the rightmost one), each of them
has its right endpoint satisfying either ϕ = ±2r or ϕ = ±r/4.
Without loss of generality, assume that at least a quarter of these right
endpoints satisfy ϕ = r/4. Denote them by b < ··· < b , with
1 M
N −1
M > 2 .
4