L^p regularity of averages over curves and bounds for associated maximal operators

0.43 MB·English

by  Malabika Pramanik

Lp REGULARITY OF AVERAGES OVER CURVES AND
BOUNDS FOR ASSOCIATED MAXIMAL OPERATORS
5
0
0
2 MALABIKAPRAMANIKANDANDREASSEEGER

n
a Abstract. WeprovethatforafinitetypecurveinR3 themaximaloperator
J generatedbydilationsisboundedonLp forsufficientlylargep. Wealsoshow
1 theendpointLp→Lp1/p regularityresultfortheaveragingoperatorsforlarge
1 p. TheproofsmakeuseofadeepresultofThomasWolffaboutdecompositions
ofconemultipliers.

]
A
C
. 1. Introduction and Statement of Results
h
t
a Let I be a compact interval and consider a smooth curve
m
γ : I R3.
[ →
We say that γ is of finite type on I if there is a natural number n, and c > 0 so

1 that for all s I, and for all ξ =1,
v ∈ | |
n
5
(1.1) γ(j)(s),ξ c.
6
h i ≥
1 Xj=1(cid:12) (cid:12)
(cid:12) (cid:12)
1
For fixed s the smallest n for which (1.1) holds is the type of γ at s. The type is
0
anupper semicontinuous function, and we refer to the supremumof the types over
5
0 s I as the maximal type of γ on I. Let χ be a smooth function supported in the
∈
/ interior of I. We define a measure µ supported on a dilate of the curve by
h t
t
a (1.2) µ ,f := f(tγ(s))χ(s)ds,
t
m h i Z
and set
:
v
i (1.3) tf(x):=f µt(x).
X A ∗
We are aiming to prove sharp Lp regularity properties of these integral operators
r
a and also Lp boundedness of the maximal operator given by
(1.4) f(x):=sup f(x).
t
M |A |
t>0
To the best of our knowledge, Lp boundedness of had not been previously
M
established for any p < . Here we prove some positive results for large p and
∞
in particular answer affirmatively a question on maximal functions associated to
helices whichhas beenaroundfora while (for example itwas explicitly formulated
in a circulated but unpublished survey by Christ [4] from the late 1980’s).
Our results rely on a deep inequality ofThomas Wolff for decompositions of the
cone multiplier in R3. To describe it consider a distribution f ′ R3 whose
∈ S
(cid:0) (cid:1)
Version1-6-05. SupportedinpartbygrantsfromtheUSNationalScienceFoundation.
1
2 M.PRAMANIKANDA.SEEGER
Fourier transform is supported in a neighborhood of the light cone ξ2 =ξ2+ξ2 at
3 1 2
level ξ 1, of width δ 1. Let Ψ be a collection of smooth functions which
3 ν
≈ ≪ { }
aresupportedin 1 δ1/2 δ-platesthat “fit”the lightcone andsatisfy the natural
× ×
sizeestimates anddifferentiability properties;fora moreprecisedescriptionsee 2.
§
Wolff [22] proved that for all sufficiently large p, say p > p , and all ǫ > 0, there
W
exists C >0 such that
ǫ,p
1
(1.5) (cid:13)(cid:13)(cid:13)Xν Ψbν ∗f(cid:13)(cid:13)(cid:13)p ≤Cǫ,pδ−21+p2−ǫ(cid:16)Xν (cid:13)(cid:13)Ψbν ∗f(cid:13)(cid:13)pp(cid:17)p.
A counterexample in [22] shows that this inequality cannot hold for all ε > 0 if
p<6 and Wolff obtained (1.5) for p 74 (a slightly better range can be obtained
≥
as was observed by Garrig´os and one of the authors [8]).
We notethatconnectionsbetweenconemultipliersandtheregularityproperties
ofcurveswith nonvanishingcurvature andtorsionhavebeen used invariousprevi-
ouspapers, firstimplicitly in the paper by Oberlin [14]who provedsharpLp L2
→
estimatesfor,say,convolutionswithmeasuresonthehelix(coss,sins,s);thesewere
extended in [9] to more general classes of Fourier integral operators. Concerning
Lp Sobolev estimates the Lp Lp boundedness follows by an easy interpola-
→ 2/3p
tion argument, but improvements of this estimate are highly nontrivial. Oberlin,
Smith and Sogge [16] used results by Bourgain [3] and Tao and Vargas [21] on
square-functions associatedto cone multipliers to show that if 2<p< then the
∞
averagesfor the helix map Lp to the Sobolev space Lp, for some α>2/(3p).
α
Weemphasizethatsharpregularityresultsforhypersurfaces havebeenobtained
frominterpolationarguments,using resultsondampedoscillatoryintegralsandan
improved L∞ bound near “flat parts” of the surface, see e.g. [6], [19], [10] and
elsewhere. However this interpolation technique does not apply to averages over
manifolds with very high codimension, in particular not to curves in Rd, d 3.
Our first result on finite type curves in R3 concerns the averaging op≥erator
in (1.3); it depends on the optimal exponent p in Wolff’s inequality
1 W
A ≡ A
(1.5).
Theorem 1.1. Suppose that γ Cn+5(I) is of maximal type n, and suppose that
∈
p +2
W
max n, <p< .
{ 2 } ∞
Then maps Lp boundedly to the Sobolev space Lp .
A 1/p
Thus the sharp Lp-Sobolev regularity properties for the helix hold for p > 38,
according to Wolff’s result. It is known by an example due to Oberlin and Smith
[15]thattheLp Lp regularityresultfailsifp<4. RecallthatWolff’sinequality
→ 1/p
(1.5) is conjectured for p (6, ), and thus establishing this conjecture would by
Theorem 1.1 imply the L∈p ∞Lp bound for p > 4. If the type n is sufficiently
→ 1/p
large then our result is sharp; it can be shown by a modification of an example by
Christ [5] for plane curves that the endpoint Ln Ln bound fails. Finally we
→ 1/n
notethatby aduality argumentandstandardfactsonSobolevspacesonecanalso
deduce sharp bounds near p=1, namely if 1<p<min n/(n 1),(p +2)/p
W W
then maps Lp boundedly to Lp . { − }
A 1/p′
Remark. There are generalizations of Theorem 1.1 which apply to variable curves;
oneassumesthattheassociatedcanonicalrelationinT∗R3 T∗R3 projectstoeach
×
AVERAGES OVER CURVES AND ASSOCIATED MAXIMAL OPERATORS 3
T∗R3 onlywithfoldsingularitiesandthatacurvatureconditionin[9]onthe fibers
ofthesingularsetissatisfied. We intendtotakeupthesemattersinaforthcoming
paper [17].
Our main result on the maximal operator is
M
Theorem 1.2. Suppose that γ Cn+5 is of maximal type n, then defines a
∈ M
bounded operator on Lp for p>max(n,(p +2)/2).
W
Again the range of p’s is only optimal if the maximal type is sufficiently large
(i.e. n (p +2)/2). The following measure-theoretic consequence (which only
W
≥
uses Lp boundedness for some p< ) appears to be new; it follows fromTheorem
∞
1.2 by arguments in [2].
Corollary 1.3. Let γ : I R3 be smooth and of finite type and let A R3 be a
set of positive measure. Le→t E be a subset of R3 with the property that⊂for every
x A there is a t(x) > 0 such that x+t(x)γ(I) is contained in E. Then E has
∈
positive outer measure.
In itself the regularity result of Theorem 1.1 does not imply boundedness of
the maximal operator, but a local smoothing estimate can be used. This we only
formulate for the nonvanishing curvature and torsion case.
Theorem 1.4. Suppose that γ C5(I) has nonvanishing curvature and torsion.
∈
Let p <p< and χ C∞((1,2)). Suppose that α<4/(3p). Then the operator
A defiWned by A∞f(x,t)=∈χ(t)0 f(x) maps Lp(R3) boundedly into Lp(R4).
At α
By interpolation with the standard L2 L2 estimate (obtained from van der
→ 1/3
Corput’s lemma) one sees that A maps Lp(R3) to Lp (R4), for some β(p)>1/p,
β(p)
if (p + 2)/2 < p < . By standard arguments the Lp boundedness of the
W
∞
maximaloperator followsinthisrange(providedthatthecurvehasnonvanishing
M
curvature and torsion).
Structureof thepaper. In 2weproveanextensionofWolff’sestimatetogeneral
§
cones which will be crucial for the arguments that follow. In 3 we prove the
§
sharp Lp Sobolev estimates for large p (Theorem 1.1). In 4 we prove a version
of the local smoothing estimate for averaging operators asso§ciated to curves in Rd
microlocalized to the nondegenerate region where γ′′(s),ξ = 0. In 5 we use
h i 6 §
the previousestimates and rescalingargumentsto proveTheorem1.4 andin 6 we
§
deduceourresultsformaximaloperators,includinganestimateforatwoparameter
family of helices.
2. Variations of Wolff’s inequality
The goal of this section is to prove a variantof Wolff’s estimate (1.5) where the
standard light-cone is replaced by a general cone with one nonvanishing principal
curvature. RatherthanredoingtheverycomplicatedproofofWolff’s inequalitywe
shall use rescaling and induction on scales arguments to deduce the general result
fromthespecialresult,assumingthevalidityof (1.5)forthelightcone,intherange
p p .
W
≥
We need to first set up appropriate notation. Let I be a closed subinterval of
[ 1,1] and let
−
α g(α)=(g (α),g (α)) R2, α I,
1 2
7→ ∈ ∈
4 M.PRAMANIKANDA.SEEGER
define a C3 curve in the plane and we assume that for positive b , b and b
0 1 2
g b ,
C3(I) 0
k k ≤
(2.1) g′(α) b1,
| |≥
g′(α)g′′(α) g′(α)g′′(α) b .
| 1 2 − 2 1 |≥ 2
We consider multipliers supported near the cone
= ξ R3 :ξ =λ(g (α),g (α),1), α I,λ>0 .
g 1 2
C { ∈ ∈ }
For each α we set
(2.2) u (α)=(g(α),1), u (α)=(g′(α),0), u (α)=u (α) u (α)
1 2 3 1 2
×
where refers to the usualcrossproduct sothat a basisof the tangentspace of
g
× C
at (g(α),1) is given by u (α),u (α) . Let λ 0, δ >0 and define the (δ,λ)-plate
1 2
at α, Rα , to be the pa{rallelepiped in} R3 give≥n by the inequalities
δ,λ
λ/2 u (α),ξ 2λ,
1
≤|h i|≤
(2.3) u (α),ξ ξ u (α) λδ1/2,
2 3 1
|h − i|≤
u (α),ξ λδ.
3
|h i|≤
For a constant A 1 we define the A-extension of the plate Rα to be the paral-
≥ δ,λ
lelepiped given by the inequalities
λ/(2A) u (α),ξ 2Aλ;
1
≤|h i|≤
u (α),ξ ξ u (α) Aλδ1/2;
2 3 1
|h − i|≤
u (α),ξ Aλδ.
3
|h i|≤
NotethattheA-extensionofa(δ,λ)-platehaswidth Aλintheradialdirection
≈
tangent to the cone, width Aλδ1/2 in the tangential direction which is perpen-
≈
dicular to the radial direction and is supported in a neighborhood of width Aλδ
≈
of the cone.
A C∞ function φ is called an admissible bump function associated to Rα if φ
δ,λ
is supported in Rα and if
δ,λ
(2.4) u (α), n1 u (α), n2 u (α), n3φ(ξ) λ−n1−n2−n3δ−n2/2δ−n3,
1 2 3
h ∇i h ∇i h ∇i ≤
(cid:12) (cid:12) 0 n +n +n 4.
(cid:12) (cid:12) 1 2 3
≤ ≤
A C∞ function φ is called an admissible bump function associated to the A-
extensionofRα ifφissupportedintheA-extensionbutstillsatisfiestheestimates
δ,λ
(2.4).
Let λ 1, δ >0, δ1/2 θ, moreover σ δ1/2. A finite collection = R N
≥ ≤ ≤ R { ν}ν=1
is called a (δ,λ,θ)-plate family with separation σ associated to g if (i) each R is
ν
of the form Rδα,νλ for some αν ∈I, (ii) ν 6= ν′ implies that |αν −αν′|≥ σ and, (iii)
max α min α θ. Given A 1 we let the A-extension of the
Rν∈R{ ν}− Rν∈R{ ν} ≤ ≥
plate family consist of the A-extensions of the plates R .
ν
R
The main result in Wolff’s paper is proved for the cone generated by g(α) =
(cos2πα,sin2πα), 1/2 α 1/2. Namely if is a (δ,λ,1)-plate family with
− ≤ ≤ R
separation √δ and for R , φ is an admissible bump function associated to R
R
∈ R
AVERAGES OVER CURVES AND ASSOCIATED MAXIMAL OPERATORS 5
then for all ε>0 there is the inequality
1/p
(2.5) (cid:13)(cid:13)RX∈RF−1[φRfbR](cid:13)(cid:13)p ≤A(ε)δp2−12−ε(cid:16)RX∈RkfRkpp(cid:17)
(cid:13) (cid:13)
if p > 74. This is equivalent with the statement (1.5) in the introduction. Our
next proposition says that this inequality for the light cone implies an analogous
inequality for a general curved cone.
Proposition 2.1. Suppose that p > 2 and that (2.5) holds for all (δ,λ,1)-plate
families associated to the circle (cos(2πα),sin(2πα)) . Then for any δ 1, λ 1,
{ } ≤ ≥
σ √δ the following holds true.
≤
Let α g(α) satisfy (2.1) and let be a (δ,λ,θ)-plate family with separation
7→ R
σ, associated to g. For each R let φ be an admissible bump function associated to
R
R. Then for ε>0 there is a constant C(ε) depending only on ε and the constants
b , b , b in (2.1) so that for f Lp(R3),
0 1 2 R
∈
1/p
(2.6) (cid:13)(cid:13)RX∈RF−1[φRfbR](cid:13)(cid:13)p ≤C(ε)δ1/2σ−1(δθ−2)2p−12−ε(cid:16)RX∈RkfRkpp(cid:17) .
(cid:13) (cid:13)
Proof. We first remark that we can immediately reduce to the case σ = √δ, by a
pidgeonhole argument and an application of the triangle inequality.
Secondly, ifΨ are bump-functions containedinthe A-extensionsofthe rectan-
R
glesR,butsatisfyingthe sameestimates(2.4)relativetothe rectanglesR,thenan
estimatesuchas(2.6)impliesasimilarestimateforthecollectionofbumpfunctions
Ψ where the constant C(ε) is replaced with C C(ε). This observation will be
R A
{ }
used extensively; it is proved by a pidgeonhole and partition of unity argument.
We now use various scaling arguments based on the formula
(2.7) −1[m(L)f](x)= −1[mf\(L∗ )](L∗−1x)
F · F ·
for any real invertible linear tranbsformation L (with transpose L∗).
Step 1. Here we still assume that g(α) = (cos2πα,sin2πα), but wish to show for
δ1/2 θ 1theimprovedestimate fora(δ,λ,θ)-plate familywithseparationδ1/2.
The≤plat≤es in R N are of the form R = Rαν where α α0 θ for
R ≡ { ν}ν=1 ν δ,λ | ν − | ≤
some α0 [0,1]. Let L be the rotation by the angle α in the (ξ ,ξ ) plane which
1 1 2
∈
leaves the ξ axis fixed. Then the family the plate family L (R ) is still a (δ,λ,θ)
3 1 ν
plate family with associated bump functions φ = φ L−1. We note that all
ν,1 Rν ◦ 1
rotated plates are contained in a larger (C λ,C λθ,C λθ2) rectangle with axes in
1 1 1
the direction of (1,0,1), (0,1,0), (1,0, 1).
−
We now use a rescaling argument from [21] and [22]. Let L be the linear
2
transformationthatmaps(1,0,1)to(1,0,1),(0,1,0)toθ−1(0,1,0)and(1,0, 1)to
−
θ−2(1,0, 1);itleavesthelightconeinvariant. Onechecksthateachparallelepiped
−
L L R is contained in a (C λ,C λδ1/2θ−1,C λδθ−2) plate R and the sets R
2 1 ν 2 2 2 ν ν
◦
formaC extensionofa(δθ−2,λ,1)familywithseparationσ =C−1δ1/2θ−1. Thus
2 e 3 e
using (2.7) we may apply the assumed result for θ = 1, and obtain the claimed
result for θ <1 (yet for the light cone).
Step 2. We shall now consider tilted cones where g is given by
(2.8) g(α)=(a+ρcosα,b+ρsinα), a + b +ρ+1/ρ K
| | | | ≤
6 M.PRAMANIKANDA.SEEGER
Suppose that we are given a (δ,λ,θ)-plate family = R associated to g,
ν
R { }
with separation √δ; moreover we are given a family of admissible bump-functions
φ associated with the plates R . Consider the linear transformation L given by
ν ν
ξ aξ ξ bξ
1 3 2 3
L(ξ)=Ξ, with Ξ = − , Ξ = − , Ξ =ξ .
1 2 3 3
ρ ρ
Then the parallelepipeds L(R ) are contained in parallelepipeds R which for a
ν ν
suitableconstantC formaC -extensionofa(δ,λ,θ)-platefamilyassociatedtothe
4 4 e
unitcircle;moreover,forsuitableC thefunctions C−1φ L−1 formanadmissible
5 5 ν◦
collectionofbumpfunctionsassociatedtothis extension. HereC ,C dependonly
4 5
on the constant K in (2.8). By scaling we obtain then estimate (2.6) for g as in
(2.8), with C(ε) equal to C(K)A(ε).
Step 3. We now use an induction on scales argument. Let β >(1/2 2/p) and let
−
W(β) denote the statement that the inequality
1/p
(2.9) −1[φ f ] B(β)(δθ−2)−β f p .
(cid:13)(cid:13)RX∈RF RbR (cid:13)(cid:13)p ≤ (cid:16)RX∈Rk Rkp(cid:17)
(cid:13) (cid:13)
holds for all g satisfying (2.1), all λ > 0, δ 1, √δ θ 1, all (δ,λ,θ)-plate-
≤ ≤ ≤
families associated to such g.
We remark that clearly W(β) holds with β = 1, with B(1) depending only on
the constants in (2.1). We shall now show that for β >1/2 2/p
−
(2.10) W(β) = W(β′) with
⇒
β′ = 2β+ 1(1 2 +ε), B(β′)=C B(β)A(ε),
3 3 2 − p 6
where C depends only on (2.1).
6
In order to show (2.10) we let = Rαν be a (δ,λ,θ)-plate family, with
R { λ,δ}
associated family of bump functions φ . We regroup the indices ν into families
ν
{ }
J , so that for ν,ν′ J we have α α δ1/3 and for ν J , ν′ J we
µ µ ν ν′ µ µ+2
∈ | − | ≤ ∈ ∈
haveα α δ1/3/2. For eachµwe pick one ν(µ) J . Thenfor allν J the
ν′ ν µ µ
− ≥ ∈ ∈
Rαν are contained in the C -extension R′ of a (δ2/3,λ)-plate at α (µ), as can be
λ,δ 7 µ ν
verifiedbyaTaylorexpansion. LetR′′ bethe 2C extensionofthatplate. Wemay
µ 7
pick a C∞-function Ψ supported in R′′ which equals 1 onR′, so that for suitable
µ µ µ
C depending only on the constants in (2.1), the functions C−1Ψ are admissible
8 8 µ
bump functions associated to the R′′. We then use assumption W(β) to conclude
µ
that
−1[φ f ] = −1[Ψ φ f ]
ν ν µ ν ν
(cid:13)(cid:13)Xν F b (cid:13)(cid:13)p (cid:13)(cid:13)Xµ νX∈JµF b (cid:13)(cid:13)p
(cid:13) (cid:13) (cid:13) (cid:13)
p 1/p
(2.11) C B(β)(δ2/3θ−2)−β −1[φ f ] .
8 ν ν
≤ (cid:16)Xµ (cid:13)(cid:13)νX∈JµF b (cid:13)(cid:13)p(cid:17)
(cid:13) (cid:13)
We claim that for each µ,
1/p
(2.12) −1[φ f ] C A(ε)(δ1/3)2/p−1/2−ε f p .
(cid:13)(cid:13)νX∈JµF νbν (cid:13)(cid:13)p ≤ 9 (cid:16)νX∈Jµk νkp(cid:17)
(cid:13) (cid:13)
Clearly a combination of (2.11) and (2.12) yields (2.10) with C =C C .
6 8 9
AVERAGES OVER CURVES AND ASSOCIATED MAXIMAL OPERATORS 7
Wefixα′ :=α andobservethatontheinterval[α′ δ1/3,α′ +δ1/3]wemay
µ ν(µ) µ− µ
approximate the curve α g(α) by its osculating circle with accuracy C δ.
10
→ ≤
The circle is given by
g (α)=g(α′)+ρn(α′)+ρ cos(α−α′µ−ϕµ),sin(α−α′µ−ϕµ)
µ µ µ ρ ρ
(cid:0) (cid:1)
where n(α) is the unit normal vector ( g′(α),g′(α))/g′(α), ρ is the reciprocal of
− 2 1 | |
the curvature of g at α′ and ϕ is the unique value between 0 and 2π for which
µ µ
g′(α′) sin(ϕ /ρ)=g′(α′ ) and g′(α′ ) cos(ϕ /ρ)=g′(α ).
| µ | µ 1 µ | µ | µ 2 µ
Inviewofthegoodapproximationpropertyweseethatforeachν J theplate
µ
∈
Rαν associatedto g is containedintheC -extensionRν ofaplate Rαν associated
δ,λ 11 δ,λ
to g . Moreover the family J can be split into no more than C subfamilies Ji
µ µ e 12e µ
wherethe α ineachsubfamily are√δ-separated. Finallythere is C sothateach
ν 13
bump function φ is the C -multiple of an admissible bump function associated
ν 13
to Rν. Here C ,C ,C depend only on the constants in (2.1). This puts us in
11 12 13
the position to apply the result from step 2, with θ =C δ1/3; we observe that the
e 14
constantK instep2controllinginparticulartheradiusofcurvaturedependsagain
only on the constants in (2.1). Thus we can deduce (2.12) and the proof of (2.10)
is complete.
Step 4. We now iterate (2.10) and replace ε by ε/2 to obtain (2.9) with
β β =(2)n+(1 (2)n)(1 2 + ε)
≡ n 3 − 3 2 − p 2
B(β )= C A(ε) n, n=1,2,...
n 15 2
(cid:0) (cid:1)
The conclusion of the proposition follows if we choose n>log(2/ε)/log(3/2). (cid:3)
One can use Proposition 2.1 and standard arguments to see that results on the
circularcone multiplier in[22]carryoverto moregeneralcones. To formulatesuch
a resultlet ρ C4(R2 0 )be positive awayfrom the origin,and homogeneousof
degree 1. Con∈sider the\F{ou}rier multiplier in R3, given by
m (ξ′,ξ )=(1 ρ(ξ′/ξ ))λ.
λ 3 − 3 +
As in [22] we obtain
Corollary 2.2. Assume that the unit sphere Σ = ξ′ R2 : ρ(ξ′) = 1 has
ρ
nonvanishing curvature everywhere. Then m is a Fou{rier∈multiplier of Lp(R}3) if
λ
λ>(1/2 2/p), p p< .
W
− ≤ ∞
Remarks.
(i) The curvature condition on Σ in the corollary can be relaxed by scaling
ρ
arguments.
(ii) The methods of Proposition 2.1 apply in higher dimensions as well. In par-
ticulartheygeneralizeWolff’sinequalityfordecompositionsoflightconesinhigher
dimensions([12])tomoregeneralellipticalconesgeneratedbyconvexhypersurfaces
with nonvanishing curvature. In particular, if ρ is a sufficiently smooth distance
function in Rd, m (ξ′,ξ )=(1 ρ(ξ′/ξ ))λ and if the unit sphere associated
λ d+1 − | d+1| +
with ρ is a convex hypersurface of Rd with nonvanishing Gaussian curvature then
m is a Fourier multiplier of Lp(Rd+1), for λ > d1/2 1/p 1/2, for the range
λ
| − |−
of p’s given in [12] for the multipliers associated with the spherical cone. After
8 M.PRAMANIKANDA.SEEGER
Proposition2.1 hadbeenobtainedL aba andPramanik[11]workedoutanalterna-
tive proof of the higher dimensional variant directly based on the methods in [22],
[12]. Their approachalsoapplies to the case ofnonellipticalconeswhichcannotbe
obtained by scaling and approximation from existing results.
3. Lp regularity
We shall first consider a “nondegenerate case”, namely we assume that s
γ(s) R3, s I [ 1,1] is of class C5 and has nonvanishing curvature an7→d
∈ ∈ ⊂ −
torsion. We assume that
5
(3.1) γ(i)(s) C , s I
0
| |≤ ∈
Xi=1
and
(3.2) det γ′(s) γ′′(s) γ′′′(s) c , s I
0
(cid:12) (cid:12)≥ ∈
(cid:12) (cid:0) (cid:1)(cid:12)
In this case we show(cid:12) Theorem 1.1 under the ass(cid:12)umption that the cutoff function χ
in (1.2) is of class C4; then we may without loss of generalizationassume that γ is
parametrized by arclength (since reparametrization introduces just a different C4
cutoff). In the end of this sectionwe shalldescribe how to extend the resultto the
finite type case.
In what follows we shall write . for two quantities , if C
1 2 1 2 1 2
E E E E E ≤ E
with a constant C only depending on the constants in (3.1), (3.2). We denote by
T(s),N(s),B(s) the Frenet frame of unit tangent, unit normal and unit binormal
vector. We recall the Frenet equations T′ = κN, N′ = κT +τB, B′ = τN,
− −
with curvature κ and τ . The assumption of nonvanishing curvature and torsion
implies that the cone generated by the binormals, B= rB(s):r > 0,s I , has
{ ∈ }
one nonvanishing principal curvature which is equal to rκ(s)τ(s) at ξ =rB(s).
Bylocalizationinsandpossiblerotationwemayassumeforthethirdcomponent
of B(s) that B (s)>1/2 for all s I. If
3
∈
(3.3) g(s)= B1(s),B2(s)
B3(s) B3(s)
(cid:0) (cid:1)
parametrizesthelevelcurveatheightξ =1thenthecurvaturepropertyofthecone
3
can be expressed in terms of the curvature of this level curve and a computation
gives
B′(s) B′(s) B′(s)
g′(s) g′(s) 1 1 2 3 κ(s)τ(s)
det 1 2 = detB′′(s) B′′(s) B′′(s)= .
(cid:18)g′′(s) g′′(s)(cid:19) (B (s))3 1 2 3 (B (s))3
1 2 3 B (s) B (s) B (s) 3
1 2 3
 
Thus the hypotheses on g in (2.1) are satisfied with constants depending only on
the constants in (3.1), (3.2).
WeshallworkwithstandardLittlewood-Paleycutoffs,andmakedecompositions
of the Fourier multiplier associated to the averages. Observe first that the contri-
butionofthe multiplier nearthe originisirrelevantinviewofthe compactsupport
of the kernel. Thus consider for k >0 the Fourier multipliers
(3.4) m (ξ)= e−ithγ(s),ξia (s,2−kξ)ds
k k
Z
AVERAGES OVER CURVES AND ASSOCIATED MAXIMAL OPERATORS 9
where we assume that a vanishes outside the annulus ξ : 1/2 < ξ < 2 and
k
{ | | }
satisfies the estimates
(3.5) ∂j∂αa(s,ξ) C , α 2,0 j 3;
| s ξ |≤ 2 | |≤ ≤ ≤
hereofcourse α = α + α + α . Thusthemultiplierm isasymboloforder0,
1 2 3 k
| | | | | | | |
with perhaps limited order of differentiability, localized to the annulus ξ 2k ).
{| |≈ }
We note that by the standard Bernstein theorem (which says that L2 [L1]
α ⊂ F
for α > 3/2) the multipliers a (s, ) and their s-derivatives up to order three are
k
Fourier multipliers of Lp(R3), unifo·rmly in s, k.
We have to establishthat for the desiredrange ofp’s the sum 2k/pm is a
k>0 k
Fourier multiplier of Lp. We may assume that the symbols ak arPe supported near
from the cone generated by the binormal vectors B(s). More precisely if θ(ξ) is
smoothawayfromtheoriginandhomogeneousofdegree0andifθhastheproperty
that
γ′′(s), ξ c>0, ξ supp(θ) supp(a ),
|h |ξ|i|≥ ∈ ∩ k
then θm = O(2−k/2) by van der Corput’s Lemma, and by the almost dis-
k ∞
k k
jointness of the supports we also have that θ 2k/2m = O(1) by van der
k k>0 kk∞
Corput’sLemma. MoreoverbystandardsingulParintegraltheory the operatorwith
Fourier multiplier θ m maps L∞ to BMO and consequently, by analytic
k>0 k
interpolation θ P2k/pm is a Fourier multiplier of Lp provided 2 p< .
k>0 k ≤ ∞
Thus by a pPartition of unity it suffices to understand the localization of the
multiplier 2k/pm to a narrow (tubular) neighborhood of the binormal cone
k>0 k
B= rB(sP):r >0,s I ,andthereforein whatfollowswemay andshallassume
{ ∈ }
that ξ in the support of a (s, ) can be expressed as
k
·
ξ =rB(σ)+uT(σ)=:Ξ(r,u,σ),
with inverse function ξ (r(ξ),u(ξ),σ(ξ)).
7→
Decomposition of the dyadic multipliers. We shall now concentrate on the
multipliersm in(3.4),andprovethe bound m .2−k/p,forp>(p +2)/2.
k k kkMp W
Here M is the usual Fourier multiplier space.
p
Wefirstdecomposefurtheroursymbolsa . Letη C∞(R)beanevenfunction
k 0 ∈ 0
supported in [ 1,1] and be equal to 1 on [ 1/2,1/2]. Let η =η (4−1 ) η . Let
1 0 0
− − · −
A 2max 1,1/τ(s):s I and set
0
≫ { ∈ }
(3.6) a (s,ξ)=a (s,ξ)η (22[k/3](u(ξ) +(s σ(ξ))2)),
k k 0
| | −
and for integers l<k/3
e
a (s,ξ)=a (s,ξ)η (22l(u(ξ) +(s σ(ξ))2))η ((s−σ(ξ))2)
(3.7) k,l k 1 | | − 0 A0u(ξ)
b (s,ξ)=a (s,ξ)η (22l(u(ξ) +(s σ(ξ))2)) 1 η ((s−σ(ξ))2) .
k,l k 1 | | − − 0 A0u(ξ)
(cid:0) (cid:1)
Thusa (s, )issupportedwheredist(ξ,B) 2−2l and s σ(ξ) .2−l,anda (s, )
k,l k
· ≈ | − | ·
is supported in a C2−2k/3 neighborhood of the binormal cone with s σ(ξ) .
2−k/3. The symbol b (s, ) is supported in a C2−2l neighborhood of t|he−bineorm| al
k,l
·
cone but now s σ(ξ) 2−l.
| − |≈
Wenotethatinviewofthepreliminarylocalizationsthesymbolsa ,b vanish
k,l k,l
for l C, moreover
≤
a (s,ξ)=a (s,ξ)+ a (s,ξ)+ b (s,ξ).
k k k,l k,l
X X
l≤k/3 l≤k/3
e
10 M.PRAMANIKANDA.SEEGER
Set
(3.8) m [a](ξ)= a(s,2kξ)e−ihγ(s),ξids.
k
Z
We shall show
Proposition 3.1. For p <p< ,
W
∞
(3.9) m [a ] C 2−4k/3p+kε,
k k Mp ε
k k ≤
(3.10) m [a ] C 2−k/p2−l/p+lε,
k ek,l Mp ε
k k ≤
(3.11) m [b ] C 2−2k/p22l/p+lε,
k k,l Mp ε
k k ≤
The constants depend only on ε, (3.1), (3.2) and (3.5).
Weshallgivetheproofof(3.10)and(3.11),andtheproofof(3.9)isareanalogous
with mainly notational changes.
For the proofs of (3.10) and(3.11) we needto further split the symbols a , b
k,l k,l
by makinganequally spaceddecompositioninto pieces supportedon2−l intervals.
Let ζ C∞ be supported in ( 1,1) so that ζ( ν) 1. We set
∈ 0 − ν∈Z ·− ≡
a (s,ξ)=ζ(2ls Pν)a (s,ξ)
k,l,ν k,l
−
andsimilarlyb (s,ξ)=ζ(2ls ν)b (s,ξ);moreoverdefinea (s,ξ)=ζ(2k/3s
k,l,ν k,l k,ν
− −
ν)a (s,ξ).
k
In order to apply Wolff’s estimate in the form of Propositieon 2.1 we need
e
Lemma 3.2. Let s =2−lν and let (T ,N ,B )=(T(s ),N(s ),B(s )). Suppose
ν ν ν ν ν ν ν
that s s 22−l. Then the following holds true:
ν
| − |≤
(i) The multipliers a (s, ), b (s, ) are supported in
k,l,ν k,l,ν
· ·
(3.12) ξ : ξ,T C2−2l, ξ,N C2−l,C−1 ξ,B C
ν ν ν
{ |h i|≤ |h i|≤ ≤|h i|≤ }
where C only depends on the constants in (3.1).
(ii) For j =0,1,2, and h =a (s, ) or b (s, )
ν k,l,ν k,l,ν
· ·
(3.13) T , jh C′22lj,
ν ν
h ∇i ≤
(3.14) (cid:12)(cid:12)(cid:0) N , (cid:1)jh(cid:12)(cid:12) C′2lj,
ν ν
h ∇i ≤
(3.15) (cid:12)(cid:12)(cid:0) B , (cid:1)jh (cid:12)(cid:12) C′.
ν ν
h ∇i ≤
(cid:12)(cid:0) (cid:1) (cid:12)
(iii) The statements analog(cid:12)ous to (i), (ii(cid:12)) hold true for a (s, ), b (s, ), with
k,ν k,ν
· ·
l replaced by [k/3]+1.
(iv)Ifh isanyofthemultipliersa (s, ),b (s, ), theenthestaetementsanal-
ν k,l,ν k,l,ν
· ·
ogous to (i)-(iii) hold for the multiplier h (ξ)2l γ′′(s),ξ . Similarly, if h denotes
ν ν
h i
any of a (s, ) or b (s, ) then h can be replaced by h 2l γ′′(s),ξ .
k,ν · k,ν · ν ν h i e
Proof. eTo see the ceontainment ofesuppak,l,ν(s, ) in theeset (3.12) we assume that
·
ξ =Ξ(r,u,σ) and expand B(σ), T(σ) about σ =s . Using the Frenet formulas for
ν
ξ in the support of a (s, ) we obtain
k,l,ν
·
ξ,T = rB(σ)+uT(σ),T =O((σ s )2)+O(u)=O(2−2l)
ν ν ν
h i h i −
and similarly ξ,N =O(2−l).
ν
h i
To show (3.13) we use the formulas
1
r =B, u=T, σ = N;
∇ ∇ ∇ uκ rτ
−