Off-diagonal estimates for the helical maximal function

The optimal $L^p \to L^q$ mapping properties for the (local) helical maximal function are obtained, except for endpoints. The proof relies on tools from multilinear harmonic analysis and, in particular, a localised version of the Bennett--Carbery--Tao restriction theorem.

1. Introduction 1.1.Main results.Let γ : I Ñ R 3 be a smooth curve, where I :" r´1, 1s, which is non-degenerate in the sense that there is a constant c 0 ą 0 such that | detpγ p1q psq, γ p2q psq, γ p3q psqq| ě c 0 for all s P I. ( This is equivalent to saying that γ has non-vanishing curvature and torsion.Prototypical examples are the helix γpsq " pcosp2πsq, sinp2πsq, sq or the moment curve γpsq " ps, s 2 {2, s 3 {6q.Given t ą 0, consider the averaging operator f px ´tγpsqq χpsq ds, defined initially for Schwartz functions f P SpR 3 q, where χ P C 8 pRq is a bump function supported on the interior of I. Furthermore, define the associated local maximal function M γ f pxq :" sup 1ďtď2 |A t f pxq|. Here we are interested in determining the sharp range of L p Ñ L q estimates for M γ .To describe the results, let T :" convtp0, 0q, p1{3, 1{3q, p1{4, 1{6qu z tp1{3, 1{3qu, so that T is a closed triangle (formed by the closed convex hull of three points) with one vertex removed.We let intpT q denote the interior of T and L denote the intersection of T with the diagonal: see Figure 1.Standard examples show that M γ fails to be L p Ñ L q bounded whenever p1{p, 1{qq R T : see §9.The following theorem therefore characterises the type set of M γ , up to endpoints.
Theorem 1.1.For all p1{p, 1{qq P intpT q Y L, there exists a constant C γ,p,q ě 1 such that the a priori estimate }M γ f } L q pR 3 q ď C γ,p,q }f } L p pR 3 q holds for all f P SpR 3 q.
For the diagonal case (that is, p1{p, 1{qq P L), the sharp range of estimates was established in [2] and [18], building on earlier work of [21].Hence, our main result is to push the range of boundedness to the region intpT q.As a consequence of Theorem 1.1 (or, more precisely, Theorem 3.1 below) and [4,Theorem 1.4], pp, q 1 qsparse bounds for the full maximal operator M full γ f pxq :" sup tą0 |A t f pxq| follow for p1{p, 1{qq P intpT q Y L. We omit the details and refer to [4] for the precise statements.
1.2.Methodology.Here we provide a brief overview of the ingredients of the proof of Theorem 1.1 and the novel features of the argument.For fixed t, the averaging operators A t correspond to convolution with an appropriate measure µ t on the t-dilate of γ.It is therefore natural to study these objects via the Fourier transform, which leads us to consider the multiplier μt pξq " ż R e ´itxγpsq,ξy χpsq ds.
Stationary phase can be used to compute the decay rate of this function in different directions in the frequency space.This involves analysing the vanishing of s-derivatives of the phase function ϕpξ, sq :" xγpsq, ξy.Following earlier work on the circular maximal function [20], it is also useful to study the Fourier transform of A t f pxq in both the x and the t variables.This leads us to consider the 4-dimensional pξ, τ q frequency space.
Broadly speaking, this approach was taken in both works [18] and [2] to study the L p Ñ L p mapping properties of M γ .However, these papers focused on different geometrical aspects of the problem.In very rough terms, the analysis of [18] centres around a 3-dimensional cone Γ 3 in pξ, τ q-space arising from the equations B s ϕpξ, sq " 0, τ " ϕpξ, sq.On the other hand, the analysis of [2] centres around a 2-dimensional cone Γ 2 in the pξ, τ q-space arising from the system of equations B s ϕpξ, sq " B 2 s ϕpξ, sq " 0, τ " ϕpξ, sq.It seems difficult to obtain almost optimal L p Ñ L q estimates using either the approach of [18] or of [2] in isolation; rather, it appears necessary to incorporate both geometries into the analysis.In order to do this, we apply a recent observation of Bejenaru [1], which provides a localised variant of the Bennett-Carbery-Tao multilinear restriction theorem [7].We describe the relevant setup in detail in §2 below; moreover, in the appendix we relate the required localised estimates to the theory of Kakeya-Brascamp-Lieb inequalities from [6].The local multilinear restriction estimate allows us to work simultaneously with the Γ 2 and Γ 3 geometries, by considering the embedded cone Γ 2 as a localised portion of Γ 3 .See Proposition 3.5 below.
On the other hand, the geometries of both Γ 2 and Γ 3 were previously exploited in a non-trivial manner in [19] and [3] using the decoupling inequalities from [10].This approach is inspired by earlier work of Pramanik-Seeger [21].Decoupling is effective for proving L p Ñ L p bounds for large p; here it is used to provide a counterpoint for interpolation with the estimates obtained via local multilinear restriction.See Proposition 3.6 below.
We let p R denote the frequency domain, which is the Pontryagin dual group of R understood here as simply a copy of R. Given f P L 1 pR d q and g P L 1 p p R d q we define the Fourier transform and inverse Fourier transform by f pξq :" ż R d e ´ixx,ξy f pxq dx and ǧpxq :" 1 e ixx,ξy gpξq dξ, respectively.For m P L 8 p p R d q we define the multiplier operator mpDq which acts initially on Schwartz functions by mpDqf pxq :" 1 e ixx,ξy mpξq f pξq dξ.
Given a list of objects L and real numbers A, B ě 0, we write A À L B or B Á L A to indicate A ď C L B for some constant C L which depends only items in the list L and our choice of underlying non-degenerate curve γ.We write A " L B to indicate A À L B and B À L A. ‚ In §4 we describe the basic properties of our operators and prove the trivial L 2 Ñ L 8 estimate.‚ In §5 we prove the trilinear L 2 Ñ L 3 estimate using the trilinear restriction theorem from §2. ‚ In §6 we prove the linear L 12 Ñ L 12 estimate using decoupling.‚ In §7 we bound a non-degenerate portion of the operator.‚ In §8 we carry out the reduction described in §3 and thereby bound the remaining degenerate portion of the operator.‚ In §9 we demonstrate the sharpness of the range T .‚ Finally, in Appendix A we present a proof of the localised trilinear restriction theorem from §2.
Acknowledgements.The first and third authors would like to thank Shaoming Guo and Andreas Seeger for discussions related to the topic of this paper over the years.

Localised trilinear restriction
The key ingredient in the proof of Theorem 1.1 is a localised trilinear Fourier restriction estimate.Here we describe the particular setup for our problem.As in [18], it is necessary to work with functions with a limited degree of regularity.Definition 2.1.Let 0 ă α ď 1 and U Ď R d be an open set.We say a function is continuously differentiable on U and, moreover, the partial derivatives satisfy the α-Hölder condition sup ξ1,ξ2PU ξ1‰ξ2

Consider an ensemble
is an open domain1 for 1 ď j ď 3. The graphs Σ j :" tpξ, Q j pξqq : ξ P U j u are hypersurfaces in p R 4 , with some limited regularity.Each Σ j has a Gauss map given by We further fix a smooth function u : U 3 Ñ R satisfying |∇upξq| ą c 0 ą 0 for all ξ P U 3 .This implicitly defines a smooth surface Z 3 :" tξ P p R 3 : upξq " 0u, which we lift to Σ 1 3 :" tpξ, Q 3 pξqq : ξ P Z 3 u.Thus, Σ 1  3 is a codimension 1 submanifold of Σ 3 , which is embedded in p R 4 .Defining it follows that tν 3 pξq, ν 1 3 pξqu forms a basis of the normal space to Σ 1 3 at pξ, Q 3 pξqq for all ξ P Z 3 .
The key localised trilinear estimate is as follows.
Theorem 2.2 (Localised trilinear restriction).With the above setup, for all ε ą 0 and all R ě 1 we have for all f j P L 1 pU j q, 1 ď j ď 3.
Here the implied constant depends on the choice of maps Q j and, in particular, the lower bound in (2.1), but is (crucially) independent of the choice of parameter µ in (2.2) and the choice of scale R.
If we consider smooth hypersurfaces rather than the C 1,1{2 class, then Theorem 2.2 is a special case of [1,Theorem 1.3].We expect that the arguments of [1] can be generalised to treat C 1,α regularity for all α ą 0. However, in Appendix A we observe that Theorem 2.2 is a rather direct consequence of the Kakeya-Brascamp-Lieb inequalities from [6] (see also [24,25]).Then for all p1{p, 1{qq P intpT q there exists some εpp, qq ą 0 such that ´ż 2 1 }mra k spD; tqf } q L q pR 3 q dt ¯1{q À p,q 2 ´k{q´εpp,qqk }f } L p pR 3 q holds for all k P N 0 , where a k is defined as in (3.3).

Initial reductions
The desired maximal bound follows from Theorem 3.1 using a standard Sobolev embedding argument; we omit the details but refer the reader to [23, Chapter XI, §3], [21, §6] or [2, §2] for similar arguments.
By results of [2], Theorem 3.1 is known to hold along the diagonal line L. By interpolation, it therefore suffices to prove an estimate at the critical vertex p1{4, 1{6q in the Riesz diagram (see Figure 1).Proposition 3.2.Under the hypotheses of Theorem 3.1, for k P N 0 and all ε ą 0, we have By the preceding discussion, our main theorem follows from Proposition 3.2.Henceforth, we focus on the proof of this critical estimate.where G 2 ps; ξq :" ř 2 j"1 |xγ pjq psq, ξy|; note that a k,0 satisfies (3.7) with an implicit constant depending on δ.We discuss this case in detail in §7.
The main difficulty is then to get to grips with the degenerate portion of the multiplier.For the above choice of 0 ă δ ă 1, this corresponds to the condition |xγ 1 psq, ξy| `|xγ 2 psq, ξy| À δ 20 |ξ| for all pξ; sq P supp ξ a ˆI; (3.9) note that this is satisfied on the support of a k :" a k ´ak,0 .To control the degenerate part, we work with a trilinear variant of Proposition 3.2, from which we deduce the corresponding linear estimate (3.6) via a standard application of the broad-narrow method from [11] (see also [16]).
To describe the trilinear setup, we introduce some notation.For 0 ă δ ă 1 as above, let Jpδq denote a covering of I by essentially disjoint intervals of length δ.Let J 3,sep pδq denote the collection of all triples J " tJ 1 , J 2 , J 3 u Ă Jpδq which satisfy the separation condition distpJ i , J j q ě 10δ for 1 ď i ă j ď 3. Given a bounded interval J Ď R, we let ψ J P C 8 c pRq satisfy supp ψ J Ď J and |B N s ψ J psq| À N |J| ´N for all N P N. Similarly to (3.1), given a symbol a P C 8 p p R 3 zt0u ˆR ˆRq, we define the multipliers adapted to an interval J P Jpδq by m J γ raspξ; tq " m J raspξ; tq :" ż R e ´itxγpsq,ξy apξ; t; sqψ J psqρptq ds.
With this setup, we prove the following estimate.
Proposition 3.3 (L 4 Ñ L 6 trilinear local smoothing).Let k P N 0 , ε ą 0 and δ ą 0. Under the hypotheses of Theorem 3.1 and further assuming (3.9), we have whenever J P J 3,sep pδq and f J P SpR 3 q for J P J.
Here we use the notation Op1q to denote an unspecified absolute constant.In applications, we work with relatively large values of δ (namely, δ " ε 1) and accordingly there is no need to precisely track the δ dependence.We will also assume without loss of generality that 0 ă δ ă c where c ą 0 is a small absolute constant, chosen to satisfy the forthcoming requirements of the argument, and k is sufficiently large depending on δ ´1.
As mentioned above, the (ostensibly weaker) trilinear estimate in Proposition 3.3 implies the linear estimate in Proposition 3.2 (under the additional assumption (3.9)) using a variant of the procedure introduced in [11].We postpone the details of this reduction to §8 below.

3.3.
Reduction to perturbations of the moment curve.At small scales, any non-degenerate curve can be thought of as a perturbation of an affine image of the moment curve γ ˝psq :" ps, s 2 {2, s 3 {6q.We refer to [2, §4] for details (which involve the affine rescalings described in §4.2 below), and just record here that it suffices to consider curves in the class Gpδ 0 q defined below for 0 ă δ 0 ă 1 sufficiently small.Definition 3.4.Given 0 ă δ 0 ă 1 and M P N, let Gpδ 0 , M q denote the class of all smooth curves γ : I Ñ R 3 that satisfy the following conditions: i) γp0q " 0 and γ pjq p0q " ⃗ e j for 1 ď j ď 3; ii) }γ ´γ˝}C M pIq ď δ 0 for all 0 ď j ď M .Here ⃗ e j denotes the jth standard Euclidean basis vector and If M " 4, then we will simply write Gpδ 0 q for Gpδ 0 , 4q Henceforth we will always assume that γ P Gpδ 0 q for δ 0 :" 10 ´10 .
It is clear that θ 2 is homogeneous of degree 0. Let upξq :" xγ 1 ˝θ2 pξq, ξy for all ξ P supp ξ a. (3.11) Since (3.10) is satisfied on the support of each we decompose each of these pieces with respect to the size of |upξq|.Given ε ą 0 and 0 ă δ ă 1, we write where a k,0 is as in ( Here we assume that k is large enough so that the decomposition (3.13) makes sense; note that Theorem 3.1 trivially holds for small values of k.In particular, we concern ourselves with k P N satisfying k ě 4 log 2 pδ ´8q.In the definition (3.15), for any x P R, we let txu denote the largest integer less or equal than x and rxs denote the smallest integer greater or equal than x.It will also be useful to introduce the notation Λpkq :" Λpkq Y tk{3u.
Note that the indexing set Λpkq depends on the chosen δ and ε, but we do not record this dependence for notational convenience.We also note that here the function β should be defined slightly differently compared with (3.2); in particular, here βprq :" ηp2 ´2rq ´ηprq (we ignore this minor change in the notation).
As mentioned in §3.2, for the extreme case ℓ " 0 we have the non-degeneracy condition (3.7) (with an implied constant depending on δ).This situation is easy to handle using known estimates: see §7 below.On the other hand, for ℓ P Λpkq, the multipliers mra k,ℓ s are degenerate in the sense that (3.9) now holds.A key aspect of this decomposition is that for ℓ P Λpkq, Taylor series expansion shows that for all pξ, sq P supp a k,ℓ p ¨; t; ¨q; (3.16) see, for example [3, (5.15)] for a detailed derivation.The weak non-degeneracy condition (3.16) will allow for improved estimates depending on the value of ℓ.Bounding these pieces, and the piece for ℓ " k{3, is the difficult part of the argument and is the focus of § §5-6 below.
3.5.Microlocalised estimates.Throughout this section we work under the hypotheses of Theorem 3.1 and, in addition, assume (3.9) holds for a specified value of δ.That is, we let γ : I Ñ R 3 be a smooth curve and suppose a P C 8 p p R 3 zt0uˆRˆRq satisfies (3.4), (3.5) and (3.9).Furthermore, we define the symbols a k,ℓ as in (3.14).
The key ingredient in the proof of Proposition 3.3 is a trilinear estimate for the multipliers associated to the localised symbols a k,ℓ .To describe this result, we work with a triple of integers ℓ J " pℓ J q JPJ indexed by J P J 3,sep pδq and write |ℓ J | :" ř JPJ ℓ J .
Proposition 3.5 is a fairly direct consequence of Theorem 2.2; we describe the proof in §5 below.To deduce the critical L 4 Ñ L 6 estimate stated in Proposition 3.3, we interpolate Proposition 3.5 with the following linear inequalities.Proposition 3.6 (L 12 Ñ L 12 local smoothing).For k P N, ε ą 0, 0 ă δ ă 1 and ℓ P Λpkq, we have We remark that 0 ă δ ă 1 plays no significant rôle in the proofs of Proposition 3.6 and Lemma 3.7 and it is used only to set up the underlying decomposition in the a k,ℓ .Similarly, ε ą 0 plays no significant rôle in Lemma 3.7.
Proposition 3.6 is a minor variant of estimates which have appeared in, for instance, [19] and [3].The result is highly non-trivial, and relies on the ℓ p decoupling inequality for the moment curve from [10].We discuss the details in §6.
Lemma 3.7, on the other hand, is elementary.It follows from basic pointwise estimates for the multipliers a k,ℓ , obtained via stationary phase.We discuss the details in §4.1.
Given the preceding results, the key trilinear L 4 Ñ L 6 local smoothing estimate is immediate.
Proof (of Proposition 3.3).By multilinear Hölder's inequality, Propositions 3.6 and 3.7 imply their trilinear counterparts.Since ˜1 4 1 6 interpolation of the three estimates immediately gives for ℓ J " pℓ J q JPJ with 0 ď ℓ J ď k{3; see Figure 1.Here we carry out the interpolation using a multilinear variant of the Riesz-Thorin theorem: see, for instance, [8, §4.4]. 2 Using the geometric decay in 2 ´ℓJ for each J P J, we sum these bounds to deduce the desired result.□ To prove Proposition 3.3, it therefore remains to establish Proposition 3.5, Proposition 3.6 and Lemma 3.7.We carry this out in § §4-6 below.
2 Alternatively, a suitable multilinear interpolation theorem can be proved by directly adapting the argument used to prove the classical Riesz-Thorin theorem.Proof (of Lemma 3.7).By the Cauchy-Schwarz inequality, we have the elementary inequality Since ∇upξq " γ 1 ˝θ2 pξq is bounded away from zero, the latter estimate is clear.On the other hand, the former estimate is a consequence of a simple stationary phase analysis.Indeed, for ℓ " k{3 we apply van der Corput's inequality with third order derivatives.For ℓ P Λpkq, we apply van der Corput's inequality with either first or second order derivatives, using the lower bound (3.16)where D λ :" diagpλ, λ 2 , λ 3 q is the diagonal matrix with eigenvalues λ, λ 2 , λ 3 .With this data, define the pσ, λq-rescaling of γ as the curve γ σ,λ P C 8 pI; R 3 q given by γ σ,λ psq :" rγs ´1 σ,λ `γpσ `λsq ´γpσq ˘. (4.2) A simple computation shows detrγ σ,λ s s " det `rγs σ`λs det `rγs σ ˘, and therefore γ σ,λ is also a non-degenerate curve.Furthermore, the rescaled curve satisfies the relations xγ pjq σ,λ psq, ξy " λ j xγ pjq pσ `λsq, prγs σ,λ q ´Jξy for all j ě 1. ( Combining this with the fact that γ σ,λ is non-degenerate, we have here the last inequality is a simple consequence of the definition (4.1).
iii) If upξq ă 0, then the equation (4.9) has precisely two solutions on I.
Taylor expansion yields the following.4.4.Stationary phase.We next use the approach in [18] and apply stationary phase to express the multipliers m J ra k,ℓ s as a product of a symbol and an oscillatory term.In what follows, we let q 2 pξq :" xγ ˝θ2 pξq, ξy and q 1 pξq :" xγ ˝θ1 pξq, ξy for any value of ξ such that the expression is well-defined.Our analysis leads to various rapidly decreasing error terms.Given R ě 1, we let RapDecpRq denote the class of functions e P C 8 p p R 3 ˆRq which satisfy |epξ; tq| À N R ´N for all N P N 0 .
Lemma 4.3.Let k P N and J P Jpδq.
ii) If upξq ą 0, we know from Lemma 4.1 that the phase function s Þ Ñ xγpsq, ξy has no critical points, and one can therefore obtain rapid decay of the portion of the multiplier where this condition holds; see [2, Lemma 8.1] for similar arguments.We thus focus on the portion of the multiplier where upξq ă 0. Arguing analogously to the proof of part i), for a given ρ ą 0 we may define b J,k ,ℓ p2 ´kξ; tq :" 2 pk´ℓq{2 2 ´ℓ ż R e ´it2 k´3ℓ Φ k,ℓ pξ;sq α J,k ,ℓ pξ; t; 2 ´ℓsqηpρ ´1sq ds where: ‚ The symbols α Moreover, s " 0 is the only critical point of s Þ Ñ Φ k,ℓ pξ; sq on the support of α J,k ,ℓ if |s| ď ρ for ρ ą 0 sufficiently small.An integration-by-parts argument similar to that in [2, Lemma 8.1] then shows (4.12) holds for some e k,ℓ P RapDecp2 k q.
By the support properties of the α J,k ,ℓ and Lemma 4. where |ωpξ; sq| À 2 k for all pξ; t; sq P supp α J,k ,ℓ ; in particular, |Φ k,ℓ pξ; sq| À 1 in the support of α J,k ,ℓ .In view of the factor s 2 in (4.16), the bound (4.15) is again a consequence of van der Corput's lemma with second-order derivatives (in the specific form of, for example, [22, Lemma 1.1.2]).□

5.
Proof of the L 2 Ñ L 3 trilinear estimate In this section we prove the key trilinear estimate from Proposition 3.5.After massaging the operator into a suitable form, this is a consequence the localised multilinear restriction inequality from Theorem 2.2.

Reduction to multlinear restriction. Define the Fourier integral operators
e ipxx,ξy´tq2pξqq b J k,k{3 p2 ´kξ, tq f pξq dξ and U J k,ℓ f px, tq :" e ipxx,ξy´tq 1 pξqq b J,k ,ℓ p2 ´kξ, tq f pξq dξ for ℓ P Λpkq.Let J P J3,sep pδq and ℓ J " pℓ J q JPJ with ℓ J P Λpkq.In light of Lemma 4.3, to prove Proposition 3.5, it suffices to show for U J :" U J k,ℓ J and f J P SpR 3 q for J P J.This reduction follows from a standard localisation argument since the kernels K J k,ℓ associated to the propagators U J k,ℓ satisfy the bounds |K J k,ℓ px, tq| À N 2 kpd´N q |x| ´N for all |x| Á 1, N ą 0, via an integration-by-parts argument.
We may remove the t-dependence from the symbols b J k,k{3 and b J,k ,ℓ using a standard Fourier series expansion argument.Owing to the L 2 -norm on the right-hand side of (5.1), we may also freely exchange f and f .Thus, after rescaling, we are led to consider operators of the form T J k,k{3 gpx, tq :" for ℓ P Λpkq.In particular, to prove the desired estimate (5.1), it suffices to show (5.4) for T J :" T J k,ℓ J and g J P SpR 3 q for J P J and ℓ J " pℓ J q JPJ with ℓ J P Λpkq.5.2.Verifying the hypotheses of Theorem 2.2.Enumerate the intervals in J as J 1 , J 2 , J 3 so that, writing ℓ i :" ℓ Ji and T i :" T Ji , we have 0 ă ℓ 1 ď ℓ 2 ď ℓ 3 ď k{3, ℓ i P Λpkq.The trilinear estimate (5.4) is a consequence of Theorem 2.2 where we exploit the additional localisation of the symbol of T 3 to the set |upξq| À 2 ´2ℓ3 . 3 In order to apply Theorem 2.2, we must verify the regularity and transversality hypotheses.
We begin by noting, as a consequence of the definition of the functions θ 2 and θ 1 , that ∇q 2 pξq " γ ˝θ2 pξq `upξq∇θ 2 pξq and ∇q 1 pξq " γ ˝θ1 pξq. (5.5) Regularity hypothesis.We first show that the functions q 2 and q 1 all satisfy (at least) the C 1,1{2 condition.It is easy to see that the function q 2 is C 1,1 in the sense that . On the other hand, the functions q 1 are less regular and only satisfy a C 1,1{2 condition, as first observed in [18, Lemma 3.5].
□ 4 Here we must be a little careful in applying the fundamental theorem of calculus because the crucial condition |upξq| " 2 ´2ℓ does not hold on a convex set.If |ξ 1 ´ξ2 | ! 2 ´2ℓ , then this presents no problem.However, if |ξ 1 ´ξ2 | Á 2 ´2ℓ , then to apply the the fundamental theorem of calculus we construct a continuous, piecewise smooth curve connecting ξ 1 to ξ 2 which consists of two linear segments of length Op2 ´2ℓ q and a curve of length Op|ξ 1 ´ξ2 |q which traverses the level set tξ P Bp0, 10q : upξq " 2 ´2ℓ u.
Transversality hypothesis.We now turn to verify the transversality hypothesis from (2.1).This involves estimating expressions of the form ˇˇˇd et ˆ∇Q 1 pξ 1 q ∇Q 2 pξ 2 q ∇Q 3 pξ 3 q ∇upξ 3 q 1 1 1 0 ˙ˇˇw here each Q j is either of the functions q 2 or q 1 .The columns where Q j " q 2 are slightly more complicated since the formula for ∇q 2 pξq in (5.5) involves multiple terms.However, we can always treat the second term as an error and effectively ignore it.Indeed, if Q j " q 2 , then we must have ℓ j " k{3 and so we consider ξ P supp b J k,k{3 .In this case, |upξq| À 2 ´2k{3`4εk{3 and, by differentiating the defining equation for θ 2 , we also have |∇θ 2 pξq| À 1.Since k is large, we can therefore think of ∇q 2 pξq as a tiny perturbation of γ ˝θ2 pξq on supp b J k,k{3 .On the other hand, for the final column we have ∇upξq " γ 1 ˝θ2 pξq. (5.9) In view of the support conditions (5.2) and ( 5.3) and the derivative formulae (5.5) and (5.9), the transversality hypothesis would follow from the bound ˇˇˇd et ˆγps 1 q γps 2 q γps 3 q γ 1 ps 3 q 1 1 1 0 for all s 1 , s 2 , s 3 P I, where here and below V pt 1 , . . ., t m q :" ź 1ďiăjďm pt i ´tj q denotes the Vandermonde determinant in the variables t 1 , . . ., t m P R. In order to make this reduction, we use the bound |θ 2 pξq ´θ1 pξq| À 2 ´ℓ ă δ 8 from Lemma 4.2.
For ℓ 3 P Λpkq, we can think of θ 2 pξq and θ 1 pξq as approximately equal; this allows us to reduce to a situation involving only three variables s 1 , s 2 , s 3 .Here we use the fact that the right-hand side of (5.10) is bounded below by (a constant multiple of) δ 5 for s i P I with distps i , J i q ă 2δ for 1 ď i ď 3 and J " pJ 1 , J 2 , J 3 q P J 3,sep pδq.By repeated application of the fundamental theorem of calculus, we may express the left-hand determinant in (5.10) as By continuity and the reductions in §3.3, the inner determinant is single-signed and bounded below in absolute value by some constant.By the observations of the previous paragraph, the left-hand side of (5.10) is comparable to the same expression but with γ replaced by the moment curve γ ˝psq :" ps, s 2 {2, s 3 {6q.Consequently, it suffices to prove (5.10) for this particular curve.However, in this case the left-hand side of (5.10) corresponds to (the absolute value of a scalar multiple of) B t V ps 1 , s 2 , s 3 , s 3 `tq| t"0 .A simple calculus exercise shows this agrees with the expression appearing in the right-hand side of (5.10), as required.For similar arguments, see [14,18].

L 12 -local smoothing
In this section we upgrade the L p -local smoothing estimates of [21] for p ě 12 by exploiting the localisation of the spatio-temporal Fourier transform of mra k,ℓ spD; tqf with respect to the 2-dimensional cone Γ 2 in pξ, τ q-space from the introduction.The arguments are similar to those of [19] and [3].Crucially, we apply a decoupling inequality from [3], which is a conic variant of the celebrated decoupling inequality for non-degenerate curves [10,15].After this step, the remainder of the argument is similar to that of [21]. 5  6.1.Decomposition along the curve.
À N,ε,p 2 ´kN }f } L p pR 3 q for all N P N.
By Lemma 4.2, if |s ´θ2 pξq| " 2 ´ℓ, then s is far from any roots θ 1 pξq of the phase function.Hence Lemma 6.1 follows from non-stationary phase, as in the analysis of the error term in Lemma 4.3 ii).Moreover, the proof is very similar to that of [2,Lemma 8.1] and we therefore omit the precise details. 6  The support properties of the symbols a µ k,ℓ are best understood in terms of the Frenet frame.Recall, given a smooth non-degenerate curve γ : I Ñ R d , the Frenet frame is the orthonormal basis te 1 psq, . . ., e d psqu resulting from applying the Gram-Schmidt process to the vectors tγ 1 psq, . . ., γ pdq psqu.With this setup, given 0 ă r ď 1 and s P I, recall the definition of the p1, rq-Frenet boxes π 1 ps; rq introduced in [2]; namely, π 1 ps; rq :" ␣ ξ P p R 3 : |xe j psq, ξy| À r 3´j for j " 1, 2, |xe 3 psq, ξy| " 1 ( . We have the following support property.Lemma 6.2.Let k P N, ℓ P Λpkq and µ P Z. Then supp ξ a µ k,ℓ Ď 2 k ¨π1 ps µ ; 2 ´ℓq.The proof is similar to that of [2, Lemma 8.2, (a)], so we omit the details. 5 Decoupling is also used in [21], but only with respect to a cone in ξ-space, leading to non-sharp regularity estimates. 6The argument is in fact entirely the same as the proof of the case tk{3uε ď ℓ ď tk{3u from [2, Lemma 8.1].
We introduce the localised multipliers m µ k,ℓ , defined by F t " m µ k,ℓ pξ; ¨q‰ pτ q :" χ k,ℓ pξ, τ q F t " mra µ k,ℓ spξ; ¨q‰ pτ q.Here F t denotes the Fourier transform acting in the t variable.A stationary phase argument allows us to pass from mra µ k,ℓ s to m µ k,ℓ .Lemma 6.3.Let 2 ď p ă 8.For all k P N, ℓ P Λpkq and µ P Z, we have The proof, which is based on a fairly straightforward integration-by-parts argument, is similar to that of [2, Lemma 8.3] and we omit the details.
To understand the support properties of the multipliers m µ k,ℓ , we introduce the primitive curve γ : Here ş s 0 γ denotes the vector in R 3 with ith component ş s 0 γ i for 1 ď i ď 3. Note that γ is a non-degenerate curve in R 4 and, in particular, | det `γ p1q psq ¨¨¨γ p4q psq ˘| " | det `γp1q psq ¨¨¨γ p3q psq ˘| for all s P I.
The proof, which is based on a fairly straightforward integration-by-parts argument, is similar to that of [2,Lemma 8.4] and we omit the details.

A decoupling inequality.
With the above observations, we can immediately apply the decoupling inequalities in [3,Theorem 4.4] associated to the primitive curve γ to isolate the contributions from the individual m µ k,ℓ pD; ¨q.Proposition 6.5.Let p ě 12.For all k P N, ℓ P Λpkq, we have Proof.If ℓ P Λpkq satisfies ℓ ď r4εks, then we bound the left-hand side trivially using the triangle and the Cauchy-Schwarz inequalities.For the case ℓ ą r4εks, we partition the family of sets π 2,γ ps µ ; 2 4εk 2 ´ℓq for |µ| ď 2 ℓ into Op2 4εk q subfamilies, each forming a p2, 2 4εk 2 ´ℓq-Frenet box decomposition in the language of [3, §4].In view of Lemma 6.4 and after a simple rescaling, the result now follows from [3, Theorem 4.4] applied with d " 3, n " 4 to the primitive curve γ. □ 6.4.Localising the input function.The Fourier multipliers m µ k,ℓ pD; tq induce a localisation on the input function f .We recall the setup from [2, §8.6].Given ℓ P Λpkq and m P Z define ∆ k,ℓ pmq :" where C ě 1 is an absolute constant, chosen sufficiently large so that the following lemma holds.
Lemma 6.6 ([2, Lemma 8.8]).If µ P Z, then there exists some mpµq P Z such that Furthermore, for each fixed k and ℓ, given m P Z there are only Op1q values of µ P Z such that m " mpµq.
Let p ě 6, ε ą 0 and k ě 1.If a k is defined as in (3.3), then By combining Theorem 6.8 with the rescaling from §4.2 we obtain the following bound for our multipliers mra µ k,ℓ s.Proposition 6.9.Let p ě 6.For all k P N, ℓ P Λpkq and µ P Z we have Proof.The argument is essentially the same proof as that of [21,Proposition 5.1].We distinguish two cases: Case: ℓ " k{3.The result follows from interpolation between the elementary estimates sup tPR }mra µ k,k{3 sp ¨; tq} M 2 pR 3 q À 2 ´k{3`Opεkq , sup tPR }mra µ k,k{3 sp ¨; tq} M 8 pR 3 q À 2 ´k{3`Opεkq .
Both these inequalities are consequences of the size of the s support of a µ k,k{3 .The first is trivial.The second can be deduced, for instance, by adapting arguments from [3, §5.6].
Thus, in view of (4.6), we have We claim γ and ã satisfy the hypotheses of Theorem 6. then easily follow, noting that the derivatives of pa k q σ,λ can be controlled following the discussion at the end of §4.2.As a consequence of (6.4), we may write ã " ř 8 j"0 ãj where each ãj is a localised symbol as defined in (3.3) and the only non-zero terms of the sum correspond to values of j satisfying 2 j " 2 k´3ℓ .In view of (6.5) and (6.6), for p ě 6 we can apply Theorem 6.8 to obtain This, together with (6.3) and an affine transformation in the spatial variables, gives the desired inequality (6.2). □ 6.6.Putting everything together.With the above ingredients, we can now conclude the proof of the L 12 local smoothing estimate.

The non-degenerate case
In the non-degenerate case ℓ " 0 we appeal to the classical (linear) Stein-Tomas restriction estimate, rather than the trilinear theory from §5.
Proof.Decomposing the symbol a into sufficiently many pieces with small ξ and s support, the non-degeneracy condition (7.1) can be strengthened to the following: there exists B ą 1 such that In particular, to prove the lemma, with the above setup it suffices to show The inequality (7.5) follows from a generalisation of the classical Stein-Tomas restriction theorem due to Greenleaf [13] (see also [23, Chapter VIII, §5 C.]).To apply this result, we need to show that q 1 is smooth over the support of b and satisfies certain curvature conditions.
Arguing as in the proof of Lemma 5.1, we see that |∇θ 1 pξq| " 1 on supp b and, furthermore, the function q 1 is easily seen to be smooth with bounded derivatives over this set.A simple computation shows that the hessian B 2 ξξ q 1 pξq is the rank 1 matrix formed by the outer product of the vectors ∇θ 1 pξq and γ 1 ˝θ1 pξq.By elementary properties of rank 1 matrices, B 2 ξξ q 1 pξq therefore has a unique non-zero eigenvalue given by κpξq :" xγ 1 ˝θ1 pξq, ∇θ 1 pξqy.We claim that |κpξq| " 1 for all ξ P supp b; (7.6) geometrically, this means that the surface formed by taking the graph of q 1 over some open neighbourhood of supp b has precisely one non-vanishing principal curvature.This is precisely the geometric condition needed to apply the result of [13] in order to deduce (7.5).To see (7.6) holds, we take the ξ-gradient of the defining equation (7.4) for θ 1 and then form the inner product with ∇θ 1 pξq to deduce that 0 " xγ 1 ˝θ1 pξq, ∇θ 1 pξqy `xγ 2 ˝θ1 pξq, ξy|∇θ 1 pξq| 2 " κpξq `vpξq|∇θ 1 pξq| 2 .
Since |vpξq| " |∇θ 1 pξq| " 1 on supp b, the claim follows. □ One can interpolate Proposition 7.1 with the diagonal L 6 Ñ L 6 local smoothing result of Pramanik-Seeger [21] (see Theorem 6.8 above) to directly deduce the desired L 4 Ñ L 6 estimate for the non-degenerate piece a k,0 introduced in (3.8).Lemma 7.2.For all k P N and ε ą 0, we have This lemma reduces the proof of Proposition 3.2 to establishing the L 4 Ñ L 6 bound in (3.6) with a k replaced with the localised symbol a k :" a k ´ak,0 .

Concluding the argument
Here we conclude the proof of Proposition 3.2 and, in particular, present the details of the trilinear reduction discussed in §3.2.
Proof (of Proposition 3.2).Fix ε ą 0 and let 0 ă δ ă 1, M ε P N and C ε ě 1 be constants, depending only on ε, and chosen to satisfy the forthcoming requirements of the proof.We proceed by inducting on the parameter k.For 0 ď k ď δ ´100 , the result is trivial and this serves as the base case.We fix k P N satisfying k ą δ ´100 and assume that for 0 ď n ď k ´1 the result holds in the following quantified sense.Induction hypothesis.Let γ P Gpδ 0 , M ε q and suppose a P C 8 p p R 3 zt0u ˆR ˆRq satisfies the symbol condition |B α ξ B i t B j s apξ; t; sq| ď |ξ| ´|α| for all α P N 3 0 and i, j P N 0 with |α|, i, j ď M ε .(8.1)For all 0 ď n ď k ´1, we have We remark that if M ε P N is chosen sufficiently large, then all the estimates proved in this paper are uniform over all curves belonging to the class Gpδ 0 , M ε q.
We now turn to the inductive step.Fix γ P Gpδ 0 , M ε q and a satisfying (8.1) and suppose a k,0 is defined as in (3.8).Provided C ε is chosen sufficiently large in terms of δ, we may apply Lemma 7.2 to deduce a favourable bound for the corresponding multiplier mra k,0 s.It remains to show for the a k symbols as defined in (3.12).For convenience, write U k f px, tq :" mra k spD, tqf pxq and U J k f px, tq :" m J ra k spD, tqf pxq for J P Jpδq.By fixing an appropriate partition of unity, By an elementary argument (see, for instance, [18, Lemma 4.1]), we have a pointwise bound Taking L 6 -norms on both sides of (8.2), we deduce that 3) The first term on the right-hand side of (8.3) can be estimated using a combination of rescaling and the induction hypothesis.To this end, let β P C 8 c p p R 3 q be a non-negative function satisfying β β " β and |ξ| " 1 for ξ P supp β, and define βk pξq :" βp2 ´kξq for k P N 0 .For J P Jpδq fix ψJ P C 8 c pRq satisfying supp ψJ Ď 4¨J, ψJ prq " 1 for r P 2 ¨J and |B N r ψJ prq| À N |J| ´N for all N P N 0 .We define the Fourier projection f J of f by fJ pξq :" χ J pξq f pξq where χ J pξq :" βk pξq ψJ ˝θ2 pξq.
By stationary phase arguments, similar to the proof of Lemma 6.1, we then have }U J k f } L 6 pR 3`1 q À }U J k f J } L 6 pR 3`1 q `2´10k }f } L 4 pR 3 q for each J P Jpδq.
The former follows from Plancherel's theorem and the finite overlap of the Fourier supports of the f J .For the latter, it suffices to show the kernel estimate sup JPJpδq }F ´1χ J } 1 À 1.To see this, we apply a rescaling as in the proof of Proposition 6.9, which transforms χ J into a function with favourable derivative bounds.
where the constant C ε ě 1 is an amalgamation of the various implied constants appearing in the preceding argument.Now suppose δ ą 0 and C ε have been chosen from the outset so as to satisfy C ε δ 3ε ď 1{4 and C ε ě 4C ε δ ´E´4 .It then follows that }U k f } L 6 pR 3`1 q ď pC ε {2q2 ´k{6`εk }f } L 4 pR 3 q , which closes the induction and completes the proof.□

Necessary conditions
In this section we provide the examples that show that M γ fails to be bounded from L p Ñ L q whenever p1{p, 1{qq R T .By a classical result of Hörmander [17], M γ cannot map L p Ñ L q for any p ą q.Failure at the point p1{3, 1{3q was already shown in [18] via a modification of the standard Stein-type example for the circular maximal function.The line joining p1{3, 1{3q and p1{4, 1{6q is critical via a Knapp-type example, whilst the line joining p0, 0q and p1{4, 1{6q is critical from the standard example for fixed time averages.9.1.The Knapp example.By an affine rescaling (as in §4.2), we may assume γ pjq p0q " e j for 1 ď j ď 3, where e j denotes the standard basis vector.Thus, if γ denotes the moment curve as in §4.2, then γpsq " γp0q `γ˝p sq `Ops 4 q for s P I. Furthermore, we may assume without loss of generality that a :" γ 3 p0q ą 0. Given δ ą 0, let f δ :" 1 R δ where R δ :" ty P R 3 : |y j | ă δ j , 1 ď j ď 3u.

1. 4 .
Organisation of the paper.‚ In §2 we present the key localised trilinear restriction estimate.‚ In §3 we describe a reduction of Theorem 1.1 to three local-smoothing-type estimates: trilinear L 2 Ñ L 3 , linear L 12 Ñ L 12 and trivial L 2 Ñ L 8 .

4 .
Basic properties of the multipliers 4.1.Elementary estimates for the multiplier.Using stationary phase arguments, we can immediately deduce Lemma 3.7.