Sobolev improving for averages over curves in \mathbf{R^4}
David Beltran, Shaoming Guo, Jonathan Hickman, Andreas Seeger
aa r X i v : . [ m a t h . C A ] F e b SOBOLEV IMPROVING FOR AVERAGES OVER CURVES IN R DAVID BELTRAN, SHAOMING GUO, JONATHAN HICKMAN, AND ANDREAS SEEGER
Abstract.
We study L p -Sobolev improving for averaging operators A γ given by convolutionwith a compactly supported smooth density µ γ on a non-degenerate curve. In particular, in 4dimensions we show that A γ maps L p p R q the Sobolev space L p { p p R q for all 6 ă p ă 8 . Thisimplies the complete optimal range of L p -Sobolev estimates, except possibly for certain endpointcases. The proof relies on decoupling inequalities for a family of cones which decompose the wavefront set of µ γ . In higher dimensions, a new non-trivial necessary condition for L p p R n q Ñ L p { p p R n q boundedness is obtained, which motivates a conjectural range of estimates. Introduction
For n ě γ : I Ñ R n be a smooth curve, where I Ă R is a compact interval, and χ P C p R q be a bump function supported on the interior of I . Consider the averaging operator A γ f p x q : “ ˆ R f p x ´ γ p s qq χ p s q d s ; (1.1)in particular, A γ f “ µ γ ˚ f , where µ γ is the measure given by the push-forward of χ p s q d s under γ .The goal of this paper is to study sharp L p -Sobolev improving bounds for the operator A γ fora wide class of curves in R . To state the main theorem, we say a smooth curve γ : I Ñ R n is non-degenerate if there is a constant c ą | det p γ p s q , ¨ ¨ ¨ , γ p n q p s qq| ě c for all s P I (1.2)or, equivalently, the n ´ γ are all bounded away from 0. Theorem 1.1. If γ : I Ñ R is non-degenerate and ă p ă 8 , then } A γ f } L p { p p R q À p,γ,χ } f } L p p R q . This result is sharp up to p “ L p Ñ L p { p bound fails whenever 2 ď p ă L Ñ L { inequalityand duality give the complete range of L p Ñ L pα estimates for all 1 ď p ď 8 , except possibly forendpoint cases.In higher dimensions no L p Ñ L p { p estimates are currently known to hold for such averagingoperators, although it is natural to conjecture that the following holds. Conjecture 1. If γ : I Ñ R n is non-degenerate and n ´ ă p ă 8 , then } A γ f } L p { p p R n q À p,γ,χ } f } L p p R n q . (1.3)If true, then the above conjectured range would be sharp, viz. Proposition 1.2. If γ : I Ñ R n is non-degenerate, then (1.3) fails whenever ď p ă n ´ . Date : February 18, 2021. Throughout, any curve is tacitly assumed to be simple (that is, γ is injective) and regular ( γ is non-vanishing). In the euclidean plane Conjecture 1 is an elementary consequence of the decay of the Fouriertransform of the measure µ γ . In higher dimensions the problem is significantly more difficult, owingto the weaker rate of Fourier decay. The n “ p “ ℓ p -decoupling’ inequalityfor the light cone. The sharp decoupling inequality was later proved by Bourgain–Demeter [4],thus establishing the bounds for the averaging operators unconditionally. Theorem 1.1 verifies the n “ p “ n “ R rather than R . Furthermore, whilstdecoupling plays an important rˆole in the current paper, the analysis in [1] relies on square functionestimates. One useful feature of decoupling (as opposed to the use of square functions) is thatdecoupling inequalities are readily iterated. We make use of this fact in a fundamental way whendecomposing the operator with respect to the different frequency cones.1.1. Corollaries.
Theorem 1.1 has a number of consequences which follow immediately fromknown arguments.
Extension to finite type curves.
Using arguments from [17], one can show that Theorem 1.1 impliesbounds for a more general class of curves. We say a smooth curve γ : I Ñ R n is of finite maximaltype if there exists d P N and a constant c ą d ÿ j “ |x γ p j q p s q , ξ y| ě c | ξ | for all s P I , ξ P R n . (1.4)For fixed s , the smallest d for which (1.4) holds for some c ą type of γ at s . Thetype is an upper semicontinuous function, and the supremum of the types over all s P I is referredto as the maximal type of γ . Corollary 1.3. If γ : I Ñ R is of maximal type d P N and max t , d u ă p ă 8 , then } A γ f } L p { p p R q À p,γ,χ } f } L p p R q . This result is sharp up to endpoints (for further discussion of endpoint cases, see § § Endpoint lacunary maximal estimates.
For the measure µ γ introduced above, define the family ofdyadic dilates µ kγ for k P Z by x µ kγ , f y “ x µ γ , f p k ¨ qy In particular, the curve is no longer a Salem set.
OBOLEV IMPROVING FOR AVERAGES OVER CURVES 3 and consider the associated convolution operators A kγ f : “ µ kγ ˚ f . If γ is of finite maximal type,then a well-known and classical result (see, for instance, [9]) states that the associated lacunarymaximal function M γ f : “ sup k P Z | A kγ f | is bounded on L p for all 1 ă p ď 8 . A difficult problem is to understand the endpoint be-haviour of these operators near L . By an off-the-shelf application of the main theorem from [18],Corollary 1.3 implies an endpoint bound for M γ in the n “ Corollary 1.4. If γ : I Ñ R is of finite maximal type, then the lacunary maximal function M γ maps the (standard isotropic) Hardy space H p R q to L , p R q . In particular, by [18, Theorem 1.1], Corollary 1.4 follows from any L p Ñ L p { p bound for theassociated averaging operator A γ for 2 ď p ă 8 (that is, one does not require L p Ñ L p { p for thesharp range of p for this application). Note that, prior to this paper, no such bounds L p -Sobolevbounds were known for n ě
4; thus the question of the H p R n q to L , p R n q boundedness oflacunary maximal associated to finite maximal type (or even non-degenerate) curves remains openfor n ě Outline of the paper.
This paper is structured as follows: ‚ In § ‚ In § L p -Sobolev improving inequalities for our averagingoperators. In particular, we establish Proposition 1.2. ‚ In §§ ‚ In § ‚ There are three appendices which deal with various auxiliary results and technical lemmasused in the main argument.
Notational conventions.
Given a (possibly empty) list of objects L , for real numbers A p , B p ě p or dimension parameter n the notation A p À L B p , A p “ O L p B p q or B p Á L A p signifies that A p ď CB p for some constant C “ C L,p,n ě p and n . In addition, A p „ L B p is used to signify that both A p À L B p and A p Á L B p hold. Given a , b P R we write a ^ b : “ min t a, b u and a _ b : “ max t a, b u . The lengthof a multiindex α P N n is given by | α | “ ř ni “ α i . Acknowledgements.
The authors thank the American Institute of Mathematics for fundingtheir collaboration through the SQuaRE program, also supported in part by the National ScienceFoundation. D.B. was partially supported by NSF grant DMS-1954479. S.G. was partially sup-ported by NSF grant DMS-1800274. A.S. was partially supported by NSF grant DMS-1764295and by a Simons fellowship. This material is partly based upon work supported by the NationalScience Foundation under Grant No. DMS-1440140 while the authors were in residence at theMathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester.2.
Reduction to perturbations of the moment curve
A prototypical example of a smooth curve satisfying the non-degeneracy condition (1.2) is the moment curve γ ˝ : R Ñ R n , given by γ ˝ p s q : “ ´ s, s , . . . , s n n ! ¯ . Indeed, in this case the determinant appearing in (1.2) is everywhere equal to 1. Moreover, atsmall scales, any non-degenerate curve can be thought of as a perturbation of an affine image of
D. BELTRAN, S. GUO, J. HICKMAN, AND A. SEEGER γ ˝ . To see why this is so, fix a non-degenerate curve γ : I Ñ R n and σ P I , λ ą r σ ´ λ, σ ` λ s Ď I . Denote by r γ s σ the n ˆ n matrix r γ s σ : “ “ γ p q p σ q ¨ ¨ ¨ γ p n q p σ q ‰ , where vectors γ p j q p σ q are understood to be column vectors. Note that this is precisely the matrixappearing in the definition of the non-degeneracy condition (1.2) and is therefore invertible by ourhypothesis. It is also convenient to let r γ s σ,λ denote the n ˆ n matrix r γ s σ,λ : “ r γ s σ ¨ D λ , (2.1)where D λ : “ diag p λ, . . . , λ n q , the diagonal matrix with eigenvalues λ , λ , . . . , λ n . Consider theportion of the curve γ lying over the subinterval r σ ´ λ, σ ` λ s . This is parametrised by the map s ÞÑ γ p σ ` λs q for s P r´ , s . The degree n Taylor polynomial of s ÞÑ γ p σ ` λs q around σ is givenby s ÞÑ γ p σ q ` r γ s σ,λ ¨ γ ˝ p s q , (2.2)which is indeed an affine image of γ ˝ . Furthermore, by Taylor’s theorem, the original curve γ agrees with the polynomial curve (2.2) to high order at σ .Inverting the affine transformation x ÞÑ γ p σ q ` r γ s σ,λ ¨ x from (2.2), we can map the portion of γ over r σ ´ λ, σ ` λ s to a small perturbation of the moment curve. Definition 2.1.
Let γ P C n ` p I ; R n q be a non-degenerate curve and σ P I, λ ą be such that r σ ´ λ, σ ` λ s Ď I . The p σ, λ q -rescaling of γ is the curve γ σ,λ P C n ` pr´ , s ; R n q given by γ σ,λ p s q : “ r γ s ´ σ,λ ` γ p σ ` λs q ´ γ p σ q ˘ . It follows from the preceding discussion that γ σ,λ p s q “ γ ˝ p s q ` r γ s ´ σ,λ E γ,σ,λ p s q where E γ,σ,λ is the remainder term for the Taylor expansion (2.2). In particular, if γ satisfies thenon-degeneracy condition (1.2) with constant c , then } γ σ,λ ´ γ ˝ } C n ` pr´ , s ; R n q À c ´ λ } γ } nC n ` p I q . Thus, if λ ą γ σ,λ is a minor perturbationof the moment curve. In particular, given any 0 ă δ ă
1, we can choose λ so as to ensure that γ σ,λ belongs to the following class of model curves . Definition 2.2.
Given n ě and ă δ ă , let G n p δ q denote the class of all smooth curves γ : r´ , s Ñ R n that satisfy the following conditions:i) γ p q “ and γ p j q p q “ ~e j for ď j ď n ;ii) } γ ´ γ ˝ } C n ` pr´ , sq ď δ .Here ~e j denotes the j th standard Euclidean basis vector and } γ } C n ` p I q : “ max ď j ď n ` sup s P I | γ p j q p s q| for all γ P C n ` p I ; R n q . Given any γ P G n p δ q , condition ii) and the multilinearity of the determinant ensures thatdet r γ s s “ det r γ ˝ s s ` O p δ q “ ` O p δ q . Thus, there exists a dimensional constant c n ą ă δ ă c n , then any curve γ P G n p δ q is non-degenerate and, moreover, satisfies det r γ s s ě { δ ą ă δ ! A γ , the above observationsfacilitate a reduction to the class of model curves. To precisely describe this reduction, it is usefulto make the choice of cutoff function explicit in the notation by writing A r γ, χ s for the operator A γ as defined in (1.1). OBOLEV IMPROVING FOR AVERAGES OVER CURVES 5
Proposition 2.3.
Let γ : I Ñ R n be a non-degenerate curve, χ P C c p R q be supported on theinterior of I and ă δ ! . There exists some γ ˚ P G n p δ q and χ ˚ P C c p R q such that } A r γ, χ s} L p p R n qÑ L pα p R n q „ γ,χ,δ,p,α } A r γ ˚ , χ ˚ s} L p p R n qÑ L pα p R n q for all ď p ă 8 and ď α ď . Furthermore, χ ˚ may be chosen to satisfy supp χ ˚ Ď r´ δ, δ s .Proof. The proof follows by decomposing the domain of γ into small intervals and applying therescaling described in Definition 2.1 on each interval. This decomposition in s induces a decompo-sition of the derived operator p ´ ∆ q α { A r γ, χ s . The upper bound then follows from the triangleinequality and the stability of the estimates under affine transformation (together with a simplepigeonholing argument).The proof of the lower bound is more subtle since one must take into account possible cancel-lation between the different pieces of the decomposition. To get around this, we observe that } A r γ, χ s} L p p R n qÑ L pα p R n q À γ,χ } A r γ, χ s} L p p R n qÑ L pα p R n q (2.3)holds whenever χ , χ P C c p R q are supported in I and χ p s q “ s P supp χ . Once this isestablished, it is possible to localise in s and rescale to deduce the desired bound.To prove (2.3) note, after possibly applying a translation and a dilation, one may write A r γ, χ s f p x q “ ˆ R f p x ´ γ p s qq ˜ χ ˝ γ p s q χ p s q d s where the function ˜ χ P C c p R n q is supported in r´ π, π s n . Consequently, by performing a Fourierseries decomposition, ˜ χ ˝ γ p s q “ p π q n ÿ k P Z n a k e i x x,k y e ´ i x x ´ γ p s q ,k y where the sequence p a k q k P Z n of Fourier coefficients is rapidly decaying. Thus, if Mod k denotes themodulation operator Mod k g p x q : “ e i x x,k y g p x q , then A r γ, χ s f p x q “ p π q n ÿ k P Z n a k ¨ Mod k ˝ A r γ, χ s ˝ Mod ´ k f p x q . By analytic interpolation, it follows that } Mod k } L pα p R n qÑ L pα p R n q À p ` | k |q α for all 0 ď α ď } A r γ, χ s f } L pα p R n q À ÿ k P Z n | a k |p ` | k |q α ¨ } A r γ, χ s ˝ Mod ´ k f } L pα p R n q À γ,χ } A r γ, χ s} L p p R n qÑ L pα p R n q } f } L p p R n q , using the rapid decay of the Fourier coefficients. (cid:3) As a consequence of Proposition 2.3, it suffices to fix δ ą γ P G p δ q and supp χ Ď I : “ r´ δ , δ s . Thus, henceforth,we work with some fixed δ , chosen to satisfy the forthcoming requirements of the proofs. For thesake of concreteness, the choice of δ : “ ´ is more than enough for our purposes. D. BELTRAN, S. GUO, J. HICKMAN, AND A. SEEGER 1 p α n n ´ n ´ n ´ n ´ Figure 1.
Conjectured range of A γ : L p p R n q Ñ L pα p R n q boundedness for γ non-degenerate. The inner triangle follows from the elementary L estimate. The goalis to establish the L p p R n q Ñ L p { p p R n q bound at the ‘kink’ point p cr “ n ´ p cr “ n ´ n ´ ). 3. Necessary conditions
General L p Ñ L pα estimates. If γ : I Ñ R n is of maximal type d , then the van der Corputlemma shows that the Fourier transform of any smooth density µ γ on γ satisfies | ˆ µ γ p ξ q| À γ p ` | ξ |q ´ { d . (3.1)This readily implies that } A γ f } L { d p R n q À γ } f } L p R n q . (3.2)Consider the case where γ is non-degenerate, so that d “ n . By interpolating against (3.2),Conjecture 1 formally implies A γ maps L p to L pα for all p ě α ă α cr p p q : “ min ! n ´ ` p ¯ , p ) , (3.3)with the equality case also holding in the restricted range p ą n ´
2. It is an interesting questionwhat happens at the endpoint in the range 2 ă p ď n ´ α cr p p q agree precisely when p corresponds to thecritical Lebesgue exponent p cr : “ n ´ , which manifests as a ‘kink’ in the L p -Sobolev diagram.By a simple scaling argument (see, for instance, [17, pp.81-82]), Conjecture 1 further impliesbounds for A γ under a finite type hypothesis. In view of Corollary 1.3, it is reasonable to conjecturethe following. Conjecture 2. If γ : I Ñ R n is of maximal type d , then the operator A γ maps L p to L pα for all p ě and α ď α cr p d ; p q : “ min ! α cr p p q , d ) (3.4) with strict inequality if min t n ´ , d u ď p ď max t n ´ , d u . The range of conjectured bounds is represented in Figure 2
Remark.
Using the fact that p I ´ ∆ q α { : L pα ` β p R n q Ñ L pβ p R n q is an isomorphism together with aduality argument, any L p Ñ L pα estimate for A γ immediately implies a corresponding L p Ñ L p α estimate. OBOLEV IMPROVING FOR AVERAGES OVER CURVES 71 p α d n ´ n ´ n ´ d d d ´ n d n ´ n ´ p α d d d ´ d Figure 2.
Conjectured range of A γ : L p p R n q Ñ L pα p R n q boundedness for γ ofmaximal type d . The upper diagram corresponds to d ă n ´ d ě n ´ α ď { d is clearly necessary. Indeed, by duality and interpolation, any L p Ñ L pα estimate implies an L Ñ L α estimate for the same value of α . However, a slight refinement of(3.1) shows that the L estimate (3.2) is sharp in the sense that the regularity exponent on theleft-hand side cannot be taken larger than 1 { d .3.2. Band-limited examples.
The remainder of this section discusses the necessity of the con-ditions (3.3) and (3.4). To begin, given λ ą
0, consider the family of band-limited Schwartzfunctions Z λ : “ f P S p R n q : supp ˆ f Ă t ξ P ˆ R n : λ { ď | ξ | ď λ u ( . By elementary Sobolev space theory, the desired necessary conditions are a consequence of thefollowing proposition.
Proposition 3.1. If γ : I Ñ R n is a smooth curve satisfying the non-degeneracy hypothesis (1.2) and p ě , then sup } A γ f } L p p R n q : f P L p X Z λ , } f } L p p R n q “ ( Á p,γ λ ´ α cr p p q . This directly implies Proposition 1.2 and, moreover, shows that the p p, α q -ranges in (3.3) andConjecture 2 are optimal up to endpoints. Proposition 3.1 is based on testing the estimate against two examples, corresponding to the twoconstraints inherent in the minimum appearing in the definition of α crit p p q .3.3. Dimensional constraint: α ď { p . The condition α ď { p is well-known and appears tobe folkloric; in lieu of a precise reference, the details are given presently. Lemma 3.2. If γ : I Ñ R n is a smooth curve and p ě , then sup } A γ f } L p p R n q : f P L p p R n q X Z λ , } f } L p p R n q “ ( Á p,γ λ ´ { p . If γ is finite type curve, then the points s for which the type of γ at s is strictly larger than d are isolated.Consequently, any necessary condition for the non-degenerate problem is automatically a necessary condition forthe finite type problem. The necessity of the additional constraint α ď { d is discussed in the previous subsection. D. BELTRAN, S. GUO, J. HICKMAN, AND A. SEEGER
Proof.
Since the operator A γ is self-adjoint and commutes with frequency projections, given 1 ď p ď } A γ f } L p p R n q : f P L p p R n q X Z λ , } f } L p p R n q “ ( Á p,γ λ ´ { p . Fix β P C c p ˆ R n q with non-negative (inverse) Fourier transform ˇ β satisfying ˇ β p q “ β Ď t ξ P ˆ R n : 1 { ď | ξ | ď u . (3.5)In addition, let ψ P C c p R n q be non-zero, non-negative and supported in the unit ball centred atthe origin and with these bump functions define f λ : “ p β λ ´ q q ˚ ψ λ where β λ ´ p ξ q : “ β p λ ´ ξ q and ψ λ p x q : “ ψ p λx q . The condition (3.5) implies f P L p p R n q X Z λ ,whilst direct calculation shows that } f } L p p R n q „ λ ´ n { p . (3.6)By a simple computation, A γ f “ K λ ˚ ψ λ where K λ p x q : “ λ n ˆ R ˇ β ` λ p x ´ γ p s qq ˘ χ p s q d s. The key claim is that, provided 0 ă c ă λ ), thepointwise inequality K λ ˚ ψ λ p x q Á λ ´ for all x P N cλ ´ p γ q (3.7)holds, where N cλ ´ p γ q denotes the cλ ´ -neighbourhood of the curve t γ p s q : s P supp χ u . To see this, choose c sufficiently small so that ˇ β is bounded away from zero on a ball of radius 10 c centred at the origin. If x P N cλ ´ p γ q , then there exists some s P supp χ such that | x ´ y ´ γ p s q| ă cλ ´ whenever | s ´ s | À γ λ ´ and | y | ď cλ ´ ,from which (3.7) follows.Combining (3.6) and (3.7), one concludes thatsup f P L p p R n qX Z λ } A γ f } L p p R n q } f } L p p R n q Á λ ´ λ ´p n ´ q{ p λ ´ n { p “ λ ´ { p , as desired. (cid:3) Remark.
More generally, suppose A Σ is an averaging operator defined as in (1.1) but now withrespect to Σ a (regular parametrisation of a) surface in R n of arbitrary dimension. Thensup } A Σ f } L p p R n q : f P L p p R n q X Z λ , } f } L p p R n q “ ( Á p, Σ λ ´ dim Σ { p . This general necessary condition follows from the proof of Lemma 3.2 mutatis mutandis . Furthergeneralisations hold for appropriate classes of variable coefficient averaging operators: see, forinstance, [3].
OBOLEV IMPROVING FOR AVERAGES OVER CURVES 9
Fourier decay constraint: α ď n ` ` p ˘ . Establishing the second condition is a muchmore involved exercise. Here, in contrast with Lemma 3.2, the non-degeneracy hypothesis (1.2)plays a rˆole via certain refinements of the Fourier decay estimate (3.1).Recall the desired bound.
Proposition 3.3. If γ : I Ñ R n is a smooth curve satisfying the non-degeneracy hypothesis (1.2) and p ě , then sup } A γ f } L p p R n q : f P L p p R n q X Z λ , } f } L p p R n q “ ( Á p,γ λ ´ n p ` p q . This conclusion was shown in three dimensions by Oberlin and Smith [14] for the model exampleof the helix in R , t ÞÑ p cos t, sin t, t q , by using DeLeeuw’s restriction theorem and an analysis of aBessel multiplier in R . Here the more general statement in Proposition 3.3 is shown by combininga sharp example of Wolff [22] for ℓ p -decoupling inequalities with a stationary phase analysis of theFourier multiplier ˆ µ γ .The proof of Proposition 3.3 is broken into stages. The worst decay cone.
At any given large scale, the decay estimate (3.1) is only sharp for ξ belonging to a narrow region around a low-dimensional cone in the frequency space. To proveProposition 3.3, it is natural to test the L p -Sobolev estimate against functions which are Fouriersupported in a neighbourhood of this ‘worst decay cone’.By Proposition 2.3 we may assume without loss of generality that γ P G n p δ q for some small0 ă δ ! χ in the definition of A γ is supported in I “ r´ δ , δ s . In viewof the van der Corput lemma, the worst decay cone should correspond to the ξ for which thederivatives x γ p j q p s q , ξ y , 1 ď j ď n ´
1, all simultaneously vanish for some s P I . In order todescribe this region, first note that x γ p n ´ q p s q , ξ y “ BB s x γ p n ´ q p s q , ξ y ˇˇˇ s “ s ξ “ ξ “ p s , ξ q “ p , ~e n q , by the reduction γ p j q p q “ ~e j for 1 ď j ď n . Consequently, provided thesupport of χ is chosen sufficiently small, by the implicit function theorem and homogeneity thereexists a constant c ą θ : Ξ Ñ I , where Ξ : “ ξ “ p ξ , ξ n q P ˆ R n : | ξ | ď c | ξ n | ( , such that s “ θ p ξ q is the unique solution in I to the equation x γ p n ´ q p s q , ξ y “ ξ P Ξ.Note that θ is homogeneous of degree one.Further consider the system of n equations in n ` x γ p j q p s q , ξ y “ ď j ď n ´ ξ n “ . (3.8)Again, by the reduction γ p j q p q “ ~e j for 1 ď j ď n , this can be solved for suitably localised ξ usingthe implicit function theorem, expressing s , ξ , ... ξ n ´ as functions of ξ n ´ . Thus (3.8) holds ifand only if ξ i “ Γ i p ξ n ´ q , ď i ď n ´ ,s “ θ p Γ p ξ n ´ q , . . . , Γ n ´ p ξ n ´ q , ξ n ´ , q , (3.9a)for some smooth functions Γ i , i “ , . . . , n ´ i p q “
0. On I we form the R n -valuedfunction τ ÞÑ Γ p τ q with the first n ´ n ´ p τ q : “ τ, Γ n p τ q : “ . (3.9b)With this definition, the formulæ in (3.9a) can be succinctly expressed as ξ “ Γ p ξ n ´ q , s “ θ ˝ Γ p ξ n ´ q . Moreover, the ‘worst decay cone’ can then be defined as the cone generated by the curve Γ, givenby C : “ λ Γ p τ q : λ ą τ P I ( . Remark.
For the model case γ p s q “ ř ni “ s i i ! e i one may explicitly compute that Γ p τ q “ ř ni “ p´ τ q n ´ i p n ´ i q ! ~e i . The Wolff example revisited.
In analogy with the example in [22], here we consider functions withFourier support on a union of balls with centres lying on the worst decay cone C . To this end,let ε ą N ε p λ q : “ Z X t s P R : | s | ď ελ { n u . The centres of the aforementioned balls are then given by ξ ν : “ λ Γ p νλ ´ { n q for all ν P N ε p λ q . (3.10)Fix η P C c p ˆ R n q satisfying η p ξ q “ | ξ | ď { η p ξ q “ | ξ | ě
1. Let 0 ă ρ ă g ν for ν P N ε p λ q via the Fourier transform byˆ g ν p ξ q : “ η ` λ ´ { n ρ ´ p ξ ´ ξ ν q ˘ . (3.11)We consider randomised sums of the functions (3.11). In particular, set g ω p x q : “ ÿ ν P N ε p λ q r ν p ω q g ν p x q for ω P r , s , (3.12)where t r ν u ν “ is the sequence of Rademacher functions. We claim ´ ˆ } g ω } pL p p R n q d ω ¯ { p „ λ ´ p ` n . (3.13)To prove this we apply Fubini’s theorem and Khinchine’s inequality (see, for instance, [19, Ap-pendix D]) to see that the left hand side is (3.13) is equal to ›››´ ˆ | g ω | p d ω ¯ { p ››› L p p R n q „ ›››´ ÿ ν P N ε p λ q | g ν | ¯ { ››› L p p R n q . The right-hand side of the last display is equal to ›››´ ÿ ν P N ε p λ q | λρ n ˇ η p λ { n ρ ¨ q| ¯ { ››› L p p R n q “ “ N ε p λ q ‰ { } λρ n ˇ η p λ { n ρ ¨ q} L p p R n q „ λ n ` ´ p , which yields (3.13). The above estimates depend on ρ , but since this parameter is chosen to bea dimensional constant (independently of λ ) this dependence is suppressed. Also note that so farthe argument is independent of the choice of the ξ ν . Asymptotics.
The next step is to study the behaviour of the multiplier ˆ µ γ near the support of theˆ g ν . The key result is Lemma 3.4 below, which relies on the asymptotics of ˆ µ γ near the worst decaycone and the observation that the functions ˆ g ν with the choice of ξ ν as in (3.10) are supportednear that cone.Set ˆ g ` ,ν p ξ q : “ η ` p λ ´ { n ρ ´ p ξ ´ ξ ν qq where η ` P C c p p R n q is such that η ` p ξ q “ | ξ | ď η ` p ξ q “ | ξ | ą {
2, so that ˆ g ν “ ˆ g ` ,ν ¨ ˆ g ν . Let φ p ξ q : “ x γ ˝ θ p ξ q , ξ y . (3.14) OBOLEV IMPROVING FOR AVERAGES OVER CURVES 11
Lemma 3.4. If ε , ρ ą are chosen sufficiently small, then for all λ ě and ν P N ε p λ q theidentity ˆ µ γ p ξ q “ e ´ iφ p ξ q m p ξ q holds on supp ˆ g ` ,ν wherei) | m p ξ q| Á λ ´ { n for ξ P supp ˆ g ` ,ν ;ii) The function a ν : “ m ´ ¨ ˆ g ` ,ν satisfies |B αξ a ν p ξ q| ď C α λ p ´| α |q{ n for all α P N n . The proof, which is based on the stationary phase method and, in particular, oscillatory integralestimates from [6], is postponed until § Lower bounds for the operator norm.
For each ν P N ε p λ q define f ν byˆ f ν p ξ q : “ ˆ g ν p ξ q ˆ µ γ p ξ q and consider the randomised sums f ω p x q : “ ÿ ν P N ε p λ q r ν p ω q f ν p x q for ω P r , s . Note that, by Lemma 3.4, the f ω are well-defined smooth function with compact support (andwith bounds depending on λ ). Furthermore, if g ω is the function defined in (3.12), then g ω “ A γ f ω . (3.15)We proceed to estimate the L p norm of f ω , uniformly in ω .We have f ν “ K ν ˚ g ν where the kernel K ν is given by K ν p x q : “ p π q n ˆ ˆ R n e i x x,ξ y ˆ µ γ p ξ q ´ ˆ g ` ,ν p ξ q d ξ. By Lemma 3.4 one may write ˆ µ γ p ξ q ´ ˆ g ` ,ν p ξ q “ e iφ p ξ q a ν p ξ q where a ν is supported where | ξ ´ ξ ν | ď ρλ { n and satisfies B αξ a ν p ξ q “ O p λ p ´| α |q{ n q . Setting E ν p ξ q : “ φ p ξ q ´ φ p ξ ν q ´ xB ξ φ p ξ ν q , ξ ´ ξ ν y ,x ν : “ ´B ξ φ p ξ ν q , (3.16)it follows that ˆ µ γ p ξ q ´ ˆ g ` ,ν p ξ q “ e iφ p ξ ν q e ´ i x x ν ,ξ ´ ξ ν y e i E ν p ξ q a ν p ξ q . Applying a change of variable, K ν p x q “ e i px x,ξ ν y` φ p ξ ν qq λ p π q n ˆ ˆ R n e i x λ { n p x ´ x ν q ,ξ y e i E ν p ξ ν ` λ { n ξ q a ν p ξ ν ` λ { n ξ q d ξ. By the homogeneity of φ , it follows that |B αξ e i E ν p ξ ν ` λ { n ξ q | À α α P N n . Onthe other hand, by Lemma 3.4 implies that ˆ ˆ R n |B αξ a ν p ξ ν ` λ { n ξ q| d ξ À α λ { n for all α P N n .Repeated integration-by-parts therefore yields | K ν p x q| À N λ p n ` q{ n p ` λ { n | x ´ x ν |q ´ N for all N P N n . Consequently, the pointwise inequality | f ν p x q| À λ p n ` q{ n p ` λ { n | x ´ x ν |q ´ N À λ p n ` q{ n ÿ ℓ ě ´ ℓN B νℓ,λ p x q holds with B νℓ,λ : “ t x P R n : | x ´ x ν | ď ℓ λ ´ { n u . Hence, } f ω } L p p R n q À λ p n ` q{ n ÿ ℓ ě ´ ℓN ››› ÿ ν P N ε p λ q B νℓ,λ ››› L p p R n q for all ω P r , s . (3.17)To estimate the terms of (3.17) for 2 ℓ ą ελ { n use the immediate bound ››› ÿ ν P N ε p λ q B νℓ,λ ››› L p p R n q À N ε p λ q ¨ ℓn { p λ ´ { p À ℓn { p λ { n ´ { p . For 2 ℓ ď ελ { n this may be improved upon using a separation property of the x ν : namely, | x ν ´ x ν | Á λ ´ { n | ν ´ ν | , (3.18)provided the parameter ε ą λ ). The property(3.18) implies that the balls B νℓ,λ , B ν ℓ,λ are disjoint for | ν ´ ν | Á ℓ . Assuming (3.18) for a momentand taking 2 ℓ ď ελ { n , we obtain ››› ÿ ν P N ε p λ q B νℓ,λ ››› L p p R n q ď ℓ ´ ÿ i “ ››› ÿ m P Z | m |ď ε ´ ℓ λ { n B ℓm ` iℓ,λ ››› L p p R n q À ℓ ´ ÿ i “ ´ ÿ m P Z | m |ď ε ´ ℓ λ { n ›› B ℓm ` iℓ,λ ›› pL p p R n q ¯ { p À ℓ p λ { n ´ ℓ q { p p ℓ λ ´ { n q n { p . Applying the preceding bounds to estimate the terms in (3.17) and choosing N ą n ´ n { p , thisleads to the uniform estimate sup ω Pr , s } f ω } L p p R n q À λ p n ` q{ n ´p n ´ q{ np . (3.19)Thus, one concludes from (3.15), (3.13) and (3.19), together with the fact that the f ω are Fouriersupported where | ξ | „ λ , thatsup f P L p X Z λ } A γ f } L p p R n q } f } L p p R n q ě ` ´ } A γ f ω } pL p p R n q d ω ˘ { p sup ω Pr , s } f ω } L p p R n q “ p ´ } g ω } pL p p R n q d ω ˘ { p sup ω Pr , s } f ω } L p p R n q Á λ ´ n p ` p q , which is the desired bound stated in Proposition 3.3.It remains to verify the crucial separation property (3.18). Recall from (3.16) and (3.10) that x ν “ ´B ξ φ ` λ Γ p νλ ´ { n q ˘ . Thus, by homogeneity, one wishes to bound x ν ´ x ν “ ´ “ pB ξ φ q ˝ Γ p νλ ´ { n q ´ pB ξ φ q ˝ Γ p ν λ ´ { n q ‰ . (3.20)In particular, it suffices to show that ddτ pB ξ φ q ˝ Γ p τ q “ ´ ~e ` O p τ q . (3.21)Indeed, applying Taylor’s theorem to (3.20) and using (3.21) to bound the linear term yields x ν ´ x ν “ ν ´ ν λ { n ¨ ~e ` O ´ p| ν |`| ν |q| ν ´ ν | λ { n ¯ “ ν ´ ν λ { n ¨ ~e ` O ´ ε | ν ´ ν | λ { n ¯ for all ν, ν P N ε p λ q . Choosing ε ą OBOLEV IMPROVING FOR AVERAGES OVER CURVES 13
Turning to the proof of (3.21), we have B ξ φ p ξ q “ γ ˝ θ p ξ q ` x γ ˝ θ p ξ q , ξ yB ξ θ p ξ q by the definitionof φ from (3.14). Since x γ ˝ θ p ξ q , ξ y “ ξ “ Γ p τ q , this yields pB ξ φ q ˝ Γ p τ q “ γ ˝ θ p Γ p τ qq and, consequently, ddτ pB ξ φ q ˝ Γ p τ q “ γ ˝ θ p Γ p τ qq ¨ xB ξ θ p Γ p τ qq , Γ p τ qy . By the initial reductions, γ p j q p q “ ~e j for 1 ď j ď n , and so ddτ pB ξ φ q ˝ Γ p τ q “ xB ξ θ p Γ p qq , Γ p qy ¨ ~e ` O p τ q . (3.22)Thus, to prove (3.21) it suffices to show that the inner product in the above display is equal to ´
1. Differentiating the defining equation x γ p n ´ q ˝ θ p ξ q , ξ y “
0, one deduces that B ξ θ p ξ q “ ´ x γ p n q ˝ θ p ξ q , ξ y γ p n ´ q ˝ θ p ξ q . Since, by uniqueness in (3.9a) and (3.9b) together with the initial reductions, Γ p q “ e n and θ p ~e n q “
0, it follows that pB ξ θ q ˝ Γ p q “ ´ ~e n ´ . On the other hand, from (3.9b) it is clear that x ~e n ´ , Γ p qy “
1. Applying these observations to the formula in (3.22) concludes the proof.3.5.
Proof of Lemma 3.4.
It remains to prove Lemma 3.4. To this end, we recall an asymptoticexpansion from [6], based on the following formula: ˆ e iλs n d s “ α n λ ´ { n for k “ , , . . . and λ ą
0, (3.23)where α n is given by α n : “ n Γ p n q sin p p n ´ q π n q if n is odd , n Γ p n q exp p i π n q if n is even . (3.24)The derivation of (3.23) relies on contour integration arguments, whilst the formula itself yieldsasymptotic expansions for integrals ´ R e iλs n χ p s q d s with χ P C : see, for instance, [20, VIII.1.3]or [13, § s ÞÑ s n , as demonstrated by the following lemma proved in [6]. We use the notation } g } C m p I q : “ max ď j ď m sup x P I | g p j q p x q| . Lemma 3.5 ([6], Lemma 5.1) . Let ă r ď , I “ r´ r, r s , I ˚ “ r´ r, r s and let g P C p I ˚ q .Suppose that r ď p ` } g } C p I ˚ q q and let η P C c p R q be supported in I and satisfy the bounds } η } ` } η } ď A , and } η } ď A . Let n ě , define I λ p η, w q : “ ˆ R η p s q exp ` iλ p n ´ ÿ j “ w j s j ` s n ` g p s q s n ` q ˘ d s and let α n be as in (3.24) . Suppose | w j | ď δλ p j ´ n q{ n , j “ , . . . , n ´ . Then there is an absoluteconstant C such that, for λ ą , | I λ p η, w q ´ η p q α n λ ´ { n | ď C r A δλ ´ { n ` A λ ´ { n p ` β n log λ qs ; here β : “ and β n : “ for n ą . Lemma 3.4 is obtained via a fairly direct application of the above result.
Proof (of Lemma 3.4).
Taylor expand the phase x γ p s q , ξ y with s “ θ p ξ q ` h to obtain x γ p θ p ξ q ` h q , ξ y “ φ p ξ q ` n ÿ j “ u j p ξ q h j j ! ` u n ` p ξ, h q h n ` p n ` q !where u j p ξ q : “ x γ p j q ˝ θ p ξ q , ξ y , for 1 ď j ď n , u n ` p ξ, h q : “ ˆ p n ` qp ´ t q n x γ p n ` q p θ p ξ q ` th q , ξ y d t. Recall that u n ´ p ξ q ” θ p ξ q , whilst u n p ξ q „ | ξ | „ λ for | ξ | ď cξ n . Thus,writing ˆ µ γ p ξ q “ e ´ iφ p ξ q m p ξ q as in the statement of the lemma, it follows that the function m is given by m p ξ q : “ ˆ R e ´ i p ř nj “ u j p ξ q hjj ! ` u n ` p h,ξ q hn ` p n ` q ! q χ p θ p ξ q ` h q d h. Thus, defining Ψ p ξ, h q : “ n ´ ÿ j “ w j p ξ q h j ` h n ` g p ξ, h q h n ` (3.25)where w j p ξ q : “ j ! ¨ u j p ξ q u n p ξ q and g p ξ, h q : “ p n ` q ! ¨ u n ` p ξ, h q u n p ξ q , one may succinctly express m as m p ξ q “ ˆ R e ´ iu n p ξ q Ψ p ξ,h q χ p θ p ξ q ` h q d h. We now turn to proving the bounds on m stated in Lemma 3.4. i) The desired pointwise lower bound on m follows from a direct application of Lemma 3.5. Inparticular, by the definition of the ξ ν we have w j p ξ ν q “ ď j ď n ´ | w j p ξ q| À ρλ p ´ n q{ n and | g p ξ, h q| À ξ P supp ˆ g ν, ` . (3.26)Thus, provided ρ and ε are chosen small enough, Lemma 3.5 can be applied to show that | m p ξ q| Á λ ´ { n for all ξ P supp ˆ g ` ,ν , (3.27)as desired. ii) It remains to show that |B αξ “ m ´ ¨ ˆ g ` ,ν ‰ p ξ q ˇˇ À α λ p ´| α |q{ n for all α P N n . (3.28)The derivative B αξ m p ξ q can be expressed as a sum of functions of the form m ακ,d p ξ q : “ ˆ R e ´ i p ř nj “ u j p ξ q hjj ! ` u n ` p h,ξ q hn ` p n ` q ! q h κ b αd p ξ, h q d h, d ` κ ě | α | , where b αd P C p ˆ R n zt u ˆ R q and homogeneous of degree ´ d in the ξ -variable. The key claim is | m ακ,d p ξ q| À α λ ´p ` κ q{ n ´ d for ξ P supp ˆ g ` ,ν . (3.29)Indeed, once (3.29) is established it can be combined with (3.27) and the Leibniz rule the deducethe desired bound (3.28). OBOLEV IMPROVING FOR AVERAGES OVER CURVES 15
The asserted bound (3.29) follows from ˇˇˇ ˆ R e ´ iu n p ξ q Ψ p ξ,h q χ p ξ, h q h κ d h ˇˇˇ À λ ´p ` κ q{ n , (3.30)where χ P C p ˆ R n zt u ˆ R q is homogeneous of degree zero with respect to ξ and vanishes unless | h | À ε . To prove (3.30) we form a dyadic decomposition of the integral. Fix ζ P C p R q suchthat ζ p h q “ | h | ă { ζ Ď r´ , s . For ℓ P N set ζ ℓ p h q “ ζ p ´ ℓ h q ´ ζ p ´ ℓ ´ h q and define J ℓ,λ p ξ q : “ ˆ R e ´ iu n p ξ q Ψ p ξ,h q ζ ℓ p λ { n h q χ p ξ, h qq h κ d h. (3.31)By just a size estimate we have | J ℓ,λ p ξ q| À p ℓ λ ´ { n q ` κ , which we use for ℓ ď C . For larger ℓ weuse integration-by-parts.Recall that the assumption ξ P supp ˆ g ` ,ν implies the bounds (3.26). Consequently, on thesupport of the integrand in (3.31), the dominant term in the formula for Ψ as given in (3.25) is h n . Moreover, ˇˇˇ BB h Ψ p ξ, h q ˇˇˇ „ p ℓ λ ´ { n q n ´ and, similarly, ˇˇˇ B i B h i Ψ p ξ, h q ˇˇˇ À min tp ℓ λ ´ { n q n ´ i , u . Also, it is not difficult to show that ˇˇˇ B i B h i “ ζ ℓ p λ { n h q χ p ξ, h qq h κ ‰ˇˇˇ À p ℓ λ ´ { n q κ ´ i . Using these bounds we derive, by N -fold integration-by-parts, | J ℓ,λ p ξ q| À N p ℓ λ ´ { n q ` κ ´ ℓnN and, by summing in ℓ , obtain (3.30). (cid:3) The Christ example.
We close this section by making an observation regarding an endpointcase. We may rule out L d p R n q Ñ L d { d p R n q boundedness under the maximal type d hypothesis for d ě
3. Note that this corresponds to the critical vertices in the lower diagram in Figure 2. Toshow the failure of the estimate, suppose that for some t with χ p t q ‰
0, there is a unit vector u with x u, γ p k q p t qy “ k “ , . . . , d ´ x u, γ p d q p t qy ‰
0. By a rotation we can assume that γ p t q “ ~e and u “ ~e , the standard coordinate vectors. The L d p R n q Ñ L d { d p R n q boundedness isequivalent with the statement that the multiplier | ξ | { d υ p ξ q ˆ R e i x γ p t q ,ξ y χ p t q d t belongs to the multiplier class M d p R n q ; here υ P C , equal to 1 for large ξ and vanishing in aneighborhood of the origin. Since p ξ ` ξ q { d | ξ | ´ { d belongs to M p p R n q for 1 ă p ă 8 we mayreplace in the display | ξ | { d with p ξ ` ξ q { d . Now apply the theorem by de Leeuw [8] on therestriction of multipliers to subspaces to see that p ξ ` ξ q { d υ p ξ , ξ , , . . . , q ˆ χ p t q e i p γ p t q ξ ` γ p t q ξ q d t is a multiplier in M d p R q which implies the L d p R q Ñ L d { d p R q boundedness of the averagingoperator associated to the plane curve p γ p t q , γ p t qq . However the latter statement can be disprovedby using the argument of Christ [7], who considered the curve p t, t d q . Initial reductions and auxiliary results
The remainder of the paper deals with the proof of Theorem 1.1. This section contains somepreliminary results, the most significant of which is the decoupling result in Theorem 4.4 whichlies at the heart of the proof.4.1.
Multiplier notation.
From the reduction described in Proposition 2.3 it suffices to consider γ P G p δ q where δ is a small parameter, as described at the end of §
2. If f belongs to a suitable a priori class, then the Fourier transform of A γ f is the product of ˆ f and the multiplierˆ µ γ p ξ q “ ˆ R e ´ i x γ p s q ,ξ y χ p s q d s. (4.1)Recall, again from the reduction described in Proposition 2.3, that we may assume χ P C c p R q satisfies supp χ Ď I “ r´ δ , δ s .Given m P L p ˆ R q , define the associated multiplier operator m p D q by m p D q f p x q : “ p π q ˆ ˆ R e i x x,ξ y m p ξ q ˆ f p ξ q d ξ so that, in this notation, A γ “ ˆ µ γ p D q . We also define the associated L p multiplier norms } m } M p p R q : “ } m p D q} L p p R qÑ L p p R q for 1 ď p ď 8 .To prove Theorem 1.1, we analyse various multipliers obtained by decomposing (4.1). To thisend, given a P C p ˆ R zt u ˆ R q , define m r a sp ξ q : “ ˆ R e ´ i x γ p s q ,ξ y a p ξ ; s q χ p s q d s. (4.2)Any decomposition of the symbol a results in a corresponding decomposition of the multiplier.4.2. Reduction to band-limited functions.
Given a symbol a P C p ˆ R zt u ˆ R q we performa dyadic decomposition in the frequency variable ξ as follows. Fix η P C c p R q non-negative andsuch that η p r q “ r P r´ , s and supp η Ď r´ , s and define β k P C c p R q by β k p r q : “ η p ´ k r q ´ η p ´ k ` r q (4.3)for each k P Z . By a slight abuse of notation we also let η , β k P C c p ˆ R q denote the functions η p ξ q : “ η p| ξ |q and β k p ξ q : “ β k p| ξ |q . One may then decompose a “ ÿ k “ a k where a k p ξ ; s q : “ " a p ξ ; s q ¨ β k p ξ q for k ě a p ξ ; s q ¨ η p ξ q for k “ . (4.4)Theorem 1.1 is a direct consequence of the following result for multipliers localised to somedyadic frequency band. Here we work with additional absolute constants 0 ă δ j ď δ for 1 ď j ď δ j : “ δ for j “ , δ : “ δ . It is also convenient to define δ : “ { Theorem 4.1.
Let γ P G p δ q and ď J ď . Suppose that a P C p ˆ R zt u ˆ R q satisfies |B αξ B Ns a p ξ ; s q| À α,N | ξ | ´| α | for all α P N and N P N (4.5) and $&% inf s P I |x γ p J q p s q , ξ y| ě δ J | ξ | inf s P I |x γ p j q p s q , ξ y| ď δ j | ξ | for ď j ď J ´ for all ξ P supp ξ a . (4.6) OBOLEV IMPROVING FOR AVERAGES OVER CURVES 17 If a k is defined as in (4.4) , then } m r a k s} M p p R q À p ´ k { p (4.7) for k ě and p ą max t p J ´ q , u . The hypothesis (4.6) implies that |x γ p j q p s q , ξ y| ď δ | ξ | for all ξ P supp a , s P I and 1 ď j ď J ´
1. (4.8)Indeed, suppose s P I realises the infimum in (4.6) and let s P I . Then the mean value theoremimplies |x γ p j q p s q , ξ y| ď |x γ p j q p s q , ξ y| ` sup t P I | γ p j ` q p t q|| s ´ s || ξ | ď δ | ξ | , (4.9)using the fact that δ j ď δ and the uniform derivative bounds for γ P G p δ q . Proof of Theorem 1.1 given Theorem 4.1.
By the reduction from § γ P G p δ q and χ P C c p R q with supp χ Ď I “ r´ δ , δ s . For 1 ď j ď E j : “ ! ξ P S : inf s P I |x γ p j q p s q , ξ y| ă δ j ) and U j : “ N δ j E j X S if 1 ď j ď N δ E j X S if j “ , where N δ j E j denotes the δ j -neighbourhood of E j and S denotes the unit sphere in ˆ R . Since U j is an open subset of S containing the compact subset clos E j , there exists a smooth function ρ j : S Ñ r , such that ρ j p ω q “ ω P clos E j and supp ρ j Ď U j . For 1 ď J ď χ J P C p S q by χ J : “ ´ J ´ ź j “ ρ j ¯ ¨ p ´ ρ J q These functions satisfy the following properties:i) If ξ P S and ξ P supp χ J , then (4.6) holds;ii) ř J “ χ J ”
1, as functions on S .Indeed, to see property i), note that if ξ P supp χ J , then ξ R clos E J which implies the first boundin (4.6). On the other hand, for 1 ď j ď J ´ ď ξ P U j and so there exists some ξ P E j with | ξ ´ ξ | ă δ j . Consequently, there exists some s P I such that |x γ p j q p s q , ξ y| ď |x γ p j q p s q , ξ y| ` | γ p j q p s q|| ξ ´ ξ | ď δ j , (4.10)which is the second bound in (4.6). For property ii), note that (4.9) can be combined with theargument in (4.10) to conclude thatsup s P I |x γ p j q p s q , ξ y| ď δ for ξ P U j , 1 ď j ď
3, and sup s P I |x γ p q p s q , ξ y| ď ` δ for ξ P U .Provided δ is sufficiently small, the non-degeneracy of γ P G p δ q implies Ş j “ U j “ H . Since ř J “ χ J “ ´ ś j “ ρ j , property ii) follows from the support conditions of the ρ j .In view of the above, we may apply Theorem 4.1 with a p ξ q : “ χ J p ξ {| ξ |q for J “ , , , J to conclude that } β k p D q A γ f } L p p R q À ´ k { p } f } L p p R q for all k ě p ą
6; note that the case k “ A γ is an averaging operator. Topass from the frequency localised estimates (4.11) to genuine L p -Sobolev bounds, one may applya Calder´on–Zygmund estimate from [16]. This argument is described in the Appendix A: seeProposition A.2. (cid:3) The hypothesis (4.5) implies that } F ´ ξ a k p ¨ ; s q} L p R q À
1, where F ´ ξ denotes the inverseFourier transform in the ξ variable. Consequently, it is not difficult to show that the p “ 8 caseof (4.7) holds for all 1 ď J ď p near to max t p J ´ q , u .Note that the proof of the J “ J “ γ P G p δ q and a P C p R zt uq satisfies the hypotheses Theorem 4.1 for J “
2, with δ : “ δ and δ : “ δ . Note,in particular, that |x γ p s q , ξ y| ě δ | ξ | for all p ξ ; s q P supp a ˆ I .Thus, the van der Corput lemma (see, for instance, [20, Chapter VIII, Proposition 2]) implies } m r a k s} M p R q “ } m r a k s} L p R q À ´ k { . On the other hand, by the triangle inequality, Fubini’s theorem, translation-invariance and integration-by-parts (see Lemma C.2), } m r a k s} M p R q ď } F ´ a k } L p R q À . Interpolation yields } m r a k s} M p p R q À ´ k { p for all 2 ď p ď 8 ,which concludes the proof for the J “ J “ J “ J “ The Frenet frame.
At this juncture it is convenient to recall some elementary conceptsfrom differential geometry which feature in our proof. Given a smooth non-denegenate curve γ : I Ñ R n , the Frenet frame is the orthonormal basis resulting from applying the Gram–Schmidtprocess to the vectors t γ p s q , . . . , γ p n q p s qu , which are linearly independent in view of the condition (1.2). Defining the functions ˜ κ j p s q : “ x e j p s q , e j ` p s qy for j “ , . . . , n ´ , one has the classical Frenet formulæ e p s q “ ˜ κ p s q e p s q , e i p s q “ ´ ˜ κ i ´ p s q e i ´ p s q ` ˜ κ i p s q e i ` p s q , i “ , . . . , n ´ , e n p s q “ ´ ˜ κ n ´ p s q e n ´ p s q . Repeated application of these formulæ shows that e p k q i p s q K e j p s q whenever 0 ď k ă | i ´ j | . Consequently, by Taylor’s theorem |x e i p s q , e j p s qy| À γ | s ´ s | | i ´ j | for 1 ď i, j ď n and s , s P I . (4.12) The choice of δ , δ is not important for the argument in the J “ Note that the ˜ κ j depend on the choice of parametrisation and only agree with the (geometric) curvature functions κ j p s q : “ x e j p s q , e j ` p s qy| γ p s q| if γ is unit speed parametrised. Here we do not assume unit speed parametrisation. OBOLEV IMPROVING FOR AVERAGES OVER CURVES 19
Furthermore, one may deduce from the definition of t e j p s qu nj “ that |x γ p i q p s q , e j p s qy| À γ | s ´ s | p j ´ i q_ for 1 ď i, j ď n and s , s P I . (4.13)In this paper, much of the microlocal geometry of the operator A γ is expressed in terms of theFrenet frame.4.4. A decoupling inequality for regions defined by the Frenet frame.
Let γ : I Ñ R n bea non-degenerate curve. Definition 4.2.
Given ď d ď n ´ and ă r ď , for each s P I let π d ´ p s ; r q denote the setof all ξ P ˆ R n satisfying the following conditions: |x e j p s q , ξ y| ď r d ` ´ j for ď j ď d , (4.14a)1 { ď |x e d ` p s q , ξ y| ď |x e j p s q , ξ y| ď for d ` ď j ď n . (4.14c) Such sets π d ´ p s ; r q are referred to as p d ´ , r q -Frenet boxes. Definition 4.3.
A collection P d ´ p r q of p d ´ , r q -Frenet boxes is a Frenet box decomposition for γ if it consists of precisely the p d ´ , r q -Frenet boxes π d ´ p s ; r q for s varying over a r -net in I . In some instances it is useful to highlight the underlying curve and write π d ´ ,γ p s ; r q for π d ´ p s ; r q . The relevance of the d ´ Theorem 4.4.
Let ď d ď n ´ , ď δ ! , ă r ď and P d ´ p r q be a p d ´ , r q -Frenet boxdecomposition for γ P G n p δ q . For all ď p ď 8 and ε ą the inequality ›› ÿ π P P d ´ p r q f π ›› L p p R n q À n,γ,ε r ´ α p p q´ ε ´ ÿ π P P d ´ p r q } f π } pL p p R n q ¯ { p holds with exponent α p p q : “ $&% ´ p if ď p ď d p d ` q ´ d p d ` q` p if d p d ` q ď p ď 8 for any tuple of functions p f π q π P P d ´ p r q satisfying supp ˆ f π Ď π . This theorem corresponds to a conic version of the Bourgain–Guth–Demeter decoupling inequal-ity for the moment curve [5]. Theorem 4.4 can be deduced from the moment curve decoupling viarescaling and induction-on-scale arguments, following a scheme originating in [17]. The details ofthis argument are presented in § The proof of Theorem 4.1: The J “ case We now turn to the proof of Theorem 1.1 proper. Recall, it remains to prove the J “ J “ J “ J “ § Preliminaries.
Suppose γ P G p δ q and a P C p ˆ R zt u ˆ R q satisfies the hypotheses Theo-rem 4.1 for J “
3, with δ : “ : δ : “ δ and δ : “ δ . Note, in particular, that |x γ p q p s q , ξ y| ě δ | ξ ||x γ p j q p s q , ξ y| ď δ | ξ | for j “ , p ξ ; s q P supp ξ a ˆ I , (5.1)as a consequence of (4.8). If a k : “ a ¨ β k , as introduced in § } m r a k sp ξ q| À ´ k { . (5.2)Arguing as for J “
2, Plancherel’s theorem and interpolation with a trivial L estimate yields } m r a k s} M p p R q À ´ k { p for all 2 ď p ď 8 .In order to obtain the improved bound } m r a k s} M p p R q À ´ k { p , we decompose the symbol a k intolocalised pieces which admit more refined decay rates than (5.2).5.2. Geometry of the slow decay cone.
The first step is to isolate regions of the frequencyspace where the multiplier m r a s decays relatively slowly. Owing to stationary phase considerations,this corresponds to a region around the coneΓ : “ ! ξ P supp ξ a : x γ p j q p s q , ξ y “ , j “ , , for some s P I ) . To analyse this region, and the corresponding decay rates for m r a s , we make the following simpleobservation. Lemma 5.1. If ξ P supp a , then the equation x γ p s q , ξ y “ has a unique solution in ¨ I . The above lemma follows from the localisation of the symbol in (5.1) and (4.6) via the meanvalue theorem. The details are left to the interested reader (see [1, Lemma 6.1] for a proof usingsimilar arguments).Using Lemma 5.1, we construct a smooth mapping θ : supp ξ a Ñ r´ , s such that x γ ˝ θ p ξ q , ξ y “ ξ P supp ξ a .It is easy to see that θ is homogeneous of degree 0. This function can be used to construct anatural Whitney decomposition with respect to the cone Γ defined above. In particular, let u p ξ q : “ x γ ˝ θ p ξ q , ξ y for all ξ P supp ξ a ; (5.3)This quantity plays a central rˆole in our analysis. If u p ξ q “
0, then ξ P Γ and so, roughly speaking, u p ξ q measures the distance of ξ from Γ.5.3. Decomposition of the symbols.
Consider the frequency localised symbols a k : “ a ¨ β k , asintroduced in § a k with respect to the size of | u p ξ q| . In particular, write a k “ t k { u ÿ ℓ “ a k,ℓ (5.4)where t k { u denotes the greatest integer less than or equal to k { a k,ℓ p ξ ; s q : “ a k p ξ ; s q β ` ´ k ` ℓ u p ξ q ˘ if 0 ď ℓ ă t k { u a k p ξ ; s q η ` ´ k ` t k { u u p ξ q ˘ if ℓ “ t k { u . (5.5) The β function should be defined slightly differently compared with (4.3) and, in particular, here β p r q : “ η p ´ r q ´ η p r q . Such minor changes are ignored in the notation. OBOLEV IMPROVING FOR AVERAGES OVER CURVES 21
The J “ Proposition 5.2.
Let ď p ď , k P N and ε ą . For all ď ℓ ď t k { u , } m r a k,ℓ s} M p p R q À ε,p ´ k { p ´ ℓ p { ´ { p ´ ε q . Proof of J “ case of Theorem 4.1, assuming Proposition 5.2. Let 4 ă p ď ε p : “ ` ´ p ˘ ą
0. Apply the decomposition (5.4) and Proposition 5.2 to deduce that } m r a k s} M p p R q ď t k { u ÿ ℓ “ } m r a k,ℓ s} M p p R q À p ´ k { p ÿ ℓ “ ´ ℓ p { ´ { p ´ ε p q À p ´ k { p . This establishes the desired result for 4 ă p ď
6. The remaining range 6 ă p ď 8 follows byinterpolation with a trivial L estimate. (cid:3) The rest of § | u p ξ q| ensures thatthe functions s ÞÑ x γ p s q , ξ y and s ÞÑ x γ p s q , ξ y do not vanish simultaneously. Quantifying thisobservation, one obtains, via the van der Corput lemma, the decay estimate | m r a k,ℓ sp ξ q| À ´ k { ` ℓ { ; (5.6)see Lemma 5.6 below. This improves upon the trivial decay rate (5.2) since ℓ varies over the range0 ď ℓ ď t k { u . Note that ℓ “ k { } m r a k,ℓ s} M p R q À ´ k { ` ℓ { . As ℓ increases this estimate becomes weaker. To compensate for this, we attempt to establishstronger estimates for the M p R q norm. This is not possible, however, for the entire multiplierand a further decomposition is required. The u p ξ q localisation means that m r a k,ℓ s is supported ina neighbourhood of the cone Γ. Consequently, one may apply a decoupling theorem for this cone(in particular, an instance of Theorem 4.4) to radially decompose the multipliers. It transpiresthat each radially localised piece is automatically localised along the curve in the physical space,and this leads to favourable M p R q bounds: see Lemma 5.5 and Lemma 5.7 below.5.4. Fourier localisation and decoupling.
The first step towards Proposition 5.2 is to radiallydecompose the symbols in terms of θ p ξ q . Fix a smooth cutoff ζ P C p R q with supp ζ Ď r´ , s such that ř k P Z ζ p ¨ ´ k q ” a k,ℓ p ξ ; s q “ ÿ µ P Z a µk,ℓ p ξ ; s q where a µk,ℓ p ξ ; s q : “ a k,ℓ p ξ ; s q ζ p ℓ θ p ξ q ´ µ q . (5.7)Given 0 ă r ď s P I , recall the definition of the p , r q - Frenet boxes π p s ; r q introduced inDefinition 4.2: π p s ; r q : “ ξ P ˆ R : |x e j p s q , ξ y| À r ´ j for j “
1, 2, |x e p s q , ξ y| „ , |x e p s q , ξ y| À ( . (5.8)Here p e j q j “ denotes the Frenet frame, as introduced in § a µk,ℓ satisfy thefollowing support properties. Lemma 5.3.
With the above definitions, supp a µk,ℓ Ď k ¨ π p s µ ; 2 ´ ℓ q for all ď ℓ ď t k { u and µ P Z , where s µ : “ ´ ℓ µ . Proof.
For ξ P supp a µk,ℓ observe that |x γ p i q ˝ θ p ξ q , ξ y| À k ´p ´ i q ℓ _ for 1 ď i ď |x γ p q ˝ θ p ξ q , ξ y| „ k . Since the Frenet vectors e i ˝ θ p ξ q are obtained from the γ p i q ˝ θ p ξ q via the Gram–Schmidt process,the matrix corresponding to change of basis from ` e i ˝ θ p ξ q ˘ i “ to ` γ p i q ˝ θ p ξ q ˘ i “ is lower triangular.Furthermore, the initial localisation implies that this matrix is an O p δ q perturbation of the identity.Consequently, provided δ ą |x e i ˝ θ p ξ q , ξ y| À k ´p ´ i q ℓ _ for 1 ď i ď , |x e ˝ θ p ξ q , ξ y| „ k . On the other hand, by (5.7) we also have | θ p ξ q ´ s µ | À ´ ℓ and so (4.12) implies that |x e i ˝ θ p ξ q , e j p s µ qy| À | θ p ξ q ´ s µ | | i ´ j | À ´p i ´ j q ℓ . Writing ξ with respect to the orthonormal basis ` e j ˝ θ p ξ q ˘ j “ , it follows that |x e j p s µ q , ξ y| ď ÿ i “ |x e i ˝ θ p ξ q , ξ y||x e i ˝ θ p ξ q , e j p s µ qy| À k ´p ´ j q ℓ _ . Thus, ξ satisfies all the required upper bounds appearing in (5.8). Provided the parameter δ ą x e p s µ q , ξ y . (cid:3) In view of the Fourier localisation described above, we have the following decoupling inequality.
Proposition 5.4.
For all ď p ď and ε ą one has ››› ÿ µ P Z m r a µk,ℓ sp D q f ››› L p p R q À ε ℓ p { ´ { p q` εℓ ´ ÿ µ P Z } m r a µk,ℓ sp D q f } pL p p R q ¯ { p . Proof.
In view of the support conditions from Lemma 5.3, after a simple rescaling, the desiredresult follows from Theorem 4.4 with d ´ “ n “ r “ ´ ℓ . (cid:3) Localisation along the curve.
The θ p ξ q localisation introduced in the previous subsectioninduces a corresponding localisation along the curve in the physical space. In particular, the maincontribution to m r a µk,ℓ s arises from the portion of the curve defined over the interval | s ´ s µ | ď ´ ℓ .This is made precise in Lemma 5.5 below.Here it is convenient to introduce a ‘fine tuning’ constant ρ ą
0. This is a small (but absolute)constant which plays a minor technical rˆole in the forthcoming arguments: taking ρ : “ ´ morethan suffices for our purposes.For 0 ď ℓ ď t k { u , µ P Z and ε ą
0, define a µ, p ε q k,ℓ p ξ ; s q : “ a k,ℓ p ξ ; s q ζ p ℓ θ p ξ q ´ µ q η p ρ ℓ p ´ ε q p s ´ s µ qq . (5.9)The key contribution to the multiplier comes from the symbol a µ, p ε q k,ℓ . Lemma 5.5.
Let ď p ă 8 and ε ą . For all ď ℓ ď t k { u , } m r a µk,ℓ ´ a µ, p ε q k,ℓ s} M p p R q À N,ε,p ´ kN for all N P N .Proof. It is clear that the multipliers satisfy a trivial L -estimate with operator norm O p Ck q forsome absolute constant C ě
1. Thus, by interpolation, it suffices to prove the rapid decay estimatefor p “ } m r a µk,ℓ ´ a µ, p ε q k,ℓ s} L p ˆ R q À N,ε ´ kN for all N P N . OBOLEV IMPROVING FOR AVERAGES OVER CURVES 23
Here the localisation of the a k,ℓ symbols ensures that | u p ξ q| À k ´ ℓ for all p ξ ; s q P supp p a µk,ℓ ´ a µ, p ε q k,ℓ q , (5.10)where u is the function introduced in (5.3). On the other hand, provided ρ is sufficiently small,the additional localisation in (5.7) and (5.9) implies, via the triangle inequality, | s ´ θ p ξ q| Á ρ ´ ´ ℓ p ´ ε q for all p ξ ; s q P supp p a µk,ℓ ´ a µ, p ε q k,ℓ q . (5.11)Fix ξ P supp ξ p a µk,ℓ ´ a µ, p ε q k,ℓ q and consider the oscillatory integral m r a µk,ℓ ´ a µ, p ε q k,ℓ sp ξ q , which hasphase s ÞÑ x γ p s q , ξ y . Taylor expansion around θ p ξ q yields x γ p s q , ξ y “ u p ξ q ` ω p ξ ; s q ¨ p s ´ θ p ξ qq , (5.12) x γ p s q , ξ y “ ω p ξ ; s q ¨ p s ´ θ p ξ qq (5.13)where the ω i arise from the remainder terms and satisfy | ω i p ξ ; s q| „ k . Provided ρ is sufficientlysmall, (5.10) and (5.11) imply that the ω p ξ ; s q ¨ p s ´ θ p ξ qq term dominates the right-hand side of(5.12) and therefore |x γ p s q , ξ y| Á k | s ´ θ p ξ q| for all p ξ ; s q P supp p a µk,ℓ ´ a µ, p ε q k,ℓ q . (5.14)Furthermore, (5.13), (5.14) and the localisation (5.11) immediately imply |x γ p s q , ξ y| À ´ k ` ℓ p ´ ε q |x γ p s q , ξ y| , |x γ p j q p s q , ξ y| À k À j ´p k ´ ℓ p ´ ε qqp j ´ q |x γ p s q , ξ y| j for all j ě p ξ ; s q P supp p a µk,ℓ ´ a µ, p ε q k,ℓ q .On the other hand, by the definition of the symbols, (5.14) and the localisation (5.11), |B Ns p a µk,ℓ ´ a µ, p ε q k,ℓ qp ξ ; s q| À N ℓN À ´p k ´ ℓ q N ´ εℓN |x γ p s q , ξ y| N for all N P N .Thus, by repeated integration-by-parts (via Lemma C.1, with R “ k ´ ℓ ` εℓ ě | m r a µk,ℓ ´ a µ, p ε q k,ℓ sp ξ q| À N ´p k ´ ℓ q N ´ εℓN for all N P N .Since 0 ď ℓ ď t k { u ď k {
3, the desired bound follows. (cid:3)
Estimating the localised pieces.
The multiplier operators m r a µ, p ε q k,ℓ sp D q satisfy favourable L and L bounds, owing to the u p ξ q and s localisation, respectively. Lemma 5.6.
For all ď ℓ ď t k { u , µ P Z and ε ą , we have } m r a µ, p ε q k,ℓ s} M p R q À ´ k { ` ℓ { . Proof. If ℓ “ t k { u , then the desired estimate follows from Plancherel’s theorem and van derCorput lemma with third order derivatives, as the localisation (5.1) implies |x γ p q p s q , ξ y| Á k for all p ξ, s q P supp a k,ℓ .For the remaining case, it suffices to show that |x γ p s q , ξ y| ` ´ ℓ |x γ p s q , ξ y| Á k ´ ℓ for all p ξ ; s q P supp a µ, p ε q k,ℓ . (5.15)Here the localisation of the symbol ensures the key property | u p ξ q| „ k ´ ℓ for all p ξ ; s q P supp a µ, p ε q k,ℓ . (5.16)Indeed, this follows from (5.5) together with the hypothesis 0 ď ℓ ă t k { u . By Taylor expansion around θ p ξ q , one has x γ p s q , ξ y “ u p ξ q ` ω p ξ ; s q ¨ p s ´ θ p ξ qq , (5.17) x γ p s q , ξ y “ ω p ξ ; s q ¨ p s ´ θ p ξ qq , (5.18)where the functions ω i arise from the remainder terms and satisfy | ω i p ξ ; s q| „ k for i “
1, 2.The analysis now splits into two cases.
Case 1: | s ´ θ p ξ q| ă ρ ´ ℓ . Provided ρ is sufficiently small, (5.16) implies that the u p ξ q termdominates in the right-hand side of (5.17) and therefore |x γ p s q , ξ y| Á k ´ ℓ . Case 2: | s ´ θ p ξ q| ě ρ ´ ℓ . In this case, (5.18) implies that |x γ p s q , ξ y| Á ρ k ´ ℓ .In either case, the desired bound (5.15) holds. (cid:3) Lemma 5.7.
For all ď ℓ ď t k { u , µ P Z and ε ą , } m r a µ, p ε q k,ℓ s} M p R q À ´ ℓ p ´ ε q . Proof.
By Lemma 5.3, we have supp a µk,ℓ Ď k ¨ π p s µ ; 2 ´ ℓ q . Consequently, an integration-by-partsargument (see Lemma C.2) reduces the problem to showing | ∇ N v j a µk,ℓ p ξ q| À N ´p k ´p ´ j q ℓ _ q N for all 1 ď j ď N P N , (5.19)where ∇ v j denotes the directional derivative in the direction of the vector v j : “ e j p s µ q .Given ξ P supp a µk,ℓ , we claim that2 ℓ | ∇ N v j θ p ξ q| À N ´p k ´p ´ j q ℓ _ q N and 2 ´ k ` ℓ | ∇ N v j u p ξ q| À N ´p k ´p ´ j q ℓ _ q N (5.20)for all N P N . Assuming that this is so, the derivative bounds (5.19) follow directly from the chainand Leibniz rule, applying (5.20).The claimed bounds in (5.20) follow from repeated application of the chain rule, provided |x γ p q ˝ θ p ξ q , ξ y| Á k , (5.21a) |x γ p K q ˝ θ p ξ q , ξ y| À K k ` ℓ p K ´ q , (5.21b) |x γ p K q ˝ θ p ξ q , v j y| À K p ´ j q ℓ _ ` ℓ p K ´ q (5.21c)hold for all K ě ξ P supp a µk,ℓ . In particular, assuming (5.21a), (5.21b) and (5.21c), thebounds in (5.20) are then a consequence of Lemma B.1 in the appendix: (5.20) corresponds to(B.2) and (B.4) whilst the hypotheses in the above display correspond to (B.1) and (B.3). Herethe parameters featured in the appendix are chosen as follows: g h A B M M e γ γ k ´ ℓ k ´ ℓ ´ k `p ´ j q ℓ _ ℓ v j See Example B.2.The conditions (5.21a), (5.21b) and (5.21c) are direct consequences of the support properties ofthe a µk,ℓ . Indeed, (5.21a) and the K ě a k . The remaining K “ x γ ˝ θ p ξ q , ξ y “ θ localisation imply |x γ p K q ˝ θ p ξ q , v j y| À K | θ p ξ q ´ s µ | p j ´ K q_ À ´pp j ´ K q_ q ℓ and this is easily seen to imply (5.21c). (cid:3) Lemma 5.6 and Lemma 5.7 can be combined to obtain the following L p bounds. OBOLEV IMPROVING FOR AVERAGES OVER CURVES 25
Corollary 5.8.
Let ď ℓ ď t k { u and ε ą . For all ď p ď 8 , ´ ÿ µ P Z } m r a µ, p ε q k,ℓ sp D q f } pL p p R q ¯ { p À ´ k { p ´ ℓ p ´ { p q` εℓ } f } L p p R q . When p “ 8 the left-hand ℓ p -sum is interpreted as a supremum in the usual manner.Proof. For p “ L bounds from Lemma 5.6 with asimple orthogonality argument. For p “ 8 the estimate is a restatement of the L bound fromLemma 5.7. Interpolating these two endpoint cases, using mixed norm interpolation (see, forinstance, [21, § (cid:3) Putting everything together.
We are now ready to combine the ingredients to concludethe proof of Proposition 5.2.
Proof of Proposition 5.2.
By Proposition 5.4, for all 2 ď p ď ε ą } m r a k,ℓ sp D q f } L p p R q “ ››› ÿ µ P Z m r a µk,ℓ sp D q f ››› L p p R q À ε ℓ p { ´ { p q` εℓ ´ ÿ µ P Z } m r a µk,ℓ sp D q f } pL p p R q ¯ { p . Moreover, for all 2 ď p ă 8 , µ P Z and all ε ą
0, Lemma 5.5 implies that } m r a µk,ℓ s} M p p R q À N,ε,p ››› ÿ µ P Z m r a µ, p ε q k,ℓ s ››› M p p R q ` ´ kN for all N P N . Combining the above, we obtain that for all 2 ď p ď ε ą } m r a k,ℓ sp D q f } L p p R q À ε,p ℓ p { ´ { p q` εℓ ´ ÿ µ P Z } m r a µk,ℓ sp D q f } pL p p R q ¯ { p ` ´ kN } f } L p p R q which together with Corollary 5.8 yields } m r a k,ℓ sp D q f } L p p R q À ε ´ k { p ´ ℓ p { ´ { p ´ ε q } f } L p p R q . Since ε ą (cid:3) We have established Proposition 5.2 and therefore completed the proof of the J “ The proof of Theorem 4.1: The J “ case The analysis used to prove the J “ J “
3. This case constitutes to the main content of Theorem 4.1.6.1.
Preliminaries.
Suppose γ P G p δ q and a P C p ˆ R zt u ˆ R q satisfies the hypotheses Theo-rem 4.1 for J “ δ : “ : δ : “ δ , δ : “ δ and δ : “ { Note, in particular, that |x γ p q p s q , ξ y| ě ¨ | ξ ||x γ p j q p s q , ξ y| ď δ | ξ | for j “ , , p ξ ; s q P supp ξ a ˆ I , (6.1)as a consequence of (4.8). We note two further consequences of this technical reduction: ‚ Recall γ p j q p q “ ~e j for 1 ď j ď | ξ | ě ¨ | ξ | and | ξ j | ď δ | ξ | for j “ , ,
3, for all ξ P supp ξ a . The choice δ : “ δ is not relevant to this part of the argument (we may simply take δ : “ δ ) but is used forconsistency with the previous section. ‚ Since γ P G p δ q , we have } γ p q } ď δ . Thus, provided δ is sufficiently small, |x γ p q p s q , ξ y| ě ¨ | ξ | for all p ξ ; s q P supp ξ a ˆ r´ , s . (6.2)Observe that this inequality holds on the large interval r´ , s , rather than just I .Henceforth, we also assume that ξ ą ξ P supp ξ a . In particular, x γ p q p s q , ξ y ą p ξ ; s q P supp ξ a ˆ r´ , s (6.3)and thus, for each ξ P supp ξ a , the function s ÞÑ x γ p s q , ξ y is strictly convex on r´ , s . Theanalysis for the portion of the symbol supported on the set t ξ ă u follows by symmetry.If a k : “ a ¨ β k , as introduced in § | m r a k sp ξ q| À ´ k { . (6.4)Thus, Plancherel’s theorem and interpolation with a trivial L estimate, as in the J “ } m r a k s} M p p R q À ´ k { p for all 2 ď p ď 8 .As in the J “ } m r a k s} M p p R q À ´ k { p , we decompose thesymbol a k into localised pieces which admit more refined decay rates than (6.4). This decomposi-tion is, however, significantly more involved than that used in the previous section.6.2. Geometry of the slow decay cones.
The first step is to isolate regions of the frequencyspace where the multiplier m r a s decays relatively slowly. Owing to stationary phase considerations,this corresponds to the regions around the conic varietiesΓ d ´ : “ t ξ P supp ξ a : x γ p j q p s q , ξ y “ , ď j ď d, for some s P I u , ď d ď . Note that Γ d ´ has codimension d ´
1, which motivates the choice of index. Since Γ Ď Γ , thedecay rate for the multiplier m r a s depends on the relative position with respect to both cones. Toanalyse this, we begin with the following observation, which helps us to understand the geometryof Γ . Lemma 6.1. If ξ P supp ξ a , then the equation x γ p q p s q , ξ y “ has a unique solution in s P r´ , s ,which corresponds to the unique global minimum of the function s ÞÑ x γ p s q , ξ y . Furthermore, thesolution has absolute value O p δ q . The above lemma quickly follows from (6.3) and the localisation of the symbol via the meanvalue theorem. A detailed proof (of a very similar result) can be found in [1, Lemma 6.1].By Lemma 6.1, there exists a unique smooth mapping θ : supp ξ a Ñ r´ , s such that x γ p q ˝ θ p ξ q , ξ y “ ξ P supp ξ a. It is easy to see that θ is homogeneous of degree 0. Define the quantities u , p ξ q : “ x γ ˝ θ p ξ q , ξ y and u p ξ q : “ x γ ˝ θ p ξ q , ξ y for all ξ P supp ξ a .Note that ξ P Γ if and only if u , p ξ q “ u p ξ q “ | u p ξ q| and | u , p ξ q| measure the distance of ξ to Γ .The next observation helps us to understand the geometry of the cone Γ . Lemma 6.2.
Let ξ P supp ξ a and consider the equation x γ p s q , ξ y “ . (6.5) i) If u p ξ q ą , then the equation (6.5) has no solution on r´ , s .ii) If u p ξ q “ , then the equation (6.5) has only the solution s “ θ p ξ q on r´ , s . OBOLEV IMPROVING FOR AVERAGES OVER CURVES 27 iii) If u p ξ q ă , then the equation (6.5) has precisely two solutions on r´ , s . Both solutionshave absolute value O p δ { q . Again, this lemma quickly follows using the information in Lemma 6.1, the localisation of thesymbol and Taylor expansion. The relevant details can be found in [1, Lemma 6.2].Using Lemma 6.2, we construct a (unique) pair of smooth mappings θ ˘ : t ξ P supp ξ a : u p ξ q ă u Ñ r´ , s with θ ´ p ξ q ď θ ` p ξ q which satisfies x γ ˝ θ ˘ p ξ q , ξ y “ ξ P supp ξ a with u p ξ q ă u ˘ p ξ q : “ x γ ˝ θ ˘ p ξ q , ξ y and u ˘ , p ξ q : “ x γ p q ˝ θ ˘ p ξ q , ξ y for all ξ P supp ξ a with u p ξ q ă ξ P Γ if and only if u ` p ξ q “ u ´ p ξ q “
0. For this reason, we introduce u p ξ q : “ $&% u ` p ξ q if | u ` p ξ q| “ min ˘ | u ˘ p ξ q| u ´ p ξ q if | u ´ p ξ q| “ min ˘ | u ˘ p ξ q| and θ p ξ q : “ θ ` p ξ q if u p ξ q “ u ` p ξ q θ ´ p ξ q if u p ξ q “ u ´ p ξ q , which clearly satisfy u p ξ q “ x γ ˝ θ p ξ q , ξ y . Roughly speaking, the quantity | u p ξ q| measures the distance of ξ from Γ . Furthermore, if ξ P Γ satisfies u , p ξ q “ u , p ξ q : “ x γ p q ˝ θ p ξ q , ξ y , then ξ P Γ . Thus, again, | u , p ξ q| may be interpreted as measuring the distance of ξ P Γ to Γ .The following lemma relates important information regarding the functions θ p ξ q , θ ˘ p ξ q andthe associated quantities u p ξ q , u , p ξ q , u ˘ p ξ q , u ˘ , p ξ q . Lemma 6.3.
Let ξ P supp ξ a with u p ξ q ă . Then the following hold:i) ˇˇ u ˘ , ` ξ | ξ | ˘ˇˇ „ | θ ˘ p ξ q ´ θ p ξ q| „ | θ ` p ξ q ´ θ ´ p ξ q| „ ˇˇ u ` ξ | ξ | ˘ˇˇ { ,ii) ˇˇ u , ` ξ | ξ | ˘ ´ u ˘ ` ξ | ξ | ˘ˇˇ À ˇˇ u ` ξ | ξ | ˘ˇˇ { ,iii) ˇˇ u ` ` ξ | ξ | ˘ ´ u ´ ` ξ | ξ | ˘ˇˇ „ ˇˇ u ` ξ | ξ | ˘ˇˇ { .Proof. i) This is almost immediate from Taylor expansion around θ p ξ q , and around θ ´ p ξ q in thelast display. The interested reader is referred to [1, Lemma 6.3] for details of a closely relatedcalculation.ii) By Taylor expansion around θ ˘ p ξ q , u , p ξ q “ u ˘ p ξ q ` u ˘ , p ξ q ¨ p θ p ξ q ´ θ ˘ p ξ qq ` ω p ξ q ¨ p θ p ξ q ´ θ ˘ p ξ qq , where | ω p ξ q| „ | ξ | . The desired estimate follows from the above expansion and part i).iii) By part i), it suffices to show | u ` p ξ q ´ u ´ p ξ q| „ | u p ξ q|| θ ` p ξ q ´ θ ´ p ξ q| . (6.6)To this end, note that u ` p ξ q ´ u ´ p ξ q “ ˆ θ ` p ξ q θ ´ p ξ q x γ p s q , ξ y d s. By Lemma 6.1, u p ξ q ď x γ p s q , ξ y ď θ ´ p ξ q ď s ď θ ` p ξ q . Thus, the upper bound in (6.6)immediately follows from the above identity and the triangle inequality. To see the lower boundin (6.6), recall from (6.3) that the function s ÞÑ φ p s q : “ x γ p s q , ξ y is strictly convex in r´ , s andthat φ ˝ θ ` p ξ q “ φ ˝ θ ´ p ξ q “
0. As θ ´ p ξ q ď θ p ξ q ď θ ` p ξ q and φ ˝ θ p ξ q “ u p ξ q , the convexity of φ implies ˆ θ ` θ ´ |x γ p s q , ξ y| d s ě | u p ξ q|| θ ` p ξ q ´ θ ´ p ξ q| , and thus (6.6) follows from the constant sign of φ p s q on r θ ´ p ξ q , θ ` p ξ qs . (cid:3) Decomposition of the symbols.
For k ě a k : “ a ¨ β k as defined in § a k in relation to the codimension 2 coneΓ corresponding to the directions of slowest decay for ˆ µ . In order to measure the distance tothis cone, we consider the two quantities u , and u introduced in the previous subsection and,in particular, form a simultaneous dyadic decomposition according to the relative sizes of each.Here it is convenient to introduce a ‘fine tuning’ constant ρ ą
0. This is a small (but absolute)constant which plays a minor technical rˆole in the forthcoming arguments: taking ρ : “ ´ morethan suffices for our purposes. Decomposition with respect to Γ . Let β, η P C c p ˆ R q be the functions used to perform a Littlewood–Paley decomposition in § β ` , β ´ P C c p R q with supp β ` Ă p , and supp β ´ Ă p´8 , q be such that β “ β ` ` β ´ . For each m P N , write η p ´ r q η p r q “ m ÿ ℓ “ β p ℓ ´ r q η p ℓ r q ` m ´ ÿ ℓ “ η p ℓ r q ` β ` p ℓ r q ` β ´ p ℓ r q ˘ ` η p m r q η p m r q . The above formula corresponds to a smooth decomposition of r´ , s ˆ r´ , s into axis-paralleldyadic rectangles: see Figure 3. We apply this decomposition with r “ ´ k u , p ξ q and r “ ρ ´ ´ k u p ξ q . This is then used to split the symbol a k as a sum a k “ t k { u ÿ ℓ “ a k,ℓ, ` a k,ℓ, ` t k { u ´ ÿ ℓ “ b k,ℓ where t k { u denotes the greatest integer less than or equal to k { a k,ℓ, p ξ ; s q : “ a k p ξ ; s q β p ´ k ` ℓ u , p ξ qq η p ρ ´ ´ k ` ℓ u p ξ qq ď ℓ ď t k { u ,a k,ℓ, p ξ ; s q : “ a k p ξ ; s q η p ´ k ` ℓ u , p ξ qq β ` p ρ ´ ´ k ` ℓ u p ξ qq if 0 ď ℓ ă t k { u a k p ξ ; s q η p ´ k ` t k { u u , p ξ qq η p ρ ´ ´ k ` t k { u u p ξ qq if ℓ “ t k { u ,b k,ℓ p ξ ; s q : “ a k p ξ ; s q η p ´ k ` ℓ u , p ξ qq β ´ p ρ ´ ´ k ` ℓ u p ξ qq ď ℓ ă t k { u . The following remarks help to motive the above decomposition:For ξ P supp ξ a k,ℓ, , the functions s ÞÑ x γ p s q , ξ y and s ÞÑ x γ p s q , ξ y do not vanish simultaneously.This is due, in part, to the lower bound on | u , p ξ q| . On the other hand, for ξ P supp ξ a k,ℓ, wehave u p ξ q ą s ÞÑ x γ p s q , ξ y is non-vanishing by Lemma 6.2. Quantifying theseobservations, one obtains the decay estimate | m r a k,ℓ,ι sp ξ q| À ´ k { ` ℓ for ι “
1, 2 (6.7) Here the β function should be defined slightly differently compared with (4.3). In particular, when acting on r we have β p r q : “ η p ´ r q ´ η p r q and when acting on r we have β p r q : “ η p ´ r q ´ η p r q . Such minor changesare ignored in the notation. OBOLEV IMPROVING FOR AVERAGES OVER CURVES 29 r r ´ m ´ m ´ m ` ´ m ` ´ m ` ´ m ` ´ m ` Figure 3.
Two parameter dyadic decomposition in the upper-left quadrant.via the van der Corput lemma. See Lemma 6.12 a) for details. This improves upon the trivialdecay rate (6.4) since ℓ varies over the range 0 ď ℓ ď t k { u . Note that ℓ “ k { ξ P supp ξ b k,ℓ , as u p ξ q ă
0, the function s ÞÑ x γ p s q , ξ y vanishes at s “ θ ˘ p ξ q by Lemma 6.2.Moreover, the lack of a lower bound for | u , p ξ q| allows for simultaneous vanishing of s ÞÑ x γ p s q , ξ y and s ÞÑ x γ p s q , ξ y , in contrast with the situation considered above. However, the lower bound on | u p ξ q| implies that the functions s ÞÑ x γ p s q , ξ y and s ÞÑ x γ p q p s q , ξ y do not vanish simultaneously.Quantifying these observations, one obtains, via the van der Corput lemma, the decay estimate | m r b k,ℓ sp ξ q| À ´ k { ` ℓ { . (6.8)Again, this improves upon the trivial decay rate (6.4) since 0 ď ℓ ă t k { u and, furthermore, ℓ “ k { b k,ℓ with respect to the codimension 1 cone Γ . Recall thatthis cone corresponds to directions of slow (but not necessarily minimal) decay for ˆ µ . We proceedby performing a secondary dyadic decomposition with respect to the function u , which measuresthe distance to Γ . Decomposition with respect to Γ . If ξ P supp ξ b k,ℓ , then u p ξ q ă θ ˘ p ξ q Pr´ , s are well-defined by Lemma 6.2. Observe that | u p ξ q| „ ρ k ´ ℓ and | u , p ξ q| À k ´ ℓ for all ξ P supp ξ b k,ℓ , and so it follows from Lemma 6.3 ii) that | u p ξ q| À k ´ ℓ for all ξ P supp ξ b k,ℓ . Consequently, provided ρ is chosen sufficiently small, b k,ℓ p ξ ; s q “ b k,ℓ p ξ ; s q η p ρ ´ k ` ℓ u p ξ qq . (6.9)For every k P N define the indexing setΛ p k q : “ ! ℓ “ p ℓ , ℓ q P Z : 0 ď ℓ ă t k { u , ℓ ď ℓ ď X k ` ℓ \) and, for each 0 ď ℓ ă t k { u , consider the fibre associated to its projection in the ℓ -variable,Λ p k, ℓ q : “ ℓ P Λ p k q : ℓ “ p ℓ , ℓ q for some ℓ P Z ( . In view of (6.9), we may decompose b k,ℓ p ξ ; s q “ b k,ℓ p ξ ; s q η p ρ ´ k ` ℓ u p ξ qq “ a k,ℓ , p ξ ; s q ` a k,ℓ , p ξ ; s q ` ÿ ℓ P Λ p k,ℓ q b k, ℓ p ξ ; s q where a k,ℓ , p ξ ; s q : “ b k,ℓ p ξ ; s q ` η p ρ ´ k ` ℓ u p ξ qq ´ η p ρ ´ ´ k ` ℓ u p ξ qq ˘ ,a k,ℓ , p ξ ; s q : “ b k,ℓ p ξ ; s q η p ρ ´ ´ k ` ℓ u p ξ qq ` ´ η p ρ ´ ℓ p s ´ θ p ξ qqq ˘ and b k, ℓ p ξ ; s q : “ b k,ℓ p ξ ; s q β p ρ ´ ´ k ` ℓ u p ξ qq η p ρ ´ ℓ p s ´ θ p ξ qqq if ℓ ă t p k ` ℓ q{ u b k,ℓ p ξ ; s q η p ρ ´ ´ k ` ℓ u p ξ qq η p ρ ´ ℓ p s ´ θ p ξ qqq if ℓ “ t p k ` ℓ q{ u for ℓ “ p ℓ , ℓ q P Λ p k q . The final decomposition . Combining the preceding definitions, we have a k “ t k { u ÿ ℓ “ ÿ ι “ a k,ℓ,ι ` ÿ ℓ P Λ p k q b k, ℓ (6.10)where for ι “ , a k,ℓ,ι ” ℓ “ t k { u . This concludes the initial frequencydecomposition.The following remarks help to motive the above decomposition:For ξ P a k,ℓ, or ξ P a k,ℓ, it transpires that the functions s ÞÑ x γ p s q , ξ y and s ÞÑ x γ p s q , ξ y donot vanish simultaneously. Quantifying these observations, one obtains the decay estimate | m r a k,ℓ,ι sp ξ q| À ´ k { ` ℓ for ι “
3, 4 , exactly as in (6.7). See Lemma 6.12 a) for details. Here, however, the attendant stationary phasearguments are a little more delicate than those used to prove (6.7) and, in particular, they relyon a careful analysis involving both Γ and Γ . The lower bounds on | u p ξ q| and | s ´ θ p ξ q| arefundamental in each case.Turning to the b k, ℓ symbols, the localisation | s ´ θ p ξ q| À ρ ´ ℓ leads to the following keyobservation. Lemma 6.4.
Let k P N and ℓ “ p ℓ , ℓ q P Λ p k q . Then |x γ p q p s q , ξ y| „ ρ { k ´ ℓ for all p ξ ; s q P supp b k, ℓ . (6.11) Proof.
The localisation of the symbol ensures the key properties | u p ξ q| „ ρ k ´ ℓ , | s ´ θ p ξ q| À ρ ´ ℓ for all p ξ ; s q P supp b k, ℓ . (6.12)By the mean value theorem we obtain x γ p q p s q , ξ y “ u , p ξ q ` ω p ξ ; s q ¨ p s ´ θ p ξ qq , (6.13)where ω satisfies | ω p ξ ; s q| „ k . Observe that (6.12) and Lemma 6.3 i) imply | u , p ξ q| „ ρ { k ´ ℓ .Consequently, provided ρ is sufficiently small, the second inequality in (6.12) implies that the u , term dominates the right-hand side of (6.13) and therefore the desired bound (6.11) holds. (cid:3) The condition (6.11) reveals that the symbol b k, ℓ essentially corresponds to a scaled version ofthe multiplier a k,ℓ from the J “ ℓ and k . Of course, the condition(6.11) immediately implies | m r b k, ℓ p ξ qs| À ´ k { ` ℓ { , (6.14) OBOLEV IMPROVING FOR AVERAGES OVER CURVES 31 ℓ ℓ k { k { k { Figure 4.
Setting ℓ “ ℓ for a k,ℓ, and ℓ “ ℓ for a k,ℓ,ι , 2 ď ι ď
4, one caninterpret the decomposition (6.10) in the p ℓ , ℓ q -plane as follows. The symbols a k,ℓ, correspond to horizontal lines in the lower triangle, whilst the symbols a k,ℓ, correspond to vertical lines in the diagonal and upper triangle whenever u p ξ q ą u p ξ q ă
0, the symbols a k,ℓ, correspond to vertical lines in the fattened diagonal,the symbols a k,ℓ, correspond to vertical lines in the upper triangle (under theadditional condition that | s ´ θ p ξ q| Á ´ ℓ ) and the symbols b k, ℓ correspond tointeger points in the upper triangle (under the additional condition that | s ´ θ p ξ q| À ´ ℓ ).as in (6.8). However, arguing as in Lemma 5.6, one may improve the decay rate to | m r b k, ℓ p ξ qs| À ´ k { `p ℓ ` ℓ q{ ; (6.15)see Lemma 6.12 b). Indeed, for each 0 ď ℓ ă t k { u , the decomposition of the a k,ℓ for the J “ § b k, ℓ above, with the identification k ÐÑ k ´ ℓ and ℓ ÐÑ ℓ ´ ℓ . The bound (6.15) corresponds to the conclusion of Lemma 5.6 once we substitute in these indices.Observe that (6.15) is indeed an improvement over the trivial decay rate (6.14) since, for ℓ fixed, ℓ varies over the range 0 ď ℓ ď t k ` ℓ u . Note that ℓ “ k ` ℓ corresponds to the critical indexwhere (6.14) and (6.15) agree. Remark.
The symbols in the above decomposition are in fact smooth. This is not entirely obvious,since the function u is defined pointwise as the minimum of | u ´ | and | u ` | . Thus, u fails to besmooth whenever u ´ p ξ q “ ˘ u ` p ξ q . However, the decomposition ensures that | u p ξ q| „ ρ k ´ ℓ and | u p ξ q| À ρ k ´ ℓ for all ξ P supp ξ a k,ℓ , or ξ P supp ξ b k, ℓ . Combining these facts withLemma 6.3, one easily deduces that | u ´ p ξ q ˘ u ` p ξ q| Á ρ { k ´ ℓ and so u is smooth on the support of either a k,ℓ , or b k,ℓ . Furthermore, these observations alsoimply that the function θ p ξ q is smooth on the supports. The symbol a k,ℓ , can be treated in asimilar manner, by writing it as a difference of the symbols b k,ℓ p ξ ; s q η p ρ ´ k ` ℓ u p ξ qq “ b k,ℓ p ξ ; s q and b k,ℓ p ξ ; s q η p ρ ´ ´ k ` ℓ u p ξ qq and showing that both are smooth. Given the above decomposition, in order to prove the J “ Proposition 6.5.
Let ď p ď , k P N and ε ą .a) For all ď ℓ ď t k { u and ď ι ď , } m r a k,ℓ,ι s} M p p R q À p,ε ´ k { p ´ ℓ p { ´ { p ´ ε q . b) For all ℓ “ p ℓ , ℓ q P Λ p k q , } m r b k, ℓ s} M p p R q À p,ε ´ p ℓ ´ ℓ qp { p ´ ε q´ ℓ p { ´ { p ´ ε q . Proof of J “ case of Theorem 4.1, assuming Propositions 6.5. Let 6 ă p ď
12 and define ε p : “ ¨ min ! ´ p , p ) ą . Apply the decomposition (6.10) to deduce that } m r a k } M p p R q ď ÿ ι “ t k { u ÿ ℓ “ } m r a k,ℓ,ι } M p p R q ` ÿ ℓ P Λ p k q } m r b k, ℓ } M p p R q . By Proposition 6.5 a), we have ÿ ι “ t k { u ÿ ℓ “ } m r a k,ℓ,ι } M p p R q À p ´ k { p ÿ ℓ “ ´ ℓ p { ´ { p ´ ε p q À p ´ k { p , Similarly, by Proposition 6.5 b), we have ÿ ℓ P Λ p k q } m r b k, ℓ } M p p R q À p ´ k { p ÿ ℓ “ ´ ℓ p { ´ { p ´ ε p q 8 ÿ ℓ “ ℓ ´ p ℓ ´ ℓ qp { p ´ ε p q À p ´ k { p . Combining these observations establishes the desired result for 6 ă p ď
12. The remaining range12 ă p ď 8 follows by interpolation with a trivial L estimate. (cid:3) The rest of § } m r a k,ℓ,ι s} M p R q À ´ k { ` ℓ { and } m r b k, ℓ s} M p R q À ´ k { `p ℓ ` ℓ q{ . As ℓ , ℓ and ℓ increase, these estimates become weaker. To compensate for this, we attempt toestablish stronger estimates for the M p R q norm. This is not possible, however, for the entiremultipliers and a further decomposition is required. The u p ξ q localisation means that m r a k,ℓ,ι s and m r b k, ℓ s are supported in a neighbourhood of the cone Γ . Furthermore, the u p ξ q localisationmeans that m r b k, ℓ s is localised in a neighbourhood of the cone Γ . Consequently, one may apply adecoupling theorem for such cones (in particular, instances of Theorem 4.4) to radially decomposethe multipliers. In the case of the m r b k, ℓ s , we first decouple with respect to the cone Γ . Afterrescaling, the localised pieces can be treated in a similar manner to the multipliers from the J “ to further decompose into smaller pieces. For both the a k,ℓ,ι and b k, ℓ , it transpires that eachradially localised piece is automatically localised along the curve in the physical space, and thisleads to favourable M p R q bounds: see Lemma 6.11 and Lemma 6.13 below. OBOLEV IMPROVING FOR AVERAGES OVER CURVES 33
Fourier localisation and decoupling.
The first step towards proving Proposition 6.5 isto further decompose the symbols a k,ℓ,ι and b k, ℓ in terms of θ p ξ q and θ p ξ q respectively. Fix ζ P C p R q with supp ζ Ď r´ , s such that ř l P Z ζ p ¨ ´ l q ”
1. For 0 ď ℓ ď t k { u , 1 ď ι ď ℓ “ p ℓ , ℓ q P Λ p k q , write a k,ℓ,ι “ ÿ µ P Z a µk,ℓ,ι and b k, ℓ “ ÿ ν P Z b νk, ℓ where a µk,ℓ,ι p ξ ; s q : “ a k,ℓ,ι p ξ ; s q ζ p ℓ θ p ξ q ´ µ q , (6.16) b νk, ℓ p ξ ; s q : “ b k, ℓ p ξ ; s q ζ p p ℓ ´ ℓ q{ θ p ξ q ´ ν q . (6.17)In the case of the b k, ℓ , we also consider symbols formed by grouping the b νk, ℓ into pieces at thelarger scale 2 ´ ℓ . Given ℓ “ p ℓ , ℓ q P Λ p k q we write Z “ Ť µ P Z N ℓ p µ q , where the sets N ℓ p µ q aredisjoint and satisfy N ℓ p µ q Ď t ν P Z : | ν ´ p ℓ ´ ℓ q{ µ | ď p ℓ ´ ℓ q{ u . For each µ P Z , we then define b ˚ ,µk, ℓ : “ ÿ ν P N ℓ p µ q b νk, ℓ and note that | θ p ξ q ´ s µ | À ´ ℓ on supp b ˚ ,µk, ℓ , where s µ : “ ´ ℓ µ . Of course, by the definition ofthe sets N ℓ p µ q , b k, ℓ “ ÿ µ P Z b ˚ ,µk, ℓ “ ÿ µ P Z ÿ ν P N ℓ p µ q b νk, ℓ . Given 0 ă r ď s P I , recall the definition of the p , r q - Frenet boxes π p s ; r q introduced inDefinition 4.2: π p s ; r q : “ ξ P ˆ R : |x e j p s q , ξ y| À r ´ j for 1 ď j ď , |x e p s q , ξ y| „ ( . (6.18)The symbols a µk,ℓ,ι and b ˚ ,µk, ℓ satisfy the following support properties. Lemma 6.6.
With the above definitions,a) supp ξ a µk,ℓ,ι Ď k ¨ π p s µ ; 2 ´ ℓ q for all ď ℓ ď t k { u , ď ι ď and µ P Z , where s µ : “ ´ ℓ µ ;b) supp ξ b ˚ ,µk, ℓ Ď k ¨ π p s µ ; 2 ´ ℓ q for all ℓ “ p ℓ , ℓ q P Λ p k q and µ P Z , where s µ : “ ´ ℓ µ . It is convenient to set up a unified framework in order to treat parts a) and b) of Lemma 6.6simultaneously. Given n , s P R , let Ξ p k, n ; s q denote the set of all ξ P supp ξ a k which lie in thedomain of θ and satisfy | θ p ξ q ´ s | À ´ n , | u , p ξ q| À k ´ n , | u p ξ q| À k ´ n . (6.19)Note in particular that:a) supp ξ a µk,ℓ,ι Ď Ξ p k, ℓ ; s µ q for 1 ď ι ď ξ b νk, ℓ Ď Ξ p k, ℓ ; s ν q and supp ξ b νk, ℓ Ď Ξ p k, ℓ ; s µ q for all ν P N ℓ p µ q .Indeed, for the respective parameter values, all the desired properties stated in (6.19) hold asan immediate consequence of the definition of the symbols, with the exception of the bounds | θ p ξ q ´ s ν | À ´ ℓ for ξ P supp ξ b νk,ℓ and | θ p ξ q ´ s µ | À ´ ℓ for ξ P supp ξ b νk,ℓ and ν P N ℓ p µ q .However, by Lemma 6.3, it follows from the localisation of the symbol that | θ p ξ q ´ s ν | À ˇˇ u ` ξ | ξ | ˘ˇˇ { ` | θ p ξ q ´ s ν | À ´ ℓ for all ξ P supp ξ b νk,ℓ , which further implies | θ p ξ q ´ s µ | À | θ p ξ q ´ s ν | ` | s ν ´ s µ | À ´ ℓ for all ξ P supp ξ b νk,ℓ , ν P N ℓ p µ q , by the condition | s µ ´ s ν | À ´ ℓ for s ν : “ ´p ℓ ´ ℓ q{ ν . Thus, all the required bounds hold.Note that the support property in b) immediately implies that supp ξ b ˚ ,µk, ℓ Ď Ξ p k, ℓ ; s µ q . Proof of Lemma 6.6.
Let n , s P R . As a consequence of the preceding discussion, it suffices toshow that Ξ p k, n ; s q Ď k ¨ π p s ; 2 ´ n q . Let ξ P Ξ p k, n ; s q and observe that the localisation of a k , the implicit definition of θ and thelatter two conditions in (6.19) imply |x γ p i q ˝ θ p ξ q , ξ y| À k ´p ´ i q n for 1 ď i ď e i ˝ θ p ξ q are obtained from the γ p i q ˝ θ p ξ q via the Gram–Schmidt process, |x e i ˝ θ p ξ q , ξ y| À k ´p ´ i q n for 1 ď i ď |x e i ˝ θ p ξ q , e j p s qy| À | θ p ξ q ´ s | | i ´ j | À ´p i ´ j q n . Writing ξ with respect to the orthonormal basis ` e j ˝ θ p ξ q ˘ j “ , it follows that |x e j p s q , ξ y| ď ÿ i “ |x e i ˝ θ p ξ q , ξ y||x e i ˝ θ p ξ q , e j p s µ qy| À k ´p ´ j q n . Thus, ξ satisfies all the required upper bounds arising from (6.18). The remaining condition |x e p s q , ξ y| Á k holds as an immediate consequence of the initial localisation of a k . (cid:3) The argument used in the proof of Lemma 6.6 can be applied to analyse the support propertiesof the individual b νk, ℓ , although in this case the geometric significance of the supporting set is onlyapparent after rescaling (see Lemma 6.8 below). Given 0 ă r , r ď s P I , define the set π p s ; r , r q : “ ξ P ˆ R : |x e j p s q , ξ y| À r ´ j for j “
1, 2 , |x e p s q , ξ y| „ , |x e p s q , ξ y| À r ( . (6.20)The multipliers b νk, ℓ satisfy the following support property. Lemma 6.7.
With the above definitions, supp ξ b νk, ℓ Ď k ´ ℓ ¨ π p s ν ; 2 ´p ℓ ´ ℓ q{ , ℓ q for all ℓ “ p ℓ , ℓ q P Λ p k q and ν P Z , where s ν : “ ´p ℓ ´ ℓ q{ ν . As with the proof of Lemma 6.6, here and in Lemma 6.13, we will work with a more generalsetup. This abstraction is not particularly useful at this stage, but it will help to unify some ofthe later arguments. Given n “ p n , n q P p , and s P R , let Ξ p k, n ; s q denote the set of all ξ P supp ξ a k which lie in the domain of θ and satisfy | θ p ξ q ´ s | À ´ n , | u p ξ q| À k ´ n ´ n , | u , p ξ q| „ k ´ n . (6.21)Note in particular that:a) supp ξ a µk,ℓ,ι Ď Ξ p k, ℓ, ℓ ; s µ q for ι “ ι “ ξ b νk, ℓ Ď Ξ ` k, ℓ ´ ℓ , ℓ ; s ν ˘ .Indeed, the definition of the symbols implies | u p ξ q| „ k ´ ℓ , | u p ξ q| À k ´ ℓ and | θ p ξ q ´ s µ | À ´ ℓ for all ξ P supp ξ a µk,ℓ,ι when ι P t , u . Consequently, by Lemma 6.3, it follows that | θ p ξ q ´ s µ | À ˇˇ u ` ξ | ξ | ˘ˇˇ { ` | θ p ξ q ´ s µ | À ´ ℓ , ˇˇ u , ` ξ | ξ | ˘ˇˇ „ ˇˇ u ` ξ | ξ | ˘ˇˇ { „ ℓ for all ξ P supp ξ a µk,ℓ,ι , which covers the required bounds for a). Turning to b), all the desiredproperties hold as an immediate consequence of the definition of the symbols, with the exception OBOLEV IMPROVING FOR AVERAGES OVER CURVES 35 of the bound | u , p ξ q| „ k ´ ℓ . However, as in a), the function u , can be estimated via Lemma 6.3using the u localisation. Proof of Lemma 6.7.
Let n “ p n , n q P p , and s P I . As a consequence of the precedingdiscussion, it suffices to show thatΞ p k, n ; s q Ď k ´ n ¨ π p s ; 2 ´ n , ´ n q . The argument in fact depends on the implicit constants in (6.21) satisfying certain size relations,but we shall ignore this minor technicality. In the case in question (namely, on the support of b νk,ℓ ), the required size relations follow provided ρ is chosen sufficiently small.Let ξ P Ξ p k, n ; s q and observe that the localisation of a k , the implicit definition of θ and thelatter two conditions in (6.21) imply |x γ p i q ˝ θ p ξ q , ξ y| À k ´p ´ i q n ´ n for i “
1, 2 , |x γ p q ˝ θ p ξ q , ξ y| „ k ´ n , |x γ p q ˝ θ p ξ q , ξ y| „ k . Since the Frenet vectors e i ˝ θ p ξ q are obtained from the γ p i q ˝ θ p ξ q via the Gram–Schmidt pro-cess, the matrix corresponding to change of basis from ` e i ˝ θ p ξ q ˘ i “ to ` γ p i q ˝ θ p ξ q ˘ i “ is lowertriangular. Furthermore, the initial localisations imply that this matrix is an O p δ q perturbationof the identity. Consequently, provided δ ą |x e i ˝ θ p ξ q , ξ y| À k ´p ´ i q n ´ n for i “
1, 2 , |x e ˝ θ p ξ q , ξ y| „ k ´ n , |x e ˝ θ p ξ q , ξ y| „ k . On the other hand, the first condition in (6.21) together with (4.12) imply |x e i ˝ θ p ξ q , e j p s qy| À | s ´ θ p ξ q| | i ´ j | À ´p i ´ j q n . Writing ξ with respect to the orthonormal basis ` e j ˝ θ p ξ q ˘ j “ , it follows that |x ξ, e j p s qy| ď ÿ i “ |x e i ˝ θ p ξ q , ξ y||x e i ˝ θ p ξ q , e j p s qy| À k ´pp ´ j q n ` n q_ . Thus, ξ satisfies all the required upper bounds arising from (6.20). The above argument can easilybe adapted to give the required lower bounds, provided the implied constant in the the hypothesis | u , p ξ q| „ k ´ n is large compared to that in the hypothesis | θ p ξ q ´ s | À ´ n . (cid:3) Fix some ℓ “ p ℓ , ℓ q P Λ p k q and µ P Z with s µ : “ ´ ℓ µ P r´ , s . To simplify notation, let σ : “ s µ , λ : “ ´ ℓ and let ˜ γ : “ γ σ,λ denote the rescaled curve, as defined in Definition 2.1, so that˜ γ p s q : “ ` r γ s σ,λ ˘ ´ ` γ p σ ` λs q ´ γ p σ q ˘ . (6.22)Given a symbol b P C c p ˆ R ˆ I q , let ˜ b be the rescaled symbol defined by the relation˜ b p ˜ ξ ; ˜ s q “ b p ξ ; s q for ˜ ξ : “ ` r γ s σ,λ ˘ J ξ and ˜ s : “ λ ´ p s ´ σ q . (6.23)Given f P S p R q , it follows by a simple changes of the variables that m r b sp D q f p x q “ λ ¨ ˜ m r ˜ b sp D q ˜ f p ˜ x q (6.24)where: ‚ The multiplier ˜ m r ˜ b s is defined in the same manner as m r ˜ b s but with the curve γ replacedwith ˜ γ and the cut-off χ ˝ replaced with χ ˝ p σ ` λ ¨ q ; ‚ ˜ f : “ f ˝ r γ s σ,λ ; ‚ ˜ x : “ ` r γ s σ,λ ˘ ´ ` x ´ γ p σ q ˘ . Let p ˜ e j q j “ denote the Frenet frame defined with respect to ˜ γ . Given 0 ă r ď s P I , recallthe definition of the p , r q - Frenet boxes (with respect to p ˜ e j q j “ ) introduced in Definition 4.2:˜ π p s ; r q : “ ξ P ˆ R : |x ˜ e j p s q , ξ y| À r ´ j for j “
1, 2, |x ˜ e p s q , ξ y| „ , |x ˜ e p s q , ξ y| À ( . Note that all these definitions depend of the choice of µ and ℓ , but it is typographically convenientto suppress this dependence.The rescaled symbols ˜ b νk, ℓ satisfy the following support properties. Lemma 6.8.
With the above definitions, supp ξ ˜ b νk, ℓ Ď k ´ ℓ ¨ ˜ π p ˜ s ν ; 2 ´ p ℓ ´ ℓ q{ q for all ℓ “ p ℓ , ℓ q P Λ p k q and ν P N ℓ p µ q , where ˜ s ν : “ ℓ p s ν ´ s µ q for s ν : “ ´p ℓ ´ ℓ q{ ν .Proof. For ˜ ξ P supp ξ ˜ b νk, ℓ , it follows from Lemma 6.7 and the definition of the rescaling in (6.23)that ξ : “ ` r γ s σ,λ ˘ ´J ˜ ξ satisfies |x e j p s ν q , ξ y| À k ´p ´ j qp ℓ ´ ℓ q{ ´ ℓ for j “
1, 2 , |x e p s ν q , ξ y| „ k ´ ℓ , |x e p s ν q , ξ y| „ k . Since the matrix corresponding to the change of basis from ` e j p s ν q ˘ j “ to ` γ p j q p s ν q ˘ j “ is lowertriangular and an O p δ q perturbation of the identity, provided δ is sufficiently small, |x γ p j q p s ν q , ξ y| À k ´p ´ j qp ℓ ´ ℓ q{ ´ ℓ for j “
1, 2 , |x γ p q p s ν q , ξ y| „ k ´ ℓ , |x γ p q p s ν q , ξ y| „ k . On the other hand, recalling that λ : “ ´ ℓ , it follows from the definition of ˜ γ from (6.22) that x ˜ γ p j q p ˜ s ν q , ˜ ξ y “ ´ jℓ x γ p j q p s ν q , ξ y for j ě . Combining the above observations, |x ˜ γ p j q p ˜ s ν q , ˜ ξ y| À k ´p ´ j qp ℓ ´ ℓ q{ ´p j ` q ℓ for j “
1, 2 , |x ˜ γ p q p ˜ s ν q , ˜ ξ y| „ k ´ ℓ , |x ˜ γ p q p ˜ s ν q , ˜ ξ y| „ k ´ ℓ . Provided δ is sufficiently small, the desired result now follows since the matrix correspondingto the change of basis from ` ˜ e i p ˜ s ν q ˘ i “ to ` ˜ γ p i q p ˜ s ν q ˘ i “ is also lower triangular and an O p δ q perturbation of the identity. (cid:3) In view of the support conditions from Lemma 6.6 and Lemma 6.8, the multipliers can beeffectively decoupled using Theorem 4.4.
Proposition 6.9.
For all ď p ď and all ε ą , the following inequalities hold:a) For all ď ℓ ď t k { u , ď ι ď , ››› ÿ µ P Z m r a µk,ℓ,ι sp D q f ››› L p p R q À ε ℓ p { ´ { p q` εℓ ´ ÿ µ P Z } m r a µk,ℓ,ι sp D q f } pL p p R q ¯ { p . b) For all ℓ “ p ℓ , ℓ q P Λ p k q , ››› ÿ µ P Z m r b ˚ ,µk, ℓ sp D q f ››› L p p R q À ε ℓ p { ´ { p q` εℓ ´ ÿ µ P Z } m r b ˚ ,µk, ℓ sp D q f } pL p p R q ¯ { p . Proof.
In view of the support conditions from Lemma 6.6, after a simple rescaling, the desiredresult follows from Theorem 4.4 with d ´ “ n “ r “ ´ ℓ , 2 ´ ℓ for parts a) and b),respectively. (cid:3) Proposition 6.10.
For all ℓ “ p ℓ , ℓ q P Λ p k q , ď p ď and ε ą , ››› ÿ ν P Z m r b νk, ℓ sp D q f ››› L p p R q À ε ℓ p { ´ { p ` ε q p ℓ ´ ℓ qp ´ { p ` ε q{ ´ ÿ ν P Z } m r b νk, ℓ sp D q f } pL p p R q ¯ { p . OBOLEV IMPROVING FOR AVERAGES OVER CURVES 37
Proof.
It suffices to show that, under the hypotheses of the proposition, for all µ P Z one has ››› ÿ ν P N ℓ p µ q m r b νk, ℓ sp D q f ››› L p p R q À ε p ℓ ´ ℓ qp ´ { p ` ε q{ ´ ÿ ν P N ℓ p µ q } m r b νk, ℓ sp D q f } pL p p R q ¯ { p . (6.25)Indeed, one may then combine the above inequality with Proposition 6.9 b) to obtain the desireddecoupling result. However, by applying a linear change of variables, (6.25) is equivalent to thesame inequality but with each m r b νk, ℓ s replaced with the rescaled multiplier ˜ m r ˜ b νk, ℓ s as defined in(6.24). In view of the support conditions from Lemma 6.8, after a simple rescaling, the desiredresult follows from Theorem 4.4 with d ´ “ n “ r “ ´ p ℓ ´ ℓ q{ . (cid:3) Localisation along the curve.
The localisation in θ p ξ q and θ p ξ q introduced in the pre-vious subsection induces a corresponding localisation along the curve in the physical space. Inparticular, the main contribution to m r a µk,ℓ,ι s and m r b νk, ℓ s arises from the portion of the curvedefined over the interval | s ´ s µ | ď ´ ℓ and | s ´ s ν | ď ´p ℓ ´ ℓ q{ , respectively. This is madeprecise by Lemma 6.11 below.For each µ, ν P Z , let s µ : “ ´ ℓ µ and s ν : “ ´p ℓ ´ ℓ q{ ν . Given ε ą ρ as introduced in § a µ, p ε q k,ℓ,ι p ξ ; s q : “ a µk,ℓ,ι p ξ q η p ρ ℓ p ´ ε q p s ´ s µ qq , (6.26) b ν, p ε q k, ℓ p ξ ; s q : “ b νk, ℓ p ξ q η p ρ p ´ ε qp ℓ ´ ℓ q{ p s ´ s ν qq . (6.27)The key contribution to the multipliers comes from the symbols a µ, p ε q k,ℓ,ι and b ν, p ε q k, ℓ respectively. Lemma 6.11.
Let ď p ă 8 and ε ą .a) For all ď ℓ ď t k { u , µ P Z and ď ι ď , } m r a µk,ℓ,ι ´ a µ, p ε q k,ℓ,ι s} M p p R q À N,ε,p ´ kN for all N P N .b) For all ℓ “ p ℓ , ℓ q P Λ p k q and ν P Z , } m r b νk, ℓ ´ b ν, p ε q k, ℓ s} M p p R q À N,ε,p ´ kN for all N P N .Proof. In both part a) and b) it is clear that the multipliers satisfy a trivial L -estimate withoperator norm O p Ck q for some absolute constant C ě
1. Thus, by interpolation, it suffices toprove the rapid decay for p “ } m r a µk,ℓ,ι ´ a µ, p ε q k,ℓ,ι s} L p ˆ R q À N,ε ´ kN and } m r b νk, ℓ ´ b ν, p ε q k, ℓ s} L p ˆ R q À N,ε ´ kN for all N P N .(6.28)This is achieved via a simple (non)-stationary phase analysis.a) Here the localisation of the a k,ℓ,ι symbols ensures that | u , p ξ q| À k ´ ℓ , | u p ξ q| À ρ k ´ ℓ for all p ξ ; s q P supp p a µk,ℓ,ι ´ a µ, p ε q k,ℓ,ι q . (6.29)On the other hand, provided ρ is sufficiently small, the additional localisation in (6.16) and (6.26)implies | s ´ θ p ξ q| Á ρ ´ ´ ℓ p ´ ε q for all p ξ ; s q P supp p a µk,ℓ,ι ´ a µ, p ε q k,ℓ,ι q . (6.30) Fix ξ P supp ξ p a µk,ℓ,ι ´ a µ, p ε q k,ℓ,ι q and consider the oscillatory integral m r a µk,ℓ,ι ´ a µ, p ε q k,ℓ,ι sp ξ q , which hasphase s ÞÑ x γ p s q , ξ y . Taylor expansion around θ p ξ q yields x γ p s q , ξ y “ u , p ξ q ` ` u p ξ q ` ω p ξ ; s q ¨ p s ´ θ p ξ qq ˘ ¨ p s ´ θ p ξ qq , (6.31) x γ p s q , ξ y “ u p ξ q ` ω p ξ ; s q ¨ p s ´ θ p ξ qq , (6.32) x γ p q p s q , ξ y “ ω p ξ ; s q ¨ p s ´ θ p ξ qq , (6.33)where the ω i arise from the remainder terms and satisfy | ω i p ξ ; s q| „ k . Provided ρ is sufficientlysmall, (6.29) and (6.30) imply that the ω p ξ ; s q ¨ p s ´ θ p ξ qq term dominates the right-hand sideof (6.31) and therefore |x γ p s q , ξ y| Á k | s ´ θ p ξ q| for all p ξ ; s q P supp p a µk,ℓ,ι ´ a µ, p ε q k,ℓ,ι q . (6.34)Furthermore, by (6.29) and (6.30), the term ω p ξ ; s q ¨ p s ´ θ p ξ qq dominates in (6.32). This, (6.33),(6.34) and the localisation (6.30) immediately imply |x γ p s q , ξ y| À ´ k ` ℓ p ´ ε q |x γ p s q , ξ y| , |x γ p q p s q , ξ y| À ´p k ´ ℓ p ´ ε qq |x γ p s q , ξ y| , |x γ p j q p s q , ξ y| À k À j ´p k ´ ℓ p ´ ε qqp j ´ q |x γ p s q , ξ y| j for all j ě p ξ ; s q P supp p a µk,ℓ,ι ´ a µ, p ε q k,ℓ,ι q .On the other hand, by the definition of the symbols, (6.34) and the localisation (6.30), |B Ns p a µk,ℓ,ι ´ a µ, p ε q k,ℓ,ι qp ξ ; s q| À N ℓN À ´p k ´ ℓ q N ´ εℓN |x γ p s q , ξ y| N for all N P N .Thus, by repeated integration-by-parts (via Lemma C.1, with R “ k ´ ℓ ` εℓ ě | m r a µk,ℓ,ι ´ a µ, p ε q k,ℓ,ι sp ξ q| À N ´p k ´ ℓ q N ´ εℓN for all N P N . Since 0 ď ℓ ď t k { u ď k {
4, the first bound in (6.28) follows.b) Here the localisation of the b k, ℓ symbols ensures that | u p ξ q| À ρ k ´ ℓ , | u p ξ q| „ ρ k ´ ℓ , | s ´ θ p ξ q| À ρ ´ ℓ (6.35)hold for all p ξ ; s q P supp p b νk, ℓ ´ b ν, p ε q k, ℓ q . Furthermore, by Lemma 6.4, |x γ p q p s q , ξ y| „ ρ { k ´ ℓ for all p ξ ; s q P supp b k, ℓ , (6.36)whilst, provided ρ is sufficiently small, the additional localisation in (6.17) and (6.27) implies | s ´ θ p ξ q| Á ρ ´ ´p ´ ε qp ℓ ´ ℓ q{ for all p ξ ; s q P supp p b νk, ℓ ´ b ν, p ε q k, ℓ q . (6.37)Fix ξ P supp ξ p b νk, ℓ ´ b ν, p ε q k, ℓ q and consider the oscillatory integral m r b νk, ℓ ´ b ν, p ε q k, ℓ sp ξ q , which hasphase s ÞÑ x γ p s q , ξ y . Taylor expansion around θ p ξ q yields x γ p s q , ξ y “ u p ξ q ` ω p ξ ; s q ¨ p s ´ θ p ξ qq , (6.38) x γ p s q , ξ y “ ω p ξ ; s q ¨ p s ´ θ p ξ qq , (6.39)where the ω i arise from the remainder terms and satisfy | ω i p ξ ; s q| „ ρ { k ´ ℓ by (6.36). Provided ρ ą |x γ p s q , ξ y| Á ρ { k ´ ℓ | s ´ θ p ξ q| for all p ξ ; s q P supp p b νk, ℓ ´ b ν, p ε q k, ℓ q . (6.40) OBOLEV IMPROVING FOR AVERAGES OVER CURVES 39
Furthermore, (6.39), (6.36), (6.40) and the localisation (6.37) imply |x γ p s q , ξ y| À ´ k ` ℓ ` p ´ ε qp ℓ ´ ℓ q{ |x γ p s q , ξ y| , |x γ p q p s q , ξ y| À k ´ ℓ À ´ p k ´ ℓ ´ p ´ ε qp ℓ ´ ℓ q{ q |x γ p s q , ξ y| , |x γ p j q p s q , ξ y| À k À j ´p k ´ ℓ ´ p ´ ε qp ℓ ´ ℓ q{ qp j ´ q |x γ p s q , ξ y| j for all j ě p ξ ; s q P supp p a µk,ℓ,ι ´ a µ, p ε q k,ℓ,ι q .On the other hand, by the definition of the symbols, (6.40) and the localisation (6.37), |B Ns p b νk, ℓ ´ b ν, p ε q k, ℓ qp ξ ; s q| À N max ρ ´ N ℓ N , p ´ ε qp ℓ ´ ℓ q N { ( À N,ρ ´p k ´ ℓ ´ p ´ ε qp ℓ ´ ℓ q{ q N |x γ p s q , ξ y| N for all N P N and all p ξ ; s q P supp p b νk, ℓ ´ b ν, p ε q k, ℓ q , using that 0 ď ℓ ď ℓ for ℓ P Λ p k q . Thus, by repeatedintegration-by-parts (via Lemma C.1 with R “ k ´ ℓ ´ p ´ ε qp ℓ ´ ℓ q{ ě | m r b νk, ℓ ´ b ν, p ε q k, ℓ sp ξ ; s q| À N,ρ ´p k ´ ℓ ´ p ℓ ´ ℓ q{ q N ´ ε p ℓ ´ ℓ q N { for all N P N . Since ℓ ď ℓ ď p k ` ℓ q{ ď ℓ ă k { ℓ P Λ p k q , the second bound in (6.28) follows. (cid:3) Estimating the localised pieces.
Each piece of the multipliers m r a µ, p ε q k,ℓ,ι s and m r b ν, p ε q k, ℓ s arising from the preceding decomposition satisfies favourable L and L bounds. Lemma 6.12. a) For ď ℓ ď t k { u , µ P Z , ď ι ď and ε ą , we have } m r a µ, p ε q k,ℓ,ι s} M p R q À ´ k { ` ℓ . b) For ℓ “ p ℓ , ℓ q P Λ p k q , ν P Z and ε ą , we have } m r b ν, p ε q k, ℓ s} M p R q À ´ k { `p ℓ ` ℓ q{ . Proof. a) If ℓ “ t k { u , then the desired bounds follow from Plancherel’s theorem and the van derCorput lemma with fourth order derivatives. For the remaining cases, it suffices to show that |x γ p s q , ξ y| ` ´ ℓ |x γ p s q , ξ y| Á k ´ ℓ for all p ξ ; s q P supp a µ, p ε q k,ℓ,ι . (6.41)We treat each class of symbol, as index by the parameter ι , individually. ι “
1. Here the localisation of the symbol ensures the key properties | u , p ξ q| „ k ´ ℓ , | u p ξ q| À ρ k ´ ℓ for all p ξ ; s q P supp a µ, p ε q k,ℓ, . (6.42)By Taylor expansion around θ p ξ q , one has x γ p s q , ξ y “ u , p ξ q ` u p ξ q ¨ p s ´ θ p ξ qq ` ω p ξ ; s q ¨ p s ´ θ p ξ qq , (6.43) x γ p s q , ξ y “ u p ξ q ` ω p ξ ; s q ¨ p s ´ θ p ξ qq (6.44)where the functions ω i arise from the remainder terms and satisfy | ω i p ξ ; s q| „ k for i “
1, 2. Theargument splits into two cases:
Case 1: | s ´ θ p ξ q| ď ρ { ´ ℓ . Provided ρ ą u , p ξ q term dominates the right-hand side of (6.43) and therefore |x γ p s q , ξ y| Á k ´ ℓ . Case 2: | s ´ θ p ξ q| ě ρ { ´ ℓ . Again provided ρ ą |x γ p s q , ξ y| Á ρ { k ´ ℓ .Thus, in either case the desired bound (6.41) holds. ι “
2. Suppose 0 ď ℓ ă t k { u and ξ P supp a µ, p ε q k,ℓ, . Recall, by Lemma 6.1, that θ p ξ q is the uniqueglobal minimum of the function s ÞÑ x γ p s q , ξ y on r´ , s . Thus, x γ p s q , ξ y ě u p ξ q „ ρ k ´ ℓ , asrequired. ι “
3. Here the localisation of the symbol ensures the key properties | u p ξ q| „ ρ k ´ ℓ , | u p ξ q| „ ρ k ´ ℓ for all p ξ ; s q P supp a µ, p ε q k,ℓ, . (6.45)The argument splits into two cases: Case 1: min ˘ | s ´ θ ˘ p ξ q| ď ρ ´ ℓ . By Taylor expansion around θ ˘ p ξ q , one has x γ p s q , ξ y “ u ˘ p ξ q ` u ˘ , p ξ q ¨ p s ´ θ ˘ p ξ qq ` ω ˘ p ξ ; s q ¨ p s ´ θ ˘ p ξ qq , (6.46)where the functions ω ˘ arise from the third order remainder term and satisfy | ω ˘ p ξ ; s q| „ k . More-over, (6.45) and Lemma 6.3 i) imply | u ˘ , p ξ q| „ ρ { k ´ ℓ . Provided ρ is sufficiently small, (6.45)implies that the u ˘ p ξ q term dominates the right-hand side of (6.46) and therefore |x γ p s q , ξ y| Á ρ k ´ ℓ . Case 2: min ˘ | s ´ θ ˘ p ξ q| ě ρ ´ ℓ . In this case, rather than analysing Taylor expansions, we usea convexity argument. Fix ξ P supp a µ, p ε q k,ℓ, and let φ : r´ , s Ñ R , φ : s ÞÑ x γ p s q , ξ y ;by (6.3), this function is strictly convex. Thus, given t P r´ , s , the auxiliary function q t : r´ , s Ñ R , q t : s ÞÑ φ p s q ´ φ p t q s ´ t for s ‰ t and q t : t ÞÑ φ p t q is increasing. Setting t : “ θ ´ p ξ q and noting that φ ˝ θ ´ p ξ q “
0, it follows that φ p s q s ´ θ ´ p ξ q ď φ ˝ θ p ξ q θ p ξ q ´ θ ´ p ξ q “ u p ξ q θ p ξ q ´ θ ´ p ξ q ă ´ ď s ď θ p ξ q , where we have used the fact that u p ξ q ă a k,ℓ, . If s P r θ p ξ q , s , then we cancarry out the same argument with respect to t “ θ ` p ξ q to obtain a similar inequality. From this,we deduce the bound |x γ p s q , ξ y| ě min ˘ | u p ξ q|| s ´ θ ˘ p ξ q|| θ p ξ q ´ θ ˘ p ξ q| for all ´ ď s ď . (6.47)Recall from (6.45) that | u p ξ q| „ ρ k ´ ℓ and therefore | θ p ξ q ´ θ ˘ p ξ q| „ ρ { ´ ℓ by Lemma 6.3 i).Substituting these bounds and the hypothesis min ˘ | s ´ θ ˘ p ξ q| ě ρ ´ ℓ into (6.47), we concludethat |x γ p s q , ξ y| Á ρ { k ´ ℓ .Thus, in either case the desired bound (6.41) holds. ι “
4. Here the localisation of the symbol ensures the key properties | u p ξ q| À ρ k ´ ℓ , | s ´ θ p ξ q| Á ρ ´ ℓ for all p ξ ; s q P supp a µ, p ε q k,ℓ, . (6.48)By Taylor expansion around θ p ξ q , we obtain x γ p s q , ξ y “ u p ξ q ` u , p ξ q ¨ p s ´ θ p ξ qq ` ω p ξ ; s q ¨ p s ´ θ p ξ qq , (6.49) x γ p s q , ξ y “ u , p ξ q ¨ p s ´ θ p ξ qq ` ω p ξ ; s q ¨ p s ´ θ p ξ qq (6.50) OBOLEV IMPROVING FOR AVERAGES OVER CURVES 41 where the functions ω and ω arise from the remainder terms and satisfy | ω i p ξ ; s q| „ k for i “ α p ξ ; s q : “ u , p ξ q ` ω p ξ ; s q ¨ p s ´ θ p ξ qq ,β p ξ ; s q : “ ω p ξ ; s q ´ ω p ξ ; s q , so that (6.49) and (6.50) can be rewritten as x γ p s q , ξ y “ u p ξ q ` ` α p ξ ; s q ` β p ξ ; s q ¨ p s ´ θ p ξ qq ˘ ¨ p s ´ θ p ξ qq , (6.51) x γ p s q , ξ y “ α p ξ ; s q ¨ p s ´ θ p ξ qq . (6.52)The argument splits into two cases: Case 1: | α p ξ ; s q| ď ρ k ´ ℓ . By the integral form of the remainder, β p ξ ; s q ¨ p s ´ θ p ξ qq “ ´ ˆ sθ p ξ q x γ p q p t q , ξ y ¨ p s ´ t q ¨ p t ´ θ p ξ qq d t. Recall from (6.3) that x γ p q p t q , ξ y ą t P r´ , s . Thus, the integrand in the above displayhas constant sign. Furthermore, (6.2) also guarantees that |x γ p q p t q , ξ y| „ k . Combining theseobservations, | β p ξ ; s q| „ k for all p ξ ; s q P supp a k,ℓ, .Thus, provided ρ is chosen sufficiently small, the hypothesis | α p ξ ; s q| ď ρ k ´ ℓ and together withthe bound | s ´ θ p ξ q| ě ρ ´ ℓ from (6.42) imply | β p ξ ; s q|| s ´ θ p ξ q| ´ | α p ξ ; s q| Á ρ k ´ ℓ . Consequently, (6.48) implies that the second term dominates the right-hand side of (6.51) andtherefore |x γ p s q , ξ y| Á ρ k ´ ℓ . Case 2: | α p ξ ; s q| ě ρ k ´ ℓ . Here (6.48) and (6.52) immediately imply |x γ p s q , ξ y| Á ρ k ´ ℓ .Thus, in either case the desired bound (6.41) holds.b) If ℓ “ t p k ` ℓ q{ u , then the desired bound follows from Plancherel’s theorem and the van derCorput lemma with third order derivatives. Indeed, by Lemma 6.4, |x γ p q p s q , ξ y| „ ρ { k ´ ℓ for all p ξ ; s q P supp b ν, p ε q k, ℓ . (6.53)For the remaining cases, it suffices to show that |x γ p s q , ξ y| ` ´p ℓ ´ ℓ q{ |x γ p s q , ξ y| Á k ´ ℓ for all p ξ ; s q P supp b ν, p ε q k, ℓ . (6.54)Here the localisation of the symbol ensures the key properties | u p ξ q| „ ρ k ´ ℓ , | u p ξ q| „ ρ k ´ ℓ , | s ´ θ p ξ q| À ρ ´ ℓ for all p ξ ; s q P supp b ν, p ε q k, ℓ . (6.55)By Taylor expansion around θ p ξ q , we obtain x γ p s q , ξ y “ u p ξ q ` ω p ξ ; s q ¨ p s ´ θ p ξ qq , (6.56) x γ p s q , ξ y “ ω p ξ ; s q ¨ p s ´ θ p ξ qq , (6.57)where the functions ω and ω arise from the remainder terms and satisfy | ω i p ξ ; s q| „ ρ { k ´ ℓ for i “
1, 2 by (6.53). The argument splits into two cases:
Case 1: | θ p ξ q ´ s | ď ρ ´p ℓ ´ ℓ q{ . Provided ρ ą | ω p ξ ; s q| „ ρ { k ´ ℓ imply that the u p ξ q term dominates the right-hand side of (6.56)and therefore |x γ p s q , ξ y| Á ρ k ´ ℓ . Case 2: | θ p ξ q ´ s | ě ρ ´p ℓ ´ ℓ q{ . In this case, the bound | ω p ξ q| „ ρ { k ´ ℓ and (6.57)immediately imply |x γ p s q , ξ y| Á ρ { k ´ ℓ ´p ℓ ´ ℓ q{ .Thus, in either case the desired bound (6.54) holds. (cid:3) Lemma 6.13. a) For all ď ℓ ď t k { u , µ P Z , ď ι ď and ε ą , we have } m r a µ, p ε q k,ℓ,ι s} M p R q À ´p ´ ε q ℓ . b) For ℓ “ p ℓ , ℓ q P Λ p k q , ν P Z and ε ą , we have } m r b ν, p ε q k, ℓ s} M p R q À ´p ´ ε qp ℓ ´ ℓ q{ . Proof.
In view of the support properties of the symbols (see Lemma 6.6 and Lemma 6.7), by anintegration-by-parts argument (see Lemma C.2), the problem is reduced to showing | ∇ N e j p s µ q a µk,ℓ,ι p ξ ; s q| À N ´p k ´p ´ j q ℓ q N , (6.58a) | ∇ N e j p s ν q b νk, ℓ p ξ ; s q| À N ´p k ´pp ´ j qp ℓ ´ ℓ q{ ` ℓ q_ q N (6.58b)for all 1 ď j ď N P N .For all N P N , we claim the following: ‚ For all ξ P supp ξ a µk,ℓ,ι , 1 ď ι ď ℓ | ∇ N e j p s µ q θ p ξ q| , ´ k ` ℓ | ∇ N e j p s µ q u p ξ q| , ´ k ` ℓ | ∇ N e j p s µ q u , p ξ q| À N ´p k ´p ´ j q ℓ q N ; (6.59) ‚ For all ξ P supp ξ a µk,ℓ,ι , 3 ď ι ď ℓ | ∇ N e j p s µ q θ p ξ q| , ´ k ` ℓ | ∇ N e j p s µ q u p ξ q| À N ´p k ´p ´ j q ℓ q N ; (6.60) ‚ For all ξ P supp ξ b νk, ℓ ,2 ℓ | ∇ N e j p s ν q θ p ξ q| , ´ k ` ℓ | ∇ N e j p s ν q u p ξ q| , ´ k ` ℓ | ∇ N e j p s ν q u , p ξ q| À N ´p k ´p ´ j q ℓ q N ; (6.61) ‚ For all ξ P supp ξ b νk, ℓ ,2 p ℓ ´ ℓ q{ | ∇ N e j p s ν q θ p ξ q| , ´ k ` ℓ | ∇ N e j p s ν q u p ξ q| À N ´p k ´pp ´ j qp ℓ ´ ℓ q{ ` ℓ q_ q N . (6.62)Once the above claims are established, the derivative bounds (6.58a) and (6.58b) follow directlyfrom the chain and Leibniz rule.In order to prove (6.59)-(6.62) we work with the unified framework introduced in § n , s P R , recall the set Ξ p k, n ; s q introduced in (6.19).In particular, if ξ P Ξ p k, n ; s q , then ξ P supp a k and ξ lies in the domain of θ and satisfies | θ p ξ q ´ s | À ´ n and | u p ξ q| À k ´ n . (6.63)From the discussion following (6.19), we know thatsupp ξ a µk,ℓ,ι Ď Ξ p k, ℓ ; s µ q for 1 ď ι ď ξ b νk, ℓ Ď Ξ ` k, ℓ ; s ν ˘ . Let ξ P Ξ p k, n ; s q and for 1 ď j ď v j : “ e j p s q . The bounds (6.59) and (6.61) amountto proving that2 n | ∇ N v j θ p ξ q| , ´ k ` n | ∇ N v j u , p ξ q| , ´ k ` n | ∇ N v j u p ξ q| À N ´p k ´p ´ j q n q N (6.64)hold for all N P N . These bounds follow from repeated application of the chain rule, provided |x γ p q ˝ θ p ξ q , ξ y| Á k , (6.65a) |x γ p K q ˝ θ p ξ q , ξ y| À K k ` n p K ´ q , (6.65b) |x γ p K q ˝ θ p ξ q , v j y| À K p K ´ j q n (6.65c) OBOLEV IMPROVING FOR AVERAGES OVER CURVES 43 hold for all K ě
2. In particular, assuming (6.65a), (6.65b) and (6.65c), the bounds in (6.64)are then a consequence of Lemma B.1 in the appendix. More precisely, the desired estimates in(6.64) correspond to (B.2) and two separate instances of (B.4) whilst the hypotheses in the abovedisplay correspond to (B.1) and (B.3). Here the parameters featured in the appendix are chosenas follows: g h A B M M e γ p q γ k ´ n k ´ n ´ k `p ´ j q n n v j γ p q γ k ´ n k ´ n ´ k `p ´ j q n n v j The conditions (6.65a), (6.65b) and (6.65c) follow directly from the definition of Ξ p k, n ; s q .Indeed, (6.65a) and the K ě a k . The K “ x γ p q ˝ θ p ξ q , ξ y “ K “ | u p ξ q| À k ´ n from (6.63). Finally,(4.13) together with the θ localisation hypothesis from (6.63) imply that |x γ p K q ˝ θ p ξ q , v j y| À K | θ p ξ q ´ s | p j ´ K q_ À ´pp j ´ K q_ q n which yields (6.65c).We next turn to (6.60) and (6.62). Given n “ p n , n q P R and s P R , recall the set Ξ p k, n ; s q introduced in (6.21). In particular, if ξ P Ξ p k, n ; s q , then ξ P supp a k and ξ lies in the domain of θ and satisfies | θ p ξ q ´ s | À ´ n and | u , p ξ q| „ k ´ n . (6.66)From the discussion following (6.21), we know thatsupp ξ a µk,ℓ,ι Ď Ξ p k, ℓ, ℓ ; s µ q for ι “
3, 4 and supp ξ b νk, ℓ Ď Ξ ` k, ℓ ´ ℓ , ℓ ; s ν ˘ . Let ξ P Ξ p k, n ; s q where n “ p n , n q for some 0 ă n ď n and for 1 ď j ď v j : “ e j p s q .The bounds (6.60) and (6.62) amount to proving that2 n | ∇ N v j θ p ξ q| , ´ k ` n ` n | ∇ N v j u p ξ q| À N ´p k ´pp ´ j q n ` n q_ q N (6.67)hold for all N P N . These bounds follow from repeated application of the chain rule, provided |x γ p q ˝ θ p ξ q , ξ y| Á k ´ n , (6.68a) |x γ p K q ˝ θ p ξ q , ξ y| À K k ` n p K ´ q´ n , (6.68b) |x γ p K q ˝ θ p ξ q , v j y| À K n p K ´ q´ n `pp ´ j q n ` n q_ (6.68c)hold for all K ě
2. In particular, assuming (6.68a), (6.68b) and (6.68c), the bounds in (6.67) arethen a consequence of Lemma B.1 in the appendix. More precisely, the desired estimates in (6.67)correspond to (B.2) and (B.4) whilst the hypotheses in the above display correspond to (B.1) and(B.3). Here the parameters featured in the appendix are chosen as follows: g h A B M M e γ γ k ´ n ´ n k ´ n ´ n ´ k `pp ´ j q n ` n q_ n v j The conditions (6.68a), (6.68b) and (6.68c) follow directly from the definition of Ξ p k, n ; s q .Indeed, (6.68a) and the K “ | u , p ξ q| „ k ´ n from (6.66). The K ě symbol a k whist the remaining K “ x γ ˝ θ p ξ q , ξ y “ θ localisation hypothesis from (6.66) imply that |x γ p K q ˝ θ p ξ q , v j y| À N | θ p ξ q ´ s | p j ´ K q_ À pp j ´ K q_ q n which, by directly comparing exponents, yields (6.68c). (cid:3) Lemma 6.12 and Lemma 6.13 can be combined to obtain the following L p bounds. Corollary 6.14.
For all ď p ď 8 and all ε ą , the following inequalities hold:a) For all ď ℓ ď t k { u and ď ι ď , ´ ÿ µ P Z } m r a µ, p ε q k,ℓ,ι sp D q f } pL p p R q ¯ { p À ´ k { p ` ℓ p { p ´ q` εℓ } f } L p p R q . b) For all ℓ “ p ℓ , ℓ q P Λ p k q , ´ ÿ ν P Z } m r b ν, p ε q k, ℓ sp D q f } pL p p R q ¯ { p À ´ k { p `p ℓ ` ℓ q{ p ´p ℓ ´ ℓ qp { ´ { p ´ ε q } f } L p p R q . When p “ 8 the left-hand ℓ p -sums are interpreted as suprema in the usual manner.Proof. For p “ L bounds from Lemma 6.12 witha simple orthogonality argument, as the supports of m r a µ, p ε q k,ℓ,ι s and m r b ν, p ε q k, ℓ s are essentially disjointfor different µ and ν respectively. For p “ 8 the estimate is a restatement of the L boundsfrom Lemma 6.13. Interpolating these two endpoint cases, using mixed norm interpolation (see,for instance, [21, § (cid:3) Putting everything together.
We are now ready to combine the ingredients to concludethe proof of Proposition 6.5.
Proof of Proposition 6.5. a) Let 1 ď ι ď
4. By Proposition 6.9 a), for all 2 ď p ď
12 and all ε ą } m r a k,ℓ,ι sp D q f } L p p R q “ ››› ÿ µ P Z m r a µk,ℓ,ι sp D q f ››› L p p R q À ε ℓ p { ´ { p q` εℓ ´ ÿ µ P Z } m r a µk,ℓ,ι sp D q f } pL p p R q ¯ { p . Moreover, for all µ P Z , Lemma 6.11 a) implies that } m r a µk,ℓ,ι s} M p p R q À N,ε,p } m r a µ, p ε q k,ℓ,ι s} M p p R q ` ´ k for all N P N .Combining the above, we obtain } m r a k,ℓ,ι sp D q f } L p p R q À ε,p ℓ p { ´ { p q` εℓ ´ ÿ µ P Z } m r a µ, p ε q k,ℓ,ι sp D q f } pL p p R q ¯ { p ` ´ k } f } L p p R q , which, together with Corollary 6.14 a), yields } m r a k,ℓ,ι sp D q f } L p p R q À ε,p ´ k { p ´ ℓ p { ´ { p ´ ε q } f } L p p R q . Since ε ą ď p ď
12 and all ε ą } m r b k, ℓ sp D q f } L p p R q À ε ℓ p { ´ { p ` ε q p ℓ ´ ℓ qp ´ { p ` ε q{ ´ ÿ ν P Z } m r b νk, ℓ sp D q f } pL p p R q ¯ { p . Moreover, for all ν P Z , Proposition 6.11 b) implies that } m r b νk, ℓ s} M p p R q À N,ε,p } m r b ν, p ε q k, ℓ s} M p p R q ` ´ kN for all N P N . OBOLEV IMPROVING FOR AVERAGES OVER CURVES 45
Combining the above, we obtain } m r b k, ℓ sp D q f } L p p R q À ε,p ℓ p { ´ { p ` ε q p ℓ ´ ℓ qp ´ { p ` ε q{ ´ ÿ ν P Z } m r b ν, p ε q k, ℓ sp D q f } pL p p R q ¯ { p ` ´ k } f } L p p R q , which, together with Corollary 6.14 b), yields } m r b k, ℓ sp D q f } L p p R q À ε,p ´ p ℓ ´ ℓ qp { p ´ ε q´ ℓ p { ´ { p ´ ε q } f } L p p R q . Since ε ą (cid:3) We have established Proposition 6.5 and therefore completed the proof of the J “ Proof of the decoupling inequalities
This section is devoted to the proof of Theorem 4.4.7.1.
Decoupling inequalities for non-degenerate curves.
The central ingredient in the proofof Theorem 4.4 is the decoupling theorem of Bourgain–Demeter–Guth [5]. We begin by recallingthe statement of (one formulation of) this result. Given a non-degenerate curve g P C d ` p I ; ˆ R d q and 0 ă r ď
1, an ‘anisotropic r -neighbourhood’ of the curve is constructed as follows. Definition 7.1.
For each s P I define the parallelepiped α p s ; r q : “ ξ P ˆ R d : ξ “ g p s q ` d ÿ j “ λ j r j g p j q p s q for some λ j P r´ , s , ď j ď d ( ; such sets are referred to as r -slabs. In some cases it is useful to highlight the choice of function g by writing α p g ; s ; r q for a r -slab α p s ; r q . Note that the formula for the parallelepiped α p s ; r q can be expressed succinctly in termsof the matrix r g s s,r introduced in (2.1). In particular, α p s ; r q “ g p s q ` r g s s,r ` r´ , s d ˘ . (7.1)An anisotropic r -neighbourhood of the curve g is formed by taking the union of all the r slabsas s varies over I . Definition 7.2.
A collection A p r q of r -slabs is a slab decomposition for g if it consists of preciselythe r -slabs α p g ; s ; r q for s varying over a r -net in I . With the above definitions, the decoupling theorem may be stated as follows.
Theorem 7.3 (Bourgain–Demeter–Guth [5]) . Let g P G d p δ q for some ă δ ! , ă r ď and A p r q be a r -slab decomposition for g . For all ď p ď d p d ` q and ε ą the inequality ››› ÿ α P A p r q f α ››› L p p R d q À ε r ´ ε ´ ÿ α P A p r q } f α } L p p R d q ¯ { (7.2) holds for any tuple of functions p f α q α P A p r q satisfying supp ˆ f α Ď α .Remark. This is a slight variant of the decoupling inequality of Bourgain–Demeter–Guth [5] whichcan be found, for instance, in [10]. It is also remarked that the result holds for general non-degenerate curves, although not in the uniform fashion described here. Note, in particular, that More precisely, the general version of the decoupling theorem here follows by combining Theorem 1.2 and Lemma3.6 from [10]. by restricting to the model curves g P G d p δ q for 0 ă δ !
1, the decoupling inequality (7.2) holdswith a constant independent of both the choice of g and δ .7.2. Geometric observations.
In order to relate Theorem 4.4 to the Bourgain–Demeter–Guthresult from Theorem 7.3, we first relate the Frenet boxes π d ´ ,γ p s ; r q to certain regions whichare more similar in form to the slabs α p g ; s, r q introduced above. The Frenet boxes π d ´ ,γ p s ; r q do not correspond precisely to slabs but to related regions referred to as plates . These plateregions are formed by extending d -dimensional slabs into n -dimensions by adjoining additionallong directions. Moreover, the plates are naturally defined in relation to a cone generated over afamily of non-degenerate curves g j : I Ñ R d . A family of cones.
Let γ P G n p δ q for 0 ă δ ! e j : r´ , s Ñ S n ´ for 1 ď j ď n bethe associated Frenet frame. Without loss of generality, in proving Theorem 4.4 we may alwayslocalise so that we only consider the portion of the curve lying over the interval I “ r´ δ, δ s . Inthis case e j p s q “ ~e j ` O p δ q for 1 ď j ď n (7.3)where, as in Definition 2.1, the ~e j denote the standard basis vectors.Here we introduce certain conic surfaces which are ‘generated’ over the curves s ÞÑ e j p s q . Thefollowing observations extend the analysis of [17], where a cone in R generated by the binormal e features prominently in the proof of the 3-dimensional analogue of Theorem 1.1.Let 2 ď d ď n ´ R n ´ d ˆ I Ñ R n defined by˜Γ p ~λ, s q : “ n ÿ j “ d ` λ j e j p s q , ~λ “ p λ d ` , . . . , λ n q . Restricting to λ d ` bounded away from zero, this is a regular parametrisation of a p n ´ d ` q -dimensional surface in R n , which is denoted Γ n,d . Indeed, by the Frenet formulæ, B ˜Γ B s p ~λ, s q “ ´ λ d ` ˜ κ d p s q e d p s q ` E d p ~λ, s q , B ˜Γ B λ j p ~λ, s q “ e j p s q , d ` ď j ď n, where E d p ~λ, s q lies in the subspace x e d ` p s q , . . . , e n p s qy . Thus, provided λ d ` is bounded awayfrom zero, the non-vanishing of ˜ κ d ensures that these tangent vectors are linearly independent. Reparametrisation.
It is convenient to reparametrise Γ n,d so that it is realised as a surface‘generated’ over an alternative family of curves which is formed by graphs. To this end, let A : I Ñ GL p n ´ d, R q be given by A p s q : “ »—– e d ` ,d ` p s q ¨ ¨ ¨ e n,d ` p s q ... ... e d ` ,n p s q ¨ ¨ ¨ e n,n p s q fiffifl ´ , where e i,j p s q denotes the j th component of e i p s q . Provided δ is chosen sufficiently small, (7.3)ensures that the above matrix inverse is well-defined and, moreover, is a small perturbation of theidentity matrix. Define the reparametrisationΓ p ~λ, s q : “ ˜Γ p A p s q ~λ, s q for all p ~λ, s q P R n ´ d ˆ I . (7.4)Consider the restriction of this mapping to the set R n,d Ă R n ´ d consisting of all vectors ~λ “p λ d ` , . . . , λ n q satisfying1 { ď λ d ` ď | λ j | ď d ` ď j ď n ; (7.5) OBOLEV IMPROVING FOR AVERAGES OVER CURVES 47 under this restriction, Γ is a regular parametrisation by the preceding observations.The mapping (7.4) can be expressed in matrix form asΓ p ~λ, s q “ “ e d ` p s q ¨ ¨ ¨ e n p s q ‰ ˝ A p s q ~λ, (7.6) “ “ G d ` p s q ¨ ¨ ¨ G n p s q ‰ ~λ, where the G j : I Ñ R n (which form the column vectors of the above matrix) are of the form G j p s q “ „ g j p s q ` ~e j for some smooth function g j : I Ñ R d . Non-degeneracy conditions.
Given a “ p a d ` , . . . , a n q P R n,d , define G a : “ n ÿ j “ d ` a j ¨ G j and g a : “ n ÿ j “ d ` a j ¨ g j , (7.7)noting G a p s q “ Γ p a , s q . The curve g a : I Ñ R d is non-degenerate. To see this, first note that B i Γ B s i p ~λ, s q can be expressed as a linear combination of vectors of the form ” e p ℓ q d ` p s q ¨ ¨ ¨ e p ℓ q n p s q ı ˝ A p i ´ ℓ q p s q ~λ, ď ℓ ď i, (7.8)where A p k q denotes the component-wise k th-derivative of A . Indeed, this follows simply by applyingthe Leibniz rule to (7.6). Consequently, B i Γ B s i p ~λ, s q must lie in the subspace generated by the columnsof the left-hand matrix in (7.8), where i is allowed to vary over the stated range. In particular,one concludes from the Frenet formulæ that B i Γ B s i p ~λ, s q P x e d ` ´ i p s q , . . . , e n p s qy for 0 ď i ď d . (7.9)On the other hand, the Frenet formulæ also show that the e d ` ´ i p s q component of B i Γ B s i p ~λ, s q arisesonly from the term in (7.8) corresponding to ℓ “ i and x B i Γ B s i p ~λ, s q , e d ` ´ i p s qy “ p´ q i ´ d ź ℓ “ d ` ´ i ˜ κ ℓ p s q ¯ x ~A p s q , ~λ y , (7.10)where ~A p s q denotes the first row of A p s q . Recall A is a small perturbation of the identity matrix.Thus, under the constraint ~λ P R n,d from (7.5), if δ is chosen sufficiently small, then (7.10) impliesthat |x B i Γ B s i p ~λ, s q , e d ` ´ i p s qy| „ ď i ď d . (7.11)Thus, combining (7.9) and (7.11), it follows that the vectors B i Γ B s i p ~λ, s q , 1 ď i ď d , are linearlyindependent. Moreover, fixing ~λ “ a and noting that G p i q a p s q “ B i Γ B s i p a , s q P R d ˆ t u n ´ d for i ě | det r g a s s | Á s P I , which is the claimed non-degeneracy condition. Note this holds uniformly over thechoice of original curve γ P G n p δ q and over a P R n,d . Frenet boxes revisited.
From the preceding observations, the vectors G p i q a p s q for 1 ď i ď d form abasis of R d ˆ t u n ´ d . Fixing ξ P ˆ R n and r ą
0, one may write ξ ´ n ÿ j “ d ` ξ j G j p s q “ d ÿ i “ r i η i G p i q a p s q (7.13) for some vector of coefficients p η , . . . , η d q P R d . The powers of r appearing in the above expressionplay a normalising rˆole below. For each 1 ď k ď d form the inner product of both sides of theabove identity with the Frenet vector e k p s q . Combining the resulting expressions with the linearindependence relations inherent in (7.9), the coefficients η k can be related to the numbers x ξ, e k p s qy via a lower anti-triangular transformation, viz. »—– x ξ, e p s qy ... x ξ, e d p s qy fiffifl “ »—– ¨ ¨ ¨ x G p d q a p s q , e p s qy ... . . . ... x G p q a p s q , e d p s qy ¨ ¨ ¨ x G p d q a p s q , e d p s qy fiffifl »—– rη ... r d η d fiffifl . (7.14)Thus, if ξ P π d ´ ,γ p s ; r q , then it follows from combining (4.14a) and (7.11) with (7.14) that | η i | À γ ď i ď d , provided δ ą π d ´ ,γ p s ; r q Ď R n,d : “ r´ , s d ˆ R n,d The identity (7.13) can be succinctly expressed using matrices. In particular, collect the func-tions g j together as an p n ´ d q -tuple g : “ p g d ` , . . . , g n q and, for s P I and r ą
0, define the n ˆ n matrix r g s a ,s,r : “ ˆ r g a s s,r g p s q n ´ d ˙ . (7.15)Here the block r g a s s,r is the d ˆ d matrix (2.1) with γ here taken to be g a as defined in (7.7), whilst g p s q is understood to be the p n ´ d q ˆ d matrix with j th column equal to g j p s q and I n ´ d is the p n ´ d q ˆ p n ´ d q identity matrix. With this notation, the identity (7.13) may be written as ξ “ r g s a ,s,r ¨ η where η “ p η , . . . , η d , ξ d ` , . . . , ξ n q .Moreover, if ξ P π d ´ ,γ p s ; r q , then the preceding observations show that η in the above equationmay be taken to lie in a bounded region and so π d ´ ,γ p s ; r q Ď č a P R n,d r g s a ,s,Cr ` r´ , s n ˘ X R n,d , (7.16)where C ě Decoupling inequalities for cones generated by non-degenerate curves.
Here thegeometric setup described in § p g d ` , . . . , g n q . Definition 7.4.
Let ď d ď n ´ , g “ p g d ` , . . . , g n q be an p n ´ d q -tuple of functions in C d ` p I ; R d q and Γ g denote the codimension d ´ cone in R n parametrised by p ~λ, s q ÞÑ n ÿ j “ d ` λ j ¨ ´ „ g j p s q ` ~e j ¯ for ~λ “ p λ d ` , . . . , λ n q P R n,d and s P I. In this case, Γ g is referred to as the cone generated by g . We now take into account the non-degeneracy condition established in (7.12). Given a “p a d ` , . . . , a n q P R n ´ d and 0 ă δ ! G a n,d p δ q of all p n ´ d q -tuples offunctions g “ p g d ` , . . . , g n q P r C d ` p I ; R d qs n ´ d OBOLEV IMPROVING FOR AVERAGES OVER CURVES 49 with the property g a : “ n ÿ j “ d ` a j ¨ g j P G d p δ q , (7.17)where G d p δ q is the class of model curves introduced in § g a relevant to our study are non-degenerate , which is aweaker condition than g a P G d p δ q (provided 0 ă δ ! §
2, we will always be able to assume the condition (7.17) holdsin what follows (see the proof of Lemma 7.9 for details of the rescaling).Given g P G a n,d p δ q , s P r´ , s and 0 ă r ď
1, define the n ˆ n matrix r g s a ,s,r as in (7.15); thatis, r g s a ,s,r : “ ˆ r g a s s,r g p s q n ´ d ˙ . (7.18)In view of (7.16), one wishes to study decoupling with respect to plates θ p s ; r q : “ r g s a ,s,r ` r´ , s n ˘ X R n,d . In some cases it will be useful to highlight the choice of function g by writing θ p g ; s ; r q for θ p s ; r q .Note that each of these plates lies in an r -neighbourhood of the cone Γ g . We think of the unionof all plates θ p s ; r q as s varies over the domain r´ , s as forming an anisotropic r -neighbourhoodof Γ g , similar to the situation for curves described in § θ p s ; r q directly, certain truncated versions are considered. Definition 7.5.
For ă r ď , a “ p a d ` , . . . , a n q P R n ´ d and K ě an p a , K q -truncated r -platefor Γ g is a set of the form θ a ,K p s ; r q : “ r g s a ,s,r ` r´ , s n ˘ X Q p a , K ´ q for some s P I and Q p a , K ´ q : “ ξ P ˆ R n : | ξ j ´ a j | ď K ´ for d ` ď j ď n ( . Definition 7.6.
A collection Θ a ,K p r q of p a , K q -truncated r -plates is an p a , K q -truncated platedecomposition for g if it consists of θ a ,K p g ; s ; r q for s varying over a r -net in I . Theorem 4.4 is a consequence of the following decoupling inequality for cones Γ g . Proposition 7.7.
Let ď d ď n ´ and ε ą . There exists some integer K ě such that forall ă r ď , a P R n ´ d and g P G a n,d p δ q for ď δ ! the following holds. If Θ a ,K p r q is an p a , K q -truncated r -plate decomposition for Γ g and ď p ď d p d ` q , then ›› ÿ θ P Θ a ,K p r q f θ ›› L p p R n q À ε r ´ ε ´ ÿ θ P Θ a ,K p r q } f θ } L p p R n q ¯ { holds for any tuple of functions p f θ q θ P Θ a ,K p r q satisfying supp ˆ f θ Ď θ . Proposition 7.7 follows from the Bourgain–Demeter–Guth result (namely, Theorem 7.3) viaan argument from [4], where decoupling estimates for the light cone in R n were obtained as aconsequence of decoupling estimates for the paraboloid in R n ´ . The key observation is that,at suitably small scales, the cone Γ g can be approximated by a cylinder over the curve g . Thisapproximation is only directly useful for relatively large r values, but rescaling and induction-on-scale arguments allow one to leverage this observation in the small r setting. Arguments of thiskind originate in [17] and have been used repeatedly in the context of decoupling theory: see, forinstance, [2, 11, 12, 15].The details of the proof of Proposition 7.7 are postponed until § Relating the decoupling regions.
Theorem 4.4 may now be deduced as a consequence ofProposition 7.7 using the geometric observations from § Proof of Theorem 4.4, assuming Proposition 7.7.
First note that it suffices to show the desireddecoupling inequality in the restricted range 2 ď p ď d p d ` q ; the estimate for the remainingrange d p d ` q ď p ď 8 then follows by an interpolation argument and a trivial estimate for p “ 8 .Let γ P G d p δ q for 0 ď δ !
1. As previously noted, we may restrict attention to the portion of γ over I “ r´ δ, δ s so that the Frenet vectors satisfy (7.3). Fix 2 ď d ď n ´
1, 0 ă r ď P d ´ p r q a Frenet box decomposition of γ .Define g “ p g d ` , . . . , g n q as in § g a are non-degenerate. Let ε ą K ě p f π q π P P d ´ p r q be a tuple of functions satisfying the Fourier support hypothesis from thestatement of Theorem 4.4. If π “ π d ´ ,γ p s ; r q P P d ´ p r q , then, recalling (7.16), we havesupp ˆ f π Ď π d ´ ,γ p s ; r q Ď č a P R n,d r g s a ,s,Cr ` r´ , s n ˘ X R n,d . (7.19)The frequency domain is decomposed according to the Q p a , K ´ q from Definition 7.5. In par-ticular, let R n,d p K q : “ K ´ Z n ´ d X R n,d so that the sets Q p a , K ´ q for a P R n,d p K q are finitely-overlapping and cover of R n,d . Form asmooth partition of unity p ψ a ,K ´ q a P R n,d p K q adapted to the sets Q p a , K ´ q and define the frequencyprojection operators P a via the Fourier transform by ` P a f ˘p : “ ψ a ,K ´ ¨ ˆ f . These operators are bounded on L p for 1 ď p ď 8 uniformly in a and K and, furthermore, f π “ ÿ a P R n,d p K q P a f π for all π P P d ´ p r q .Since R n,d p K q À n,δ,ε
1, by the triangle inequality and the L p boundedness of the P a , it sufficesto show that ›› ÿ π P P d ´ p r q P a f π ›› L p p R n q À n,δ,ε r ´p { ´ { p q´ ε ´ ÿ π P P d ´ p r q } P a f π } pL p p R n q ¯ { p (7.20)uniformly in a P R n,d p K q . However, recalling (7.19), each function P a f π has frequency supportin the set θ a ,K p s, Cr q “ r g s a ,s,Cr ` r´ , s n ˘ X Q p a , K ´ q and so an ℓ version of (7.20) follows as a consequence of Proposition 7.7. The desired ℓ p -decoupling (7.20) follows by applying H¨older’s inequality to the ℓ -sum. (cid:3) Strictly speaking, Proposition 7.7 requires the additional hypothesis g P G a n,d p δ q . However, by a rescalingargument (see the proof of Lemma 7.9), the decoupling result generalises to arbitrary g for which g a is non-degenerate(albeit no longer with a uniform constant). OBOLEV IMPROVING FOR AVERAGES OVER CURVES 51
Proof of Proposition 7.7.
It remains to prove the decoupling Proposition 7.7. This isachieved using the argument outlined at the end of § Definition 7.8 (Decoupling constant) . For ď d ď n ´ , ă r ď , p ě , ă δ ! , a P R n ´ d and K ě let D a n,d p K ; r q denote the infimum over all C ě for which ›› ÿ θ P Θ a ,K p r q f θ ›› L p p R n q ď C ´ ÿ θ P Θ a ,K p r q } f θ } L p p R n q ¯ { holds whenever:i) Θ a ,K p r q is an p a , K q -truncated r -plate decomposition for Γ g for some g P G a n,d p δ q ,ii) p f θ q θ P Θ a ,K p r q is a tuple of functions satisfying supp ˆ f θ Ď θ . Thus, in this notation, Proposition 7.7 states that for all ε ą K ě n and ε , such that D a n,d p K ; r q À ε r ´ ε for all a P R n ´ d (7.21) Remark.
The definition of the decoupling constants also depends on p and δ but, for simplicity,these parameters are omitted in the notation.In conjunction to Theorem 7.3, one needs a simple scaling lemma. Lemma 7.9 (Generalised Lorentz rescaling) . If ă r ă ρ ă , then D a n,d p K ; r q À D a n,d p K ; ρ q D a n,d p K ; r { ρ q . Temporarily assuming this result, Proposition 7.7 follows by a simple induction-on-scale argu-ment.
Proof of Proposition 7.7.
Let ε ą
0, 0 ă δ ! a “ p a d ` , . . . , a n q P R n ´ d be given. Hence-forth, K “ K p ε q ě n and ε , which is chosensufficiently large to satisfy the forthcoming requirements of the proof. It will be shown, by aninduction-on-scale in the r parameter, that (7.21) holds for all 0 ă r ď p K q ´ d ă r ď
1, then it follows from the triangle and Cauchy–Schwarz inequalities that D a n,d p K ; r q ď C p ε q for some constant C p ε q ě n and ε . This serves as the base case of an inductiveargument.It remains to establish the inductive step. To this end, fix some 0 ă r ď p K q ´ d and assumethe following holds. Induction hypothesis. If r ˝ ě r , then D a n,d p K ; r ˝ q ď C p ε q r ´ ε ˝ .Given 0 ă r ă ρ ă {
2, one may combine the generalised Lorentz rescaling lemma with theinduction hypothesis to conclude that D a n,d p K ; r q À D a n,d p K ; ρ q D a n,d p K ; r { ρ q ď C p ε q ρ ε r ´ ε D a n,d p K ; ρ q . (7.22)Fix ρ : “ K ´ d . Favourable bounds for D a n,d p K ; ρ q can be obtained in this case via an appeal toTheorem 7.3. Let proj d : ˆ R n Ñ ˆ R d denote the orthogonal projection onto the coordinate planespanned by ~e , . . . , ~e d . The key observation is that any p a , K q -truncated ρ -plate θ a ,K p g ; s ; ρ q onΓ g essentially projects into a ρ -slab α p g a ; s ; ρ q on g a “ ř nj “ d ` a j ¨ g j under this mapping, where α p g a ; s ; ρ q is as defined in Definition 7.1. In particular,proj d θ a ,K p g ; s ; ρ q Ď α p g a ; s ; Cρ q (7.23) for some choice of constant C ě n . To see this, fix ξ P θ a ,K p g ; s ; ρ q and notethat ξ “ r g s a ,s,ρ ¨ η for some η P r´ , s n whilst | ξ j ´ a j | ď { K for d ` ď j ď n , (7.24)by Definition 7.5. By the definition of the matrix r g s a ,s,ρ in (7.18), it follows that ξ “ r g a s s,ρ ¨ η ` n ÿ j “ d ` η j g j p s q ,ξ j “ η j for d ` ď j ď n where ξ : “ proj d ξ and η : “ proj d η P r´ , s d . In particular, ξ ´ g a p s q “ r g a s s,ρ ´ η ` r g a s ´ s,ρ ¨ n ÿ j “ d ` p η j ´ a j q g j p s q ¯ and, for the choice of ρ “ K ´ d specified above, ˇˇ r g a s ´ s,ρ ¨ n ÿ j “ d ` p η j ´ a j q g j p s q ˇˇ ď }r g a s ´ s,ρ } op ¨ n ÿ j “ d ` | ξ j ´ a j || g j p s q|À ρ ´ d K ´ ď . The second inequality follows from the hypothesis g a P G d p δ q from (7.17), which implies that }r g a s ´ s,ρ } op À ρ ´ d (with a uniform constant), and the condition (7.24). Recalling Definition 7.1, itfollows that ξ P α p g a ; s ; Cρ q , as claimed.Let Θ a ,K p ρ q be an p a , K q -truncated ρ -plate decomposition of Γ g and p f θ q θ P Θ a ,K p ρ q be a tuple offunctions satisfying supp ˆ f θ Ď θ . For any 2 ď p ď d p d ` q and ˜ ε ą
0, by (7.23) and Theorem 7.3it follows that ›› ÿ θ P Θ a ,K p ρ q f θ p ¨ , x q ›› L p p R d q À ˜ ε ρ ´ ˜ ε ´ ÿ θ P Θ a ,K p ρ q } f θ p ¨ , x q} L p p R d q ¯ { for all x P R n ´ d . Taking the L p -norm of both sides of this inequality with respect to x and usingMinkowski’s inequality to bound the resulting right-hand side, one deduces that D a n,d p K ; ρ q À ˜ ε ρ ´ ˜ ε . (7.25)Taking ˜ ε : “ ε { D a n,d p K ; r q ď ` C ε ρ ε { ˘ C p ε q r ε , where the C ε factor arises from the various implied constants in the above argument. Thus, if K is chosen from the outset to be sufficiently large, depending only on n and ε , then C ε ρ ε { “ C ε K ´ εd { ď (cid:3) It remains to prove the Lorentz rescaling lemma. Before presenting the argument, it is usefulto introduce an extension of Definition 2.1 to tuples of curves g . Definition 7.10.
Let g “ p g d ` , . . . , g n q P G a n,d p δ q and g a : “ ř nj “ d ` a j ¨ g j , as in (7.17) and (7.18) . Define the p a ; b, ρ q -rescaling of g to be the p n ´ d q -tuple g a ,b,ρ “ p g a ,b,ρ,d ` , . . . , g a ,b,ρ,n q P r C d ` p I, R d qs n ´ d given by g a ,b,ρ p t q “ r g a s ´ b,ρ ` g p b ` ρt q ´ g p b q ˘ . (7.26) OBOLEV IMPROVING FOR AVERAGES OVER CURVES 53
Here g a ,b,ρ p t q and g p t q are understood to be the d ˆ p n ´ d q matrices whose columns are thecomponent functions of g a ,b,ρ and g , respectively, evaluated at t P I . As a consequence of this definition, the function g a ,b,ρ : “ n ÿ j “ d ` a j ¨ g a ,b,ρ,j is precisely the p b, ρ q -rescaling of g a : “ ř nj “ d ` a j ¨ g j . Thus, the notation g a ,b,ρ used here isconsistent in the sense that g a ,b,ρ “ p g a q b,ρ and, furthermore, since g a ,b,ρ : “ p g a ,b,ρ q a , one has r g a ,b,ρ s a ,u,h : “ ˆ r g a ,b,ρ s u,h g a ,b,ρ p s q n ´ d ˙ . (7.27) Proof of Lemma 7.9.
Fix g P G a n,d p δ q , an p a , K q -truncated r -plate decomposition Θ a ,K p r q for Γ g and let p f θ q θ P Θ a ,K p r q be a tuple of functions satisfying supp ˆ f θ Ď θ . By a simple pigeonholingargument, there exists an p a , K q -truncated ρ -plate decomposition Θ a ,K p ρ q such that ›› ÿ θ P Θ a ,K p r q f θ ››› L p p R n q À ›› ÿ θ P Θ a ,K p ρ q f θ ››› L p p R n q where f θ : “ ÿ θ P Θ a ,K p r q θ Ă θ f θ for all θ P Θ a ,K p ρ q .Since supp ˆ f θ Ď θ , by definition ›› ÿ θ P Θ a ,K p r q f θ ››› L p p R n q À D a n,d p K ; ρ q ´ ÿ θ P Θ a ,K p ρ q } f θ } L p p R n q ¯ { . (7.28)The goal here is to show that } f θ } L p p R n q À D a n,d p K ; r { ρ q ´ ÿ θ P Θ a ,K p ρ q θ Ă θ } f θ } L p p R n q ¯ { (7.29)for each θ P Θ a ,K p ρ q . Indeed, once this is established, by combining (7.28) and (7.29) with thedefinition of D a n,d p K ; r q , one deduces the desired result.Fix an p a , K q -truncated ρ -plate θ p b ; ρ q P Θ a ,K p ρ q and recall that θ p b ; ρ q “ ξ P ˆ R n : pr g s a ,b,ρ q ´ ξ P r´ , s n ( X Q p a , K ´ q for Q p a , K ´ q as defined in Definition 7.5. Note that the preimage of θ p b ; ρ q under the r g s a ,b,ρ mapping is the set ` r g s a ,b,ρ ˘ ´ θ p b ; ρ q “ r´ , s n X Q p a , K ´ q . On the other hand, the p a , K q -truncated r -plate θ p s ; r q ” θ a ,K p s ; r q is transformed under ` r g s a ,b,ρ ˘ ´ into ξ P ˆ R n : ` r g s a ,s,r ˘ ´ ¨ r g s a ,b,ρ ξ P r´ , s n ( X Q p a , K ´ q . (7.30)The key observation is that ` r g s a ,s,r ˘ ´ ¨ r g s a ,b,ρ “ ` r g a ,b,ρ s a , s ´ bρ , rρ ˘ ´ (7.31)so that (7.30) corresponds to an p a , K q -truncated r { ρ -plate for the cone generated over the rescaledcurve tuple g a ,b,ρ . Once this established, (7.29) follows easily by a change of variable. Indeed, taking θ “ θ p b, ρ q and defining the functions ˜ f θ and ˜ f θ for θ P Θ a ,K p r q via the Fourier transformby ` ˜ f θ ˘p : “ ˆ f θ ˝ r g s a ,b,ρ and ` ˜ f θ ˘p : “ ˆ f θ ˝ r g s a ,b,ρ , by a linear change of variable the desired inequality (7.29) is equivalent to } ˜ f θ } L p p R n q À D a n,d p K ; r { ρ q ´ ÿ θ P Θ a ,K p ρ q θ Ă θ } ˜ f θ } L p p R n q ¯ { . However, since the preceding observations show that the ˜ f θ are Fourier supported on p a , K q -truncated r { ρ -plates for the cone generated over the rescaled curve tuple g a ,b,ρ , this bound followsdirectly from the definition of the decoupling constant.To prove (7.31), first note that it suffices to show r g s a ,b,ρ ¨ r g a ,b,ρ s a , s ´ bρ , rρ “ r g s a ,s,r . Recalling (7.27) (taking u : “ p s ´ b q{ ρ and h : “ r { ρ ) and carrying out the block matrix multipli-cation, this is equivalent to the pair of identities r g a s b,ρ ¨ r g a ,b,ρ s s ´ bρ , rρ “ r g a s s,r , (7.32a) r g a s b,ρ ¨ g a ,b,ρ ` s ´ bρ ˘ ` g p b q “ g p s q . (7.32b)Note that (7.32a) is an identification of d ˆ d matrices, whilst (7.32b) is an identification of d ˆp n ´ d q matrices.Recall the definition of the matrix r g a ,b,ρ s x “ ” g p q a ,b,ρ p x q ¨ ¨ ¨ g p d q a ,b,ρ p x q ı . From the discussion following Definition 7.10, the curve g a ,b,ρ is as defined in Definition 2.1 and,in particular, is given by g a ,b,ρ p t q “ r g a s ´ b,ρ ` g a p b ` ρt q ´ g a p b q ˘ . Combining these definitions with the chain rule, r g a ,b,ρ s x “ r g a s ´ b,ρ ¨ r g a s b ` ρx,ρ for x P R with b ` ρx P r´ , s .Taking x “ s ´ bρ and right multiplying the above display by D r { ρ immediately implies (7.32a). Onthe other hand, (7.32b) follows directly from the definition (7.26). (cid:3) Appendix A. Reduction to a frequency localised estimate
Here we discuss the passage from frequency localised used in § A Calder´on–Zygmund estimate.
For each k P N we are given operators T k defined onSchwartz function f P S p R n q by T k f p x q : “ ˆ R n K k p x, y q f p y q d y where each K k is a continuous and bounded kernel (with no other quantitative assumptions). Let ζ P S p R n q , define ζ k : “ kn ζ p k ¨ q and set P k f : “ ζ k ˚ f. OBOLEV IMPROVING FOR AVERAGES OVER CURVES 55
Theorem A.1 ([16]) . Let ε ą and ă p ă p ă 8 . Assume for some A ě the operators T k satisfy sup k ą k { p } T k } L p p R n qÑ L p p R n q ď A, sup k ą k { p } T k } L p p R n qÑ L p p R n q ď A .
Furthermore, assume that there exists B ě , and for each cube Q a measurable set E Q such that | E Q | ď B max t diam p Q q n ´ , diam p Q q n u and such that, for every k P N and every cube Q with k diam p Q q ě , sup x P Q ˆ R n z E Q | K k p x, y q| d y ď A max tp k diam p Q qq ´ ε , ´ kε u . (A.1) Then for every r ą we have ›››´ ÿ k “ kr { p | P k T k f k | r ¯ { r ››› L p p R n q À ´ ÿ k “ } f k } pL p p R n q ¯ { p for any sequence of functions p f k q k “ P ℓ p p L p q , where the implicit constant depends only on A , B , r , ε , p , p , n and ζ . A.2.
Application.
We consider a regular curve given by t ÞÑ γ p t q , t P r , s , and let A γ be as in(1.1); that is, A γ f “ µ γ ˚ f .Let β ˝ P C c p ˆ R n q be, say, radial, supported in t { ă | ξ | ă u and β ˝ p ξ q ą ´ { ď | ξ | ď { and define L j f : “ β ˝ p ´ j D q f . We make the assumption that for some p ě } L j A γ f } L p p R n q ď C ´ j { p } f } L p p R n q , j ě . (A.2)For a non-degenerate curve in R such inequalities were proved in the previous sections.Theorem A.1 facilitates the following reduction. Proposition A.2.
Assumption (A.2) implies that A γ maps L p boundedly to L p { p for p ă p ă 8 . This result can be used to complete the argument described in § Proof of Proposition A.2.
Let Γ : “ t γ p t q : t P I u where I is a compact interval containing thesupport of χ . Let Q be a cube with center x Q and define E Q : “ t y P R n : dist p y ´ x Q , Γ q ď
10 diam p Q qu . Thus if Q is small then E Q is a tubular neighborhood of x Q ` Γ of width O p diam p Q qq . It is nothard to see that there is a constant C (depending on B ) such thatmeas p E Q q ď C diam p Q q n ´ if diam p Q q ď , diam p Q q n if diam p Q q ě . Let υ P C c p R n q be supported in t x : | x | ă { u , with the property that p υ p ξ q ą β ˝ , p υ p q “ ∇ p υ p q “ Let υ k : “ kn υ p k ¨ q and define T k f p x q : “ υ k ˚ µ γ ˚ f and K k p x, y q : “ υ k ˚ µ γ p x ´ y q . By the support properties of υ k we have K k p x, y q “ x P Q , y P R n z E Q and 2 k diam p Q q ą To construct such a function take any compactly supported real valued u P C c p R n q with ´ R n u “
1, form u C “ C d u p C ¨ q for sufficiently large C to ensure p u C ą β and then take the Laplacian, υ “ ∆ u C , to alsoget the condition p υ p q “ so that (A.1) holds trivially.We claim that the assumption (A.2) also impliessup k P N k { p } T k } L p p R n qÑ L p p R n q ă 8 . To see this, choose r β ˝ P C c p ˆ R n q supported in t ξ : 1 { ă | ξ | ă u such that ÿ j P Z β ˝ p ´ j ξ q r β ˝ p ´ j ξ q “ ξ ‰ r L j : “ r β ˝ p ´ j D q . Thus, } T k } L p Ñ L p ď ÿ j P Z } L j r L j T k } L p Ñ L p ď ÿ j P Z } L j A γ } L p Ñ L p } r L j υ k } . By straightforward calculations, using scaling and the cancellation and Schwartz properties of υ and F ´ r β ˝ s , one has } r L j υ k } “ O p ´| j ´ k | q for all j P Z . Using this together with the hypothesis(A.2), we get } T k } L p Ñ L p ď ÿ j P Z min t ´ j { p , u ´| j ´ k | À ´ k { p ;note for j ă } L j A γ f } p À } f } p . Since } T k } L Ñ L “ O p q , interpolationtherefore yields sup k P N k { p } T k } L p p R n qÑ L p p R n q ă 8 for all p ď p ď 8 .Let β P C c p ˆ R n q be supported in t ξ : 1 { ă | ξ | ă u such that β ˝ β “ β ˝ and p ă p ă 8 .Defining f k : “ β p ´ k D q f , by Theorem A.1 we obtain ›››´ ÿ k “ r k { p | β ˝ p ´ k D q A γ f |s r ¯ { r ››› L p p R n q “ ›››´ ÿ k “ kr { p | P k T k f k | r ¯ { r ››› L p p R n q À ´ ÿ k “ } f k } pL p p R n q ¯ p À ´ ÿ k “ } β p ´ k D q f } pL p p R n q ¯ p . (A.3)Since ℓ r ã Ñ ℓ for r ď ℓ ã Ñ ℓ p for 2 ď p , we deduce that ›››´ ÿ k “ r k { p | β ˝ p ´ k D q A γ f |s ¯ { ››› L p p R n q À ›››´ ÿ k “ | β p ´ k D q f | ¯ { ››› L p p R n q . This, together with the obvious low frequency L p estimates, yield the asserted Sobolev bound viaLittlewood–Paley inequalities. (cid:3) Remark.
Using Besov and Triebel-Lizorkin spaces one gets from (A.3) the stronger inequality A γ : B p,p Ñ F { pp,r for all r ą OBOLEV IMPROVING FOR AVERAGES OVER CURVES 57
Appendix B. Derivative bounds for implicitly defined functions
The following lemma is a particular instance of a more general lemma on derivative bounds forimplicitly defined functions found in [1, Appendix C].
Lemma B.1.
Let Ω Ď R n be an open set, I Ď R an open interval, g : I Ñ R n a C mapping and y : Ω Ñ I a C mapping such that x g ˝ y p x q , x y “ for all x P Ω .For e P S n ´ let ∇ e denote the directional derivative operator with respect to x in the direction of e . Suppose A, M , M ą are constants such that $’&’% |x g ˝ y p x q , x y| ě AM , |x g p N q ˝ y p x q , x y| À N AM N |x g p N q ˝ y p x q , e y| À N AM M N for all N P N and all x P Ω (B.1)
Then the function y satisfies | ∇ N e y p x q| À N M N M ´ for all x P Ω and all N P N . (B.2) Furthermore, for any C function h : I Ñ R n for which there exists some constant B ą satisfying |x h p N q ˝ y p x q , x y| À N BM N |x h p N q ˝ y p x q , e y| À N BM M N for all N P N and all x P Ω , (B.3) one has ˇˇ ∇ N e x h ˝ y p x q , x y ˇˇ À N BM N for all x P Ω and all N P N . (B.4)The following example illustrates how Lemma B.1 is applied in practice in this article. Example B.2 (Application to the case J “ . Let γ P G p δ q , and θ : ˆ R zt u Ñ I satisfying x γ ˝ θ p ξ q , ξ y “ . We apply the previous result with g “ γ and h “ γ . If B ď A the conditions (B.1) and (B.3) read succinctly as $’&’% |x γ p q ˝ θ p ξ q , ξ y| ě AM , |x γ p ` N q ˝ θ p ξ q , ξ y| À N BM N |x γ p ` N q ˝ θ p ξ q , e y| À N BM M N for all N P N and all ξ P Ω Ă ˆ R zt u , which imply | ∇ N e θ p ξ q| À N M N M ´ and | ∇ N e x γ ˝ θ p ξ q , ξ y| À N BM N . for all N P N and all ξ P Ω Ă ˆ R zt u .The applications in the different cases J “ are similar, with the choices p g, h q “ p γ p q , γ q , p g, h q “ p γ p q , γ q and p g, h q “ p γ , γ q . Appendix C. Integration-by-parts lemmas
C.1.
Non-stationary phase.
For a P C c p R q supported in an interval I Ă R and φ P C p I q ,define the oscillatory integral I r φ, a s : “ ˆ R e iφ p s q a p s q d s. The following lemma is a standard application of integration-by-parts.
Lemma C.1 (Non-stationary phase) . Let R ě and φ, a be as above. Suppose that for each j P N there exist constants C j ě such that the following conditions hold on the support of a :i) | φ p s q| ą , ii) | φ p j q p s q| ď C j R ´p j ´ q | φ p s q| j for all j ě ,iii) | a p j q p s q| ď C j R ´ j | φ p s q| j for all j ě .Then for all N P N there exists some constant C p N q such that | I r φ, a s| ď C p N q ¨ | supp a | ¨ R ´ N . Moreover, C p N q depends on C , . . . , C N but is otherwise independent of φ and a and, in particular,does not depend on r . A detailed proof of this lemma can be found in [1, Appendix D].C.2.
Kernel estimates.
The following lemma, which is used to obtain L bounds for the mul-tipliers, is based on integration-by-parts in the ξ variable. Lemma C.2.
Let a P C c p ˆ R n ˆ I q , σ ą , λ j ą for ď j ď n and t v , . . . , v n u be anorthonormal basis of R n . Suppose the following conditions hold:i) |t s P R : p ξ ; s q P supp a for some ξ P ˆ R n u| ď σ ,ii) supp ξ a Ď t ξ P ˆ R n : |x ξ, v j y| ď λ j for ď j ď n u ,iii) | ∇ N v j a p ξ ; s q| À N λ ´ Nj for all p ξ ; s q P ˆ R n ˆ R , ď j ď n and N P N .Then } m r a s} M p R n q À σ. Here ∇ v : “ v ¨ ∇ denotes the directional derivative with respect to ξ in the direction of v P S n ´ . Proof of Lemma C.2.
For f P S p R n q we have m r a sp D q f “ K r a s ˚ f where the kernel K r a s is givenby K r a sp x q “ ˆ R F ´ a p ¨ ; s qp x ` γ p s qq χ p s q d s. Here F ´ denotes the inverse Fourier transform in the ξ variable. Consequently, } m r a sp D q} M p R n q ď } K r a s} L p R n q ď ˆ R } F ´ a p ¨ ; s q} L p R n q χ p s q d s. By the hypothesis i) on the s -support, the problem is therefore reduced to showingsup s P R } F ´ a p ¨ ; s q} L p R n q À . (C.1)However, the conditions ii) and iii), combined with a standard integration-by-parts argument,imply | F ´ a p ¨ ; s qp x q| À N ´ n ź j “ λ j ¯´ ` n ÿ j “ λ j |x x, v j y| ¯ ´ N for all p x ; s q P R n ˆ R and all N ě (cid:3) References [1] David Beltran, Shaoming Guo, Jonathan Hickman, and Andreas Seeger,
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Local smoothing type estimates on L p for large p , Geom. Funct. Anal. (2000), no. 5, 1237–1288.MR 1800068 David Beltran: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison,WI, 53706, USA.
Email address : [email protected] Shaoming Guo: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison,WI, 53706, USA.
Email address : [email protected] Jonathan Hickman: School of Mathematics, James Clerk Maxwell Building, The King’s Buildings,Peter Guthrie Tait Road, Edinburgh, EH9 3FD, UK.
Email address : [email protected] Andreas Seeger: Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madi-son, WI, 53706, USA.
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