L^p Regularity Estimates for a Class of Integral Operators with Fold Blowdown Singularities
aa r X i v : . [ m a t h . C A ] J a n L p REGULARITY ESTIMATES FOR A CLASS OF INTEGRALOPERATORS WITH FOLD BLOWDOWN SINGULARITIES
GEOFFREY BENTSEN
Abstract.
We prove sharp L p regularity results for a class of generalized Radon trans-forms for families of curves in a three-dimensional manifold associated to a canonicalrelation with fold and blowdown singularities. The proof relies on decoupling inequali-ties by Wolff and Bourgain-Demeter for plate decompositions of thin neighborhoods ofcones and L estimates for related oscillatory integrals. Introduction
Let M be the family of all lines in R . Given a function f ∈ C ∞ ( R ), its X-ray transformis a function defined on M given by Xf ( l ) = Z l f, l ∈ M Since M is a 4 dimensional manifold, recovering f from Xf is an overdetermined prob-lem, it is natural to ask for which 3 dimensional submanifolds F ⊂ M the restriction X F f = Xf (cid:12)(cid:12) F can be inverted. We study a class of these restricted X-ray transforms ini-tially formulated in the complex setting by Gelfand and Graev ([7]) to give an essentiallycomplete characterization of when inversion is possible. Definition 1.1 (Gelfand Admissibility) . Given a three-dimensional line complex F , let Γ Q be the cone of lines in F through the point Q . We say that F is Gelfand-admissible if Γ P is tangent to Γ Q along the line between the points P and Q for every P in the cone Γ Q . This class of restricted X-ray transforms has been studied by many authors, includingGreenleaf-Uhlmann who, in [11], showed that Gelfand admissibility, along with the con-dition that the cone of lines through each point is curved, is sufficient for the inversion of X F , extending the results of Gelfand-Graev to the real setting. Various estimates havebeen proven for this collection of restricted X-ray transforms. For instance, L -Sobolevestimates were proven by Greenleaf-Uhlmann in [8], and L p → L q estimates were provenby Greenleaf-Seeger in [9]. In this paper we are interested in finding L p -Sobolev estimatesfor X F and similar operators. It is instructive to look at the following model case.Let I be a compact interval and suppose that γ : I → R is a smooth regular curvewith nonvanishing curvature (i.e. γ ′ ( s ) , γ ′′ ( s ) = 0). For a Schwartz function f ∈ S ( R ) Date : January 21, 2021.2010
Mathematics Subject Classification. nd α ∈ I define A f ( x ′ , α ) = Z f ( x ′ + sγ ( α ) , s ) χ ( s ) χ ( α ) ds, where χ and χ are smooth real-valued functions supported in the interior of [1 ,
2] and I respectively. Pramanik and Seeger, in [18], proved that for sufficiently small p the operator A maps boundedly from L p ( R ) into L p − /p ( R ), where L ps ( R ) is the standard Sobolevspace on R with respect to Lebesgue measure. This result was proven by studyingdyadic decompositions of the adjoint operator A ∗ and using ℓ p -decoupling inequalitiesfor the cone, originally proven by Wolff in [24] and later extended to the optimal rangeby Bourgain-Demeter [3]. Applying the Bourgain-Demeter decoupling result yields theboundedness of A ∗ from L p ( R ) into L p /p ( R ) for p >
4. This estimate is the best possiblefor the range of p , although it is unknown whether the range of p can be extended to p ≥ /p derivatives in L p for a class of integral operators in R with folding canonical relations, generalizing their previous result in [19], which consideredaverages over translations of curves in R . We use similar techniques to Pramanik andSeeger to generalize the results of [18] to more general integral operators associated tofold and blowdown singularities. This class of integral operators includes the adjointsof generic Gelfand-admissible restricted X-ray transforms, and also subsumes the mainresult of [2] for averaging operators over curves in the Heisenberg group.To define our class of integral operators, we recall the formalism of [20] and [11].Let Ω L , Ω R be three-dimensional manifolds and consider families of curves M x ⊂ Ω R parametrized by and smoothly depending on x ∈ Ω L . Let dσ x be the arclength measureon M x , and χ ∈ C ∞ c ( R × R ). We define the generalized Radon transform operator R : C ∞ c (Ω R ) → C ∞ (Ω L ) by R f ( x ) = Z M x f ( y ) χ ( x, y ) dσ x ( y ) . We assume that M x are sections of a manifold M ⊂ Ω L × Ω R , so that the projections(1.1) M Ω L Ω R have surjective differentials; note this ensures that R is bounded on L and L ∞ . Thesurjectivity assumption on the projections (1.1) also ensures that M x and M y = { x ∈ Ω L : ( x, y ) ∈ M} are smooth immersed curves in Ω R and Ω L respectively.The operator R can be realized as a Fourier integral operator of order − / I − (Ω L , Ω R ; ( N ∗ M ) ′ ), where( N ∗ M ) ′ = (cid:8) ( x, ξ, y, η ) : ( x, ξ, y, − η ) ∈ N ∗ M (cid:9) with N ∗ M the conormal bundle of M . The assumptions on the projections (1.1) implythat C = ( N ∗ M ) ′ ⊂ ( T ∗ Ω L \ L ) × ( T ∗ Ω R \ R ) , here 0 L and 0 R are the zero sections of the cotangent spaces T ∗ Ω L and T ∗ Ω R respectively.Moreover, C is a homogeneous canonical relation, i.e. if ω L and ω R are the canonical two-forms on T ∗ Ω L and T ∗ Ω R respectively, then C is Lagrangian with respect to ω L − ω R .As is known from the theory of Fourier integral operators (see [12, 14]) The L -Sobolevregularity properties of R are governed by by the geometry of the projections(1.2) C T ∗ Ω L T ∗ Ω Rπ L π R Since C is Lagrangian the differential ( Dπ L ) P is invertible if and only if ( Dπ R ) P is invertible([12]). For averaging operators over curves in dimensions larger than 2 the projections π L and π R fail to be diffeomorphisms, meaning that for every point ( x, y ) ∈ M there is a P = ( x, ξ, y, η ) ∈ ( N ∗ M ) ′ such that ( Dπ L ) P and ( Dπ R ) P are not invertible. However, wecan restrict how singular the maps π L and π R are on C . Following the survey papers [4]and [10], we recall the definitions of a Whitney fold and a blowdown. Definition 1.2.
Suppose g : X → Y is a C ∞ map between C ∞ manifolds of corank ≤ ,and the set L = { P ∈ X : det( dg ) P = 0 } is an immersed hypersurface. We say V , anonzero smooth vector field on X , is a kernel field of g if V | P ∈ ker( dg ) P for all P ∈ L .We say g is a Whitney fold if for every kernel field V of g and every P ∈ L we have V (det dg ) = 0 at P .We say g is a blowdown if every kernel field V of g , when restricted to L , is everywheretangential to L . Note this implies that V k (det dg ) (cid:12)(cid:12) P = 0 for all k ∈ N and all P ∈ L . In [20], Pramanik and Seeger proved R maps L p ( R ) into L p /p ( R ) boundedly for p > π L and π R are Whitney folds. Theyconjectured that only the Whitney fold assumption on π L is necessary for their result. Inthis paper we consider a “worst” case, where π R is instead a blowdown. Theorem 1.1.
Let
M ⊂ Ω L × Ω R be a four-dimensional manifold such that the projections M → Ω L and M → Ω R are submersions. Assume that the only singularities on π L :( N ∗ M ) ′ → T ∗ Ω L are Whitney folds, and that π R : ( N ∗ M ) ′ → T ∗ Ω R is a blowdown. Let L be the conic submanifold on which dπ L and dπ R drop rank by one, and let ̟ be theprojection of ( N ∗ M ) ′ onto the base M . Suppose that the restriction of ̟ to L , (1.3) ̟ : L 7→ M is a submersion. Then R extends to a continuous operator R : L p comp (Ω R ) → L p /p, loc (Ω L ) , < p < ∞ . Theorem 1.1 generalizes the results of [2] and [18], and the sharpness examples in bothpapers show that the regularity index s = 1 /p cannot be improved, and that the result failsfor p <
4. Note that the assumption on the projection ̟ ensures a curvature conditionon the fibers of L , first formulated in [9], and proven for R in [20]. This curvature ensuresthat ℓ p -decoupling can be applied. he layout of this paper is as follows. In Section 2 we introduce some example operatorsfor which Theorem 1.1 applies. In Sections 3 and 4 we begin the proof of Theorem 1.1by relating it to an estimate of oscillatory integrals in Proposition 4.1. This is the mainestimate of the paper, proven through the interpolation of a decoupling inequality and an L estimate in Sections 5 and 6 respectively. While the L boundedness of R has beenestablished by the work of Greenleaf and Seeger in [9], these estimates rely on a Strichartz-type argument that does not yield the quantitative estimates that we need to interpolatewith the ℓ p -decoupling estimates in Section 5. The work of Comech in [5] establishesthese quantitative estimates if π R is of finite type but does not cover the case when π R isa blowdown, which is what we prove in Section 6 in a general setting. Fortunately it isnot necessary to prove the endpoint L estimate (see [16]) in order to interpolate. Finallyin Section 7 we finish the proof of Theorem 1.1 with a Calder´on-Zygmund type estimateproven in [17]. 2. Some Examples
Now we elaborate on some examples to which Theorem 1.1 applies. The notation inthis section is self-contained.2.1.
Averages along curves in H . Define the Heisenberg group H to be R with thegroup operation x ⊙ y = (cid:0) x + y , x + y , x + y + ( x y − x y ) (cid:1) . Let γ : [0 , → R be a smooth regular curve whose tangent vector is nowhere parallel to(0 , , γ ( t ) = ( t, γ ( t ) , γ ( t )). Let µ be asmooth measure supported on γ ([0 , f ∈ S ( R ) we define Af ( x ) = Z f ( γ ( t ) − ⊙ x ) dµ ( t ) . Secco, in [21], developed a group-invariant notion for higher derivatives of γ and formu-lated two conditions which serve as right- and left-invariant analogues of nonvanishingcurvature and torsion. These conditions aredet (cid:16) γ ′′ ( t ) γ ′′ ( t ) γ ′′′ ( t ) γ ′′′ ( t ) (cid:17) + ( γ ′′ ( t )) = 0(2.1) det (cid:16) γ ′′ ( t ) γ ′′ ( t ) γ ′′′ ( t ) γ ′′′ ( t ) (cid:17) − ( γ ′′ ( t )) = 0 . (2.2)In [2], the author showed that if (2.1) holds for all t ∈ supp( χ ) and (2.2) does not hold forany t ∈ supp( χ ) then A maps boundedly from L pcomp ( R ) into L p /p ( R ) for p >
4. Underthis condition, the operator A is a Fourier integral operator where π L is a fold and π R isa blowdown. An example of a curve satisfying this condition is γ ( t ) = ( t, t , t ).We next check that A satisfies the final condition of Theorem 1.1. The associatedincidence manifold M is given by M = (cid:8) ( x, y ) : Φ( x, y ) = 0 } , here Φ( x, y ) = (cid:16) Φ ( x,y )Φ ( x,y ) (cid:17) = (cid:16) x − y − γ ( x − y ) x − y − γ ( x − y )+ 12 x γ ( x − y ) − x ( x − y ) (cid:17) . The twisted conormal bundle is given by( N ∗ M ) ′ = (cid:8)(cid:0) x, ( τ · Φ) x , y, − ( τ · Φ) y (cid:1) : Φ( x, y ) = 0 (cid:9) and L is the submanifold of ( N ∗ M ) ′ defined by( τ , τ ) ⊥ (cid:0) γ ′′ ( x − y ) , γ ′′ ( x − y ) − x γ ′′ ( x − y ) (cid:1) . The condition that (2.1) holds for all t ∈ supp( χ ) and (2.2) does not hold for any t ∈ supp( χ ) implies γ ′′ ( t ) = 0 for all t ∈ supp( χ ), hence the restriction of ( N ∗ M ) ′ to L amounts to a restriction of the τ variables to a 1-dimensional linear subspace for each( x, y ) ∈ M . Thus the projection ̟ defined in Theorem 1.1 is a submersion and werecover the result from [2] that A : L pcomp ( R ) → L p /p ( R ) for p > Restricted X-ray transforms in R . Let M be the space of lines in R , let F ⊂ M be a 3 dimensional line complex such that the cone of lines through each point is curved,and as defined in the introduction let X F f ( l ) = Z l f, l ∈ F . We recall the parametrization of the Lagrangian of X F from a survey paper of Phong, [14],to verify that X F satisfies the assumptions of Theorem 1.1. As shown in [11], the maps π L and π R are respectively a blowdown and a Whitney fold, so we only need to verify thatthe projection ̟ | L is a submersion in this case. We can view M locally as a submanifoldof T R , identifying each line l with a point P and a direction γ . As a consequence T ∗ M (resp. its subspace T ∗ F ) can be identified with the restriction of T ∗ ( T R ) to T M (resp. T F ), viewed as functionals on T ( T R ). The defining relation for X F is given by Z = { (( P, γ ) , Q ) : ( P, γ ) ∈ F , Q ∈ l } = { (( P, γ ) , Q ) : ( Q − P ) ∧ γ = 0 } , and its twisted conormal bundle, using the formalism above, is given by N ∗ Z = { ((( P, γ ); γ ∧ τ, ( Q − P ) ∧ τ ); ( Q ; τ ∧ γ )) (cid:12)(cid:12) T l F : ( Q − P ) ∧ γ = 0 } . At this point we make a more concrete characterization of T l F and T l M . Fixing l =( P, γ ) ∈ F , let e = γ and pick e , e such that e , e , e form an orthonormal basis ofvectors on R . With s parametrizing arclength on l , the line l can be deformed to anotherline in M by P + sγ P + sγ + ( a s + b ) e + ( a s + b ) e , where a i , b i are any constants. Thus the Jacobi fields e , se , e , se can be viewed as abasis for T l M . Given a Jacobi field X ( s ) = ( a s + b ) e + ( a s + b ) e , we can view thedeformation above using the identification l = ( P, γ ) ∈ T R as( P, γ ) ( P + X (0) , γ + X ′ ) . Thus a tangent vector in T l M can be identified as a pair ( X (0) , X ′ ) lying in T ∗ ( T R ).The Gelfand admissibility condition (Definition 1.1) states that along the line l , thenormal space to F is proportional to a fixed vector. This implies we can pick a unit Jacobifield X ( s ) that is normal to T l F and is proportional to a fixed vector perpendicular to γ . hoose e to be this vector, and choose a, b ∈ R so that a + b = 1 and X ( s ) = ( a − sb ) e .Recall that M is a symplectic manifold with symplectic form given by ω (cid:16) X i =1 ( a i s + b i ) e i , X i =1 ( c i s + d i ) e i (cid:17) = X i =1 b i c i − a i d i . Then using X ( s ) we can form a symplectic basis for T l M , given by X ( s ) = ( a − sb ) e , X ( s ) = ( as + b ) e , X ( s ) = e , X ( s ) = se Writing l = ( P, γ ) the basis for T l F is given by { ( X i (0) , X ′ ) } i =1 . Let { Φ i } i =1 be the dualbasis in T ∗ l F . Parametrizing Q by its distance t from P , i.e. Q = P + tγ , we can rewrite τ ∧ γ = τ e + τ e and τ ∧ ( Q − P ) = t ( τ e + τ e ). Then the twisted conormal bundleis given by N ∗ Z = { ( P, γ ); − ( at + b ) τ Φ − τ Φ − tτ Φ ; P + tγ ; τ e + τ e ) : τ , τ , t ∈ R } . We can parametrize (
P, γ ) = P i =1 α i ( X i (0) , X ′ i ); thus we can parametrize N ∗ Z by t, τ , τ , α i ( X i (0) , X ′ i ) , i = 1 , ,
3. Using this formalism we can describe dπ L . Using theabove parametrization we can identify π L with the map( { α i ( X i (0) , X ′ i ) } i =1 , t, τ , τ ) (cid:16) X i =3 α i ( X i (0) , X ′ i ) , − ( at + b ) τ Φ − τ Φ − tτ Φ (cid:17) , and thus analytically dπ L = (cid:18) I B (cid:19) , where B = − aτ − ( at + b ) 00 0 − − τ − t . The determinant of this matrix is τ ( at + b ), so if we make the generic assumptionthat 2 at + b = 0, L is exactly the subvariety of N ∗ Z on which τ = 0. The projection ̟ : N ∗ Z → Z from Theorem 1.1 maps(( P, γ ); − ( at + b ) τ Φ − τ Φ − tτ Φ ; Q ; τ e + τ e ) (( P, γ ); Q ) . Since (
P, γ ; Q ) is parametrized by only α i ( X i (0) , X ′ i ) and t we see that ̟ | L is a submersion.Thus again we see that Theorem 1.1 generalizes the results of [18] and covers restrictedX-ray transforms for Gelfand-admissible line complexes as long as the cones Γ Q are curved.3. Initial Setup
Using basic facts on generalized Radon transforms we can simplify our operator R .By localization we may assume that the Schwartz kernel of R is supported in a smallneighborhood of a base point P ◦ = ( x ◦ , y ◦ ) ∈ M . On that neighborhood the manifold M can be expressed locally by a defining function Φ = (Φ , Φ ) ⊺ : Ω L × Ω R → R . In otherwords, M = { ( x, y ) : Φ( x, y ) = 0 } in a neighborhood of P ◦ . Thus using the Fourier nversion formula the Schwartz kernel of R is given by an oscillatory integral distribution,formally written as(3.1) χ ( x, y ) δ ◦ Φ( x, y ) = (2 π ) − Z Z e τ · Φ( x,y ) χ ( x, y ) dτ. Following the procedure found in [20], by local changes of variables and possible redefini-tion of χ , we can write R locally as the oscillatory integral operator R f ( x ) = Z Z e iτ · ( S ( x,y ) − y ′ ) χ ( x, y ) f ( y ) dτ dy. The twisted conormal bundle associated to R is given by( N ∗ M ) ′ = { ( x, ξ, y, η ) : y i = S i ( x, y ) , i = 1 , , ξ = τ S ( x, y ) + τ S ( x, y ) ,η = ( τ , τ , − τ S y ( x, y ) − τ S ( x, y )) } . Thus parametrizing ( N ∗ M ) ′ by the coordinates ( x , x , x , τ , τ , y ), the projection π L :( N ∗ M ) ′ → T ∗ Ω L is identified with the map˜ π L : ( x , x , x , τ , τ , y ) ( x, τ S x ( x, y ) + τ S x ( x, y )) . Then we see D ˜ π L = (cid:18) I × ∂ x i ∂ x j ( τ · S ) B (cid:19) , where B = (cid:0) S x S x ( τ · S ) xy (cid:1) . Thus we seedet D ˜ π L = det( S x S x τ S xy + τ S xy ) = τ ∆ + τ ∆ , where ∆ i ( x, y ) = det( S x S x S ixy ) (cid:12)(cid:12)(cid:12) x,y , i = 1 , . We define L = { ( x, ξ, y, η ) ∈ C : det D ˜ π L = 0 } . Then L is a conic submanifold of( N ∗ M ) ′ defined by τ ∆ ( x, y ) + τ ∆ ( x, y ) = 0 . Similarly, we can identify π R : ( N ∗ M ) ′ → T ∗ Ω R with˜ π R : ( x , x , x , τ , τ , y ) (cid:0) S ( x, y ) , y , τ, − (cid:0) τ S y ( x, y ) + τ S y ( x, y ) (cid:1)(cid:1) . Let N ( x, y ) = S x ( x, y ) ∧ S x ( x, y ). We see that a kernel field for ˜ π R is given by V R = h N ( x, y ) , ∇ x i . Indeed, we see that h N ( x, y ) , ( τ · S y ) x i = τ · ∆, and thus vanishes on L . Note this impliesthat − ∆ S xy + ∆ S xy ∈ Span( S x , S x ). Since π R is a blowdown, V R is parallel to L , whichimplies V kR τ · S y = 0 on L for all k ≥ T ∗ Ω L of L . Let Σ x be the fibers of π L ( L ), given byΣ x = { ( τ · S ) x ( x, y ) : τ · ∆( x, y ) = 0 } = {± ρ Ξ( x, y ) : ρ > } , where Ξ( x, y ) = − ∆ ( x, y ) S x ( x, y ) + ∆ ( x, y ) S x ( x, y ) . Then we see two consequences, one related to our assumption on ̟ . emma 3.1 ([20], § . If π L is a fold and ̟ is a submersion, then | ∆ | 6 = 0 near L , and Σ x is a two-dimensional cone that has one non-vanishing principal curvature given by ρ h Ξ y ,y , N i . Lemma 3.2.
The direction normal to Σ x at a point specified by ( y , ρ ) is given by N ( x, y ) .Proof. Let a ∈ R be fixed. The tangent space of Σ a at a point parametrized by ( y , ρ ) isspanned by T ( a, y ) = Ξ( a, y )˜ T ( a, y ) = Ξ y ( a, y ) , so a normal vector at a point ( ρ, y ) is given by T ∧ ˜ T = Ξ ∧ Ξ y = (∆ ∆ y − ∆ ∆ y )( S x ∧ S x )+ (∆ S x − ∆ S x ) ∧ (∆ S xy − ∆ S xy ) . Since − ∆ S xy + ∆ S xy ∈ Span( S x , S x ) for fixed ( x, y ), the expression in the final lineof the calculation of T ∧ T is either 0 or a scalar multiple of the vector S x ∧ S x = N ,hence the sum is a multiple of N ( a, y ). (cid:3) Initial Decomposition
We localize in | τ | then localize away from the singular variety L , following the ideas ofPhong and Stein in [15]. Let χ ∈ C ∞ c ( R ) be equal to 1 on [ ,
2] and supported on [ , P k ∈ Z χ (2 k · ) ≡
1. For k ≥ χ k ( | τ | ) = χ (2 − k | τ | ). For 0 ≤ ℓ ≤ k/ a k,ℓ, ± ( x, y , τ ) = (cid:26) χ (2 ℓ − k ( ± τ · ∆( x, y ))) ℓ < ⌊ k ⌋ − P k> ℓ χ (2 ℓ − k ( ± τ · ∆( x, y ))) ℓ = ⌊ k ⌋ and define R k,ℓ, ± f ( x ) = χ ( x ) Z e iτ · ˜Φ( x,y ) χ ( y ) f ( y ) χ k ( | τ | ) a k,ℓ, ± ( x, y , τ ) dy dτ. (4.1)We will suppress the dependence on ± . We prove the following estimate. Proposition 4.1.
For p > there exists ε ( p ) > such that for all ℓ ≤ ⌊ k/ ⌋ , kR k,ℓ k L p → L p ≤ C p − ( k + ℓε ) /p . This proposition follows by interpolation with L estimates, L ∞ estimates, and a de-coupling inequality. Let I be a collection of intervals of length 2 − ℓ with disjoint interiorsintersecting a small neighborhood of 0. Then for a function f : R → R supported inthe unit cube and any I ∈ I , let f I ( y ) := f ( y ) I ( y ), so that f = P I ∈I f I with almostdisjoint supports in y . The necessary L estimate is the following. roposition 4.2. Let R k,ℓ be defined as above. For every ε > , kR k,ℓ k L → L . ( ℓ − k ) / ℓε , ℓ ≤ k/ . (4.2) Moreover, by almost disjoint supports of the functions f I , (cid:13)(cid:13)(cid:13)X I ∈I R k,ℓ f I (cid:13)(cid:13)(cid:13) L . ( ℓ − k ) / ℓε (cid:16) X I ∈I k f I k L (cid:17) / , ℓ ≤ k/ , . (4.3)Proposition 4.2 will be proven in Section 6 following methods of almost-orthogonalityfound in the proof of the Calder´on-Vaillancourt theorem (see [13], § Proposition 4.3.
For every ε > (cid:13)(cid:13)(cid:13)X I ∈I R k,ℓ f I (cid:13)(cid:13)(cid:13) L p . ε ℓ (1 / − /p + ε ) (cid:16) X I ∈I kR k,ℓ f I k pL p (cid:17) /p + 2 − k k f k L p for ≤ p ≤ . Following a similar approach to [1] and [20], we prove Proposition 4.3 in Section 5using an inductive argument, at each step combining l p decoupling with suitable changesof variables. Proof that Propositions 4.2 and 4.3 imply Proposition 4.1.
We begin by proving an L ∞ estimate for R k,ℓ , namely thatsup I ∈I kR k,ℓ f I k ∞ . − ℓ sup I ∈I k f I k ∞ (4.4) kR k,ℓ f k ∞ . k f k ∞ . (4.5)To see (4.4) we estimate the Schwartz kernel of R k,ℓ (call it R k,ℓ ( x, y )) by integratingby parts in the τ variables, distinguishing the directions (∆ , ∆ ) and ( − ∆ , ∆ ). Thisshows that | R k,ℓ ( x, y ) | ≤ C N U ( x, y ) U ( x, y ), where U ( x, y ) = 2 k − ℓ (1 + 2 k − ℓ | ∆ ( y − S ) + ∆ ( y − S ) | ) N U ( x, y ) = 2 k (1 + 2 k | − ∆ ( y − S ) + ∆ ( y − S ) | ) N We integrate in y ′ first, then in y , which is supported in an interval of length 2 − ℓ . Toprove (4.5) the same argument holds, but we integrate over a larger interval in y .Interpolating (4.4) with (4.3) we obtain (cid:16) X I ∈I kR k,ℓ f I k pp (cid:17) /p . ε ℓ (3 /p − ε ) − k/p (cid:16) X I ∈I k f I k pp (cid:17) /p , ≤ p ≤ ∞ . (4.6)Combining this estimate with Proposition 4.3 we obtain(4.7) kR k,ℓ f k p . ε ℓ ( ε +2 /p − / − k/p (cid:16) X I ∈I k f I k pp (cid:17) /p + 2 − k k f k p , ≤ p ≤ . ote that the power of 2 ℓ in (4.7) is negative if 4 < p ≤ ε is sufficiently small. Afurther interpolation with the L ∞ estimate (4.5) yields Proposition 4.1 for p > (cid:3) Decoupling
We mirror the structure of the decoupling estimates in [20], working out a model casefirst then reducing the general case to the model case by changes of variables. In themodel case, the functions S i are replaced by S i satisfying simplifying assumptions at theorigin. Additionally, the blowdown condition in this model case implies some additionalassumptions near the origin.5.1. A Model Case.
Consider C ∞ maps ( w, z ) S i ( w, z ) defined on a neighborhoodof [ − r, r ] for some r ∈ (0 , n ∈ N define M n > M n ≥ k S k C n +5 ([ − r,r ] ) + k S k C n +5 ([ − r,r ] ) , where the C n norm is the supremum of all derivatives orders 0 to n . We assume that for w ∈ [ − r, r ] ,(5.2) ( S , S , S z ) (cid:12)(cid:12)(cid:12) ( w, = ( w , w , w );we also assume(5.3) S w,z (0 ,
0) = 0 , and(5.4) S w z (0 ,
0) = κ . As the functions S , S play the part of S , S in our model case, we can analyze thegeometry of the canonical relation associated to S , S . Define ∆ i S = det( S w S w S iwz ).In this model case the singularity surface L S is given by the restriction µ ∆ S ( w, z ) + µ ∆ S ( w, z ) = 0. We can define the analogue of the right projection ˜ π R : ( w, µ, z ) ( S ( w, z ) , z , µ, − ( µ S z ( w, z ) + µ S z ( w, z ))), and a kernel field for this map at thepoint P parametrized by ( w, z , µ ) is given by V R ( w, z ) = h S w ( w, z ) ∧ S w ( w, z ) , ∇ w i . We assume a blowdown on ˜ π R , i.e. that V R is parallel to L S , implying that V NR [ µ ∆ S + µ ∆ S ] (cid:12)(cid:12)(cid:12) ( w,z ) ,µ ⊥ ∆ S ( w,z ) = 0for all N >
0. Since S w ( w,
0) = e and S w ( w,
0) = e , we see that V R ( w,
0) = ∂ w . Theabove conditions imply that ∂ Nw S w z ( w,
0) = 0 , ∀ N ≥ ∂ Nw ∆ S ( w,
0) = 0 , ∀ N ≥ . (5.6) ecall that the fibers of the singular manifold L S are given for fixed w by˜Σ w = { µ S w ( w, z ) + µ S w ( w, z ) : µ ∆ S ( w, z ) + ∆ S ( w, z ) = 0 } = {± ρ Ξ S ( w, z ) : ρ > , | z | ≤ r } , where Ξ S ( w, z ) is given by − S w ( w, z )∆ S ( w, z ) + S w ( w, z )∆ S ( w, z ). Thus ˜Σ is acone parametrized by ( ρ, z ) given by {± ρ Ξ S (0 , z ) : ρ > , | z | ≤ r } =: Σ . Recall from Section 3 that S w ∧ S w (0 , b ) =: N ( b ) is normal to Σ at the point P parametrized by ( ρ ′ , b ). Thus T P Σ has an orthogonal basis given by T ( b ) = Ξ S (0 , b ) T ( b ) = T ( b ) ∧ N ( b ) . For
A > δ ≪ A,b ( δ ) be set of ξ ∈ R such that A − ≤ |h T ( b ) | T ( b ) | , ξ i| ≤ A |h T ( b ) | T ( b ) | , ξ i| ≤ Aδ |h N ( b ) | N ( b ) | , ξ i| ≤ Aδ . The sets Π
A,b ( δ ) are unions of A × Aδ × Aδ -boxes with long, middle, and short sidesparallel to T ( b ) , T ( b ), and N ( b ) respectively. We will refer to Π A,b ( δ ) as a plate. Becausethe cone ˜Σ is curved we can apply decoupling to the plates Π A,b ( δ ). Theorem 5.1 ([3]) . Let ε > and A > . There exists a constant C ( ε, A ) such that thefollowing holds for < δ < δ < .Let B = { b ν } Mν =1 be a set of points in an interval J ⊂ [ − , of length δ such that | b ν − b ν ′ | ≥ δ for b ν , b ν ′ ∈ B , ν = ν ′ . Let ≤ p ≤ . Let f ν ∈ L p ( R ) such that theFourier transform of f ν is supported in Π A,b ( δ ) . Then (cid:13)(cid:13)(cid:13)X ν f ν (cid:13)(cid:13)(cid:13) p ≤ C ( ε, A )( δ /δ ) / − /p + ε (cid:16) X ν k f ν k pp (cid:17) /p . Let ( w, z ) α ( w, z ) be a C ∞ function satisfying for | ( w, z ) | ∞ < r , M − ≤ | α ( w, z ) | ≤ M (5.7) |∇ w α ( w, z ) | ≤ M (5.8)Let ( w, z, µ ) ζ ( w, z, µ ) belong to a bounded family of C ∞ functions supported where | ( w, z ) | ∞ ≤ r and 1 / ≤ | µ | ≤ T k,ℓ be an operator with Schwartz kernel(5.9) 2 k Z e i k h µ, S ( w,z ) − z ′ i η (cid:0) ℓ α ( w, z )( µ ∆ S ( w, z ) + µ ∆ S ( w, z )) (cid:1) ζ ( w, z ) η ( | µ | ) dµ. The operator T k,ℓ will play the role of R k,ℓ after a nonlinear change of variables, while α ( w, z ) is introduced in the localization as a byproduct of those changes of variables. roposition 5.1. Let < ε ≤ , k ≫ , ≤ ℓ ≤ k/ , δ ∈ (2 − ℓ (1 − ε ) , − ℓε ) , and δ > δ ≥ max { − ℓ (1 − ε/ , δ − ℓε/ } . Define ε = ( δ /δ ) . Let J be an interval oflength δ containing , and I J be a collection of intervals of length δ with disjoint interiorand whose interiors all intersect J . Let σ ∈ C ∞ c ( R ) be supported ( − , and define σ ℓ,ε ( w ) = σ (2 ℓ w , ℓ w , ε − w ) . Then for ≤ p ≤ , g ∈ L p ( R ) with g I ( y ) = g ( y ) I ( y ) ,and any N ∈ N , (cid:13)(cid:13)(cid:13) σ ℓ,ε X I ∈I J T k,ℓ g I (cid:13)(cid:13)(cid:13) p . ε ( δ /δ ) / − /p + ε X I ∈I J (cid:13)(cid:13)(cid:13) σ ℓ,ε T k,ℓ g I (cid:13)(cid:13)(cid:13) pp ! /p + C ( ε, N )2 − kN − ℓ ε k g k p . The idea here is to show that the Fourier transforms of σ ℓ,ε P I ∈I J T k,ℓ g I are concen-trated on the plates Π A,b I ( δ ) for some b I ∈ I and some large enough A >
Derivatives of S and ∆ . Some approximations will be helpful to write down. Forthe rest of Section 5.1 we omit the subscript dependence on S . Because of (5.2) we mayconclude that for any multiindex β of length at least 1, D βw S w (cid:12)(cid:12) ( w, = 0(5.10) D βw S w (cid:12)(cid:12) ( w, = 0(5.11) D βw S wz (cid:12)(cid:12) ( w, = 0 . (5.12)For w ∈ [ − r, r ] , ∆ ( w,
0) = 1(5.13) ∆ ( w,
0) = S w z ( w, z (0 ,
0) = S w z (0 , z (0 ,
0) = S w z (0 ,
0) = κ (5.16)and thus Ξ( w,
0) = − ∆ ( w, S w ( w,
0) + ∆ ( w, S w ( w,
0) = e − S w z ( w, e (5.17) Ξ w n (0 ,
0) = − S w n +13 z (0 , e = 0 , n ≥ z (0 ,
0) = − κ e + S w z (0 , e . (5.19)Using these, T ( b ) = Ξ(0 , b )(5.20) = Ξ(0 ,
0) + b Ξ z (0 ,
0) + O ( b )(5.21) = − κ be + (1 + b S w z (0 , e + O ( b )(5.22) nd N ( b ) = S w (0 , b ) ∧ S w (0 , b )(5.23) = ( e + be + O ( b )) ∧ ( e + O ( b ))= − be + e + O ( b ) . From these we see that T ( b ) = T ( b ) ∧ N ( b ) = (1 + b S w z (0 , b )) e + κ be + be + O ( b ) . (5.24)Let β = ( β w , β w , β w , β z ) be a multi-index and let D β ( w,z ) denote a derivative of order | β | = β w + β w + β w + β z in the variables w, z . By using the upper bounds M n ,trilinearity of determinants, and differentiation rules for products we can estimate(5.25) | D β ( w,z ) ∆ i | ≤ | β | M | β | . Similarly, by differentiating products,(5.26) | D β ( w,z ) Ξ | ≤ | β | M | β | . Plate Localization.
Lemma 5.1.
Let ε > , and δ , δ , ε be as in Proposition 5.1. Assume that − ℓ ≪ r , M − ℓ ≤ − , ≤ | µ | ≤ , | w ′ | ≤ − ℓ , | w | ≤ ε , | b | ≤ δ , and | z − b | ≤ δ .If (5.27) | µ ∆ ( w, z ) + µ ∆ ( w, z ) | ≤ M − ℓ , then there exists A ( ε ) > such that µ S w ( w, z ) + µ S w ( w, z ) ∈ Π A ( ε ) ,b ( δ ) . More specifically, A ( ε ) − | T ( b | ) ≤ |h T ( b ) , µ S w ( w, z ) + µ S w ( w, z ) i| ≤ A ( ε ) | T ( b ) | (5.28) |h T ( b ) , µ S w ( w, z ) + µ S w ( w, z ) i| ≤ A ( ε ) | T ( b ) | δ . (5.29) |h N ( b ) , µ S w ( w, z ) + µ S w ( w, z ) i| ≤ A ( ε ) | N ( b ) | δ . (5.30) Note that the constant A ( ε ) does not depend on δ , δ .Proof. Throughout this proof we use Taylor expansions with appropriate error remainders.Therefore, for any i = 1 , , ... the function R i ( w, z ) is C ∞ and uniformly bounded by 1.The estimate in (5.28) is clearly true for some A > ε . We start withthe proof of (5.30). Let G = ⌈ ε − ⌉ . Employing a Taylor expansion about ( w, z ) = (0 , b ),and reorganizing terms using that 2 − ℓ ≤ δ , 2 − ℓ δ ≤ δ , ε δ ≤ δ ,and ε G ≤ δ , we see hat h N ( b ) , µ · S w ( w, z ) i = G X n =0 G − n X | α | =0 h N ( b ) , ∇ w (cid:16) ( ∂ z ) n ( ∂ w ) α [ µ S + µ S ] (cid:17) (0 , b ) i (5.31) + M G δ R ( w, µ, z − h N ( b ) , µ S w + µ S w (0 , b ) i (5.32) + ( z − b ) h N ( b ) , µ S wz + µ S wz (0 , b ) i + X i =1 w i h N ( b ) , µ S ww i + µ S ww i (0 , b ) i + I + II + III + M G δ R ( w, µ, z ) , where I = G X n =1 w n n ! h N ( b ) , µ S ww n + µ S ww n (0 , b ) i II = G X n =2 w n − ( z − b ) n ! h N ( b ) , µ S ww n − z + µ S ww n − z (0 , b ) i III = G X n =2 2 X i =1 w n − w i n ! h N ( b ) , µ S ww n − w i + µ S ww n − w i (0 , b ) i . Clearly the first term in (5.31) vanishes by the definition of N ( b ) (see (5.23)). The secondterm in the expansion is( z − b ) h N ( b ) , µ S wz + µ S wz (0 , b ) i = ( z − b )( µ ∆ (0 , b ) + µ ∆ (0 , b )) . Now, since | w ′ | , | z − b | ≤ δ , applying a Taylor expansion and using trilinearity of deter-minants, and differentiation of products we get µ ∆ (0 , b ) + µ ∆ (0 , b ) = (cid:0) µ ∆ ( w, z ) + µ ∆ ( w, z ) (cid:1) + G X n =1 w n n ! (cid:16) µ ∆ w n ( w, z ) + µ ∆ w n ( w, z ) (cid:17) + 3 G M G δ R ( w, z ) . By (5.27) the first term is bounded by M − ℓ . For each 1 ≤ n ≤ G , from (5.6) and(5.13) we have ∆ iw n ( w,
0) = 0 for i = 1 ,
2, and so by trilinearity of determinants, anddifferentiation of products, expanding about z = 0 we get | ∆ iw n ( w, z ) | ≤ (cid:12)(cid:12)(cid:12) ∆ iw n ( w,
0) + 3 n M n z (cid:12)(cid:12)(cid:12) ≤ n M n δ . Thus | µ ∆ (0 , b ) + µ ∆ (0 , b ) | ≤ M − ℓ + 3 G M G ε δ + 3 G M G δ ≤ G +1 M G δ , and the second term in (5.31) is bounded by 3 G +1 M G δ . ext we deal with the first order w ′ derivatives in (5.31). We approximate about z = 0.For i = 1 ,
2, using the estimates (5.10) and (5.11), we get | w i h S w (0 , b ) ∧ S w (0 , b ) , µ S ww i (0 , b ) + µ S ww i (0 , b ) i|≤ | w i | h |h S w (0 , ∧ S w (0 , , µ S ww i (0 , µ S ww i (0 , i| + 3 M bR (0 , b ) i ≤ − ℓ (0 + 3 M δ ) . Note that the condition δ ≥ max { M − ℓ (1 − ε/ , − ℓε/ δ } from Proposition 5.1 impliesthat 2 − ℓ δ ≤ δ .Finally, we estimate I , II , and III . All rely on the blowdown condition at the origin.First we estimate I . For all n ≥
1, we expand about the origin to obtain h S w (0 , b ) ∧ S w (0 , b ) , µ S ww n (0 , b ) + µ S ww n (0 , b ) i = h S w (0 , ∧ S w (0 , , µ S ww n (0 ,
0) + µ S ww n (0 , i + b h det( S wz S w µ S ww n + µ S ww n ) (cid:12)(cid:12)(cid:12) (0 , + det( S w S wz µ S ww n + µ S ww n ) (cid:12)(cid:12)(cid:12) (0 , + det( S w S w µ S ww n z + µ S ww n z ) (cid:12)(cid:12)(cid:12) (0 , i + 3 M n b R (0 , b )Using the estimates (5.10), (5.11), (5.23), (5.12), and (5.5), we observe h S w (0 , ∧ S w (0 , , µ S ww n (0 ,
0) + µ S ww n (0 , i = 0det( S wz S w µ S ww n + µ S ww n ) (cid:12)(cid:12)(cid:12) (0 , = 0det( S w S wz µ S ww n + µ S ww n ) (cid:12)(cid:12)(cid:12) (0 , = 0det( S w S w µ S ww n z + µ S ww n z ) (cid:12)(cid:12)(cid:12) (0 , = µ S w n +13 z (0 ,
0) + µ S w n +13 z (0 ,
0) = 0 . This implies | I | ≤ M G G X n =1 ε n δ n ! ≤ M G ε δ ≤ M G δ . Next we estimate II . For n ≥
2, we expand about the origin to obtain h S w (0 , b ) ∧ S w (0 , b ) , µ · S ww n − z (0 , b ) i = det( S w S w µ S ww n − z + µ S ww n − z ) (cid:12)(cid:12) (0 , + 3 M G bR (0 , b ) . Thus the calculation from I the determinant vanishes, and thus | II | ≤ M G G X n =2 ε n − δ δ n ! ≤ M G ε δ δ ≤ M G δ . inally we estimate III . Again using the calculations from I , for n ≥ i = 1 , h S w (0 , b ) ∧ S w (0 , b ) , µ · S ww n − w i (0 , b ) i = h S w (0 , ∧ S w (0 , , µ · S ww n − w i (0 , i + 3 M G bR (0 , b )= µ · S w i w n (0 ,
0) + 3 M G bR (0 , b )= 3 M G bR (0 , b ) . This implies that | III | ≤ M G G X n =2 ε n − − ℓ δ n ! ≤ M G ε − ℓ δ ≤ M G δ . Since | N ( b ) | ≥ / A ( ε ) ≥ ⌈ /ε ⌉ +2 M ⌈ /ε ⌉ .Having proven (5.30), we prove (5.29). Using (5.24), define T ∗ ( b ) = (1 + b S w z (0 , e + κ be + be and note that | T ( b ) − T ∗ ( b ) | ≤ M δ . Next, we will approximate µ by the projection of µ ∆ ( w, z ) + µ ∆ ( w, z ) onto L S . In particular, let µ ◦ = ± | µ || ∆( w,z ) | ( − ∆ ( w, z ) , ∆ ( w, z )) , so that µ ◦ ∆ ( w, z ) + µ ◦ ∆ ( w, z ) = 0, | µ | = | µ ◦ | , and where the sign is picked so that | µ − µ ◦ | ≤ | µ | M − ℓ . This is possible since | µ ∆ + µ ∆ | ≤ M − ℓ and | ∆( w, z ) | 6 = 0. Then µ ◦ S w ( w, z ) + µ ◦ S w ( w, z ) = | µ || ∆( w,z ) | Ξ( w, z ) , and thus (cid:12)(cid:12)(cid:12) µ S w ( w, z ) + µ S w ( w, z ) − | µ || ∆( w, z ) | Ξ( w, z ) (cid:12)(cid:12)(cid:12) ≤ | µ − µ ◦ || S w | ≤ M − ℓ . We approximate by a Taylor expansion about the origin, using the fact that ε δ ≤ δ , | w ′ | ≤ − ℓ ≤ δ , δ ≤ δ , and ε G ≤ δ ≤ δ . Reorganizing, we obtain h T ∗ ( b ) , Ξ( w, z ) i = G X n =0 G − n X | α | =0 h T ∗ ( b ) , ( ∂ z ) n ( ∂ w ) α Ξ i (cid:12)(cid:12)(cid:12) (0 , + 4 G M G δ R ( w, z )= h T ∗ ( b ) , Ξ(0 , i + z h T ∗ ( b ) , Ξ z (0 , i + G X n =1 w n n ! h T ∗ ( b ) , Ξ w n (0 , i + 4 G M G δ R ( w, z ) . Using (5.17), (5.18), and (5.19) h T ∗ ( b ) , Ξ(0 , i = κ b h T ∗ ( b ) , Ξ z (0 , i = − κ (1 + b ( S w z (0 , − S w z (0 , b ))) h T ∗ ( b ) , Ξ w n (0 , i = 0 , n ≥ . hus |h T ∗ ( b ) , Ξ( w, z ) i| ≤ κ δ + κ M δ + 4 G M G δ and therefore we can estimate |h T ( b ) , µ S w ( w, z ) + µ S w ( w, z ) i| ≤ M δ + 8 M − ℓ + κ δ + κ M δ + 4 G M G δ ≤ κ (1 + 4 G +2 M G ) δ . Thus picking(5.33) A ( ε ) ≥ max { ⌈ /ε ⌉ +2 M ⌈ /ε ⌉ , κ (1 + 4 ⌈ /ε ⌉ +2 M ⌈ /ε ⌉ ) } the Lemma is proven. (cid:3) Proof of Proposition 5.1.
Fix an I ∈ I J and pick b I ∈ I . Let m A,b I ,δ be be amultiplier equal to 1 on Π A,b I ( δ ) which vanishes on Π A,b I ( δ ). Let P k,A,b I ,δ f V ( ξ ) = m A,b I ,δ (2 k ξ ) ˆ f ( ξ ) . Then by Bourgain-Demeter decoupling on the cone, (cid:13)(cid:13)(cid:13)X I P k,A,b I ,δ T k,ℓ f I (cid:13)(cid:13)(cid:13) p ≤ C ( ε, A )( δ /δ ) / − /p + ε (cid:16) X I (cid:13)(cid:13) T k,ℓ f I (cid:13)(cid:13) pp (cid:17) /p , for 2 ≤ p ≤
6. The Schwartz kernel of the operator f (Id − P k,A,b I ,δ ( δ )) T f is givenby a sum of operators P ∞ n =0 K n,k,ℓ ( w, z ), where K n,k,ℓ ( w, z ) = 2 k Z Z Z e i ( h w − v,ξ i +2 k µ · ( S ( v,z ) − z ′ )) σ ( v, w, z, µ, ξ ) dv dξdµ, and the symbol σ is given by σ ( v, w, z, µ, ξ ) = (cid:0) − m A ( ε ) ,b I ,δ (2 k ξ ) (cid:1) η (2 − n | ξ | ) η ( | µ | ) σ ℓ,ε ( v ) × η (2 ℓ α ( v, z ) µ · ∆( v, z )) χ ( v, z ) . Note that the symbol of K n,k,ℓ is supported where | ξ | ∼ n for n ≥ n = 0, | µ | ∼ | v | + | z | ≤ r , and for a priori unbounded w . We nowwill show that K n,k,ℓ is negligible and decays rapidly in n . First, we introduce the detailsof integration by parts in the ξ variables. Note that ∇ ξ Φ = w − v , ∇ jξ Φ = 0 for j ≥ |∇ jξ σ | ≤ C j min { A ( ε ) − δ k , n } − j . Thus integrating by parts many times in the ξ variables yields an estimate of C N { A ( ε ) − δ k , n }| w − v | ) N , allowing us to integrate in w .Assume that 2 | n − k | > C . Then |∇ v Φ | = | − ξ + 2 k ∇ v ( µ · S ( v, z )) | ≥ (cid:12)(cid:12)(cid:12) | ξ | − | k ∇ v ( µ · S ( v, z )) | (cid:12)(cid:12)(cid:12) ≥ C max { k , n } . e also see that |∇ jv Φ | ≤ A ( ε )2 k for j ≥
2, and |∇ jv σ | ≤ ℓj for j ≥
1. Since ℓ < k/ v variables, and then using our estimate from integrating byparts in the ξ variables, we get | K n,k,ℓ ( w, z ) | ≤ Z Z Z A ( ε )max { k , n } N | w − v | ) N dv dξ dµ. For | n − k | < C we use the support of ξ off of the plate Π A ( ε ) ,b I ( δ ).Suppose |h T ( b I ) | T ( b I ) | , ξ i| ≥ A ( ε )2 k δ . Define ∂ T ( b I ) = h T ( b I ) , ∇ v ·i . Then by (5.29) | ∂ T ( b I ) Φ | ≥ A ( ε )2 k δ . We can also estimate for j ≥ | ∂ jT ( b I ) σ | ≤ C j A ( ε )2 ℓj , and for j ≥ | ∂ jT ( b I ) Φ | ≤ C j A ( ε )2 k ≤ C j A ( ε )2 ℓ ( j − δ . Thus integrating by parts in the T ( b I ) direction N times and then M times in the ξ variables we obtain the estimate | K n,k,ℓ ( w, z ) | . N,M,ε
Z Z | ξ |∼ k Z | µ |∼ k − ℓ δ ) N k | w − v | ) M dv dξ dµ. Since 2 k − ℓ δ ≥ kε/ , integrating by parts in the T ( b I ) direction ∼ /ε times gives therequired estimate.Assume that |h N ( b I ) , ξ i| ≥ A ( ε )2 k δ . Define ∂ N ( b I ) = h N ( b I ) , ∇ v i . Then by (5.30) | ∂ N ( b I ) Φ | ≥ A ( ε )2 k δ . We claim that | ∂ jN ( b I ) σ | ≤ C j A ( ε ) max { ℓ δ , ε − } j . To see this, use the approximation N ( b I ) = − b I e + e + C ( b I ) b I , where | C ( b I ) | < M ,from (5.31). Then we can see from the definition of σ , commuting differential operatorswith constant coefficients, bounds on α , and differentiation of products and compositions,that | ∂ jN ( b I ) σ | ≤ C j A ( ε ) max { ε − , ℓ δ , ℓ δ , ℓ max ≤ i ≤ j µ · ∆ v i ( v, z ) } j Using (5.13), (5.6), and a Taylor expansion about z = 0 we see that ∂ jv µ · ∆( v, z ) = µ · ∆ v j ( v,
0) + A ( ε ) δ = A ( ε ) δ . Thus the claim is proven. To use integration by parts we also need to estimate | ∂ jN ( b I ) Φ | ≤ C j A ( ε )2 k max { ℓ δ , ε − } j − δ , j ≥ . sing (5.31) again, | ∂ jN ( b I ) Φ | = | ( h ( b I , ,
1) + O ( b I ) , ∇ v i j Φ |≤ O ( b I ) sup n =1 ,...,j |∇ nv Φ | + b I Φ v v j − ( v, z ) + Φ v j ( v, z ) ≤ C j b I + b I | µ · S v v j − ( v, z ) | + | µ · S v j ( v, z ) | . Next, we estimate from the value at ( v, v v j − ( v, z ) = µ · S v v j − ( v,
0) + 2 M j b I R ( v, z )Φ v j ( v, z ) = µ · S v j ( v,
0) + µ · S v j z ( v,
0) + 2 M j b I R ( v, z ) . From (5.10), (5.11), and (5.12) we see that µ · S v v j − ( v,
0) = 0 ,µ · S v j ( v,
0) = 0 , S v j z ( v,
0) = 0 . Moreover (5.5) ensures that S v j z ( v,
0) = 0 , j ≥ . Hence for j ≥ | ∂ jN ( b I ) Φ | ≤ C j A ( ε ) δ ≤ C j A ( ε ) ε − j +11 δ , so by N integration by parts in the v variables, followed by M integration by parts in the ξ variables, | G n,k,ℓ ( w, z ) | . N,M,ε
Z Z | ξ |∼ k Z | µ |∼ (cid:16) { k − ℓ δ /δ , k δ ε } (cid:17) N | w − v | ) M dvdξdµ. Since δ ≥ − ℓ (1 − ε/ and δ ≥ − ℓε/ δ , we have2 k − ℓ δ /δ ≥ k − ℓ + ℓε/ ≥ kε/ k δ ε ≥ k − ℓ + ℓε/ ≥ kε/ . So if N ∼ /ε we obtain the desired estimate. Thus by Minkowski’s inequality, the sizeof the support of σ ℓε , and the above estimates on K n,k,ℓ ( w, z ), we bound (cid:13)(cid:13)(cid:13) X I ∈I J (Id − P k,A ( ε ) ,b I ,δ )[ σ ℓ,ε T k,ℓ g I ] (cid:13)(cid:13)(cid:13) p = (cid:13)(cid:13)(cid:13) X I ∈I J X n ≥ Z K n,k,ℓ ( · , z ) g I ( z ) dz (cid:13)(cid:13)(cid:13) p ≤ X n ≥ { k , n } N − ℓ ε × (cid:16) Z (cid:12)(cid:12)(cid:12) | w | ) M Z g ( z ) dz (cid:12)(cid:12)(cid:12) p dw (cid:17) /p ≤ − Nk − ℓ ε || g || p . This finishes the proof of Proposition 5.1. .2. Families of changes of variables.
We use the family of changes of variables usedin [20] to reduce the general case to the model case. Let P ◦ = ( a ◦ , y ◦ ) ∈ M , with y ◦ = ( S ( a ◦ , b ◦ ) , S ( a ◦ , b ◦ ) , b ◦ ). For r > , q > Q ( r ) = { ( x , x , x ) : | x − a ◦ | ≤ r } and I ( r ) = { y : | y − b ◦ | ≤ r } . For i = 1 ,
2, let S i be smooth functions in a neighborhood of Q (2 r ) × I (2 r ), for some r >
0. We assume that ∆ ( x, y ) = det( S x , S x , S xy ) = 0 on Q (2 r ) × I (2 r ). Choose M >
M > k S k C ( Q (2 r ) × I (2 r )) + max ( x,y ) ∈ Q (2 r ) × I (2 r ) | ∆ ( x, y ) | − For a ∈ Q (2 r ), b ∈ I (2 r ), letΓ ( x, y ) = det( S x S xy S xy )Γ ( x, y ) = det( S xy S x S xy ) , and let ρ ( a, b ) ∈ R be defined by( ρ , ρ , ρ ) := 1∆ ( a, b ) ( − Γ ( a, b ) , Γ ( a, b ) , ∆ ( a, b )) . For ( x, y ) , ( a, y ) ∈ Q ( r ) × I (2 r ), consider the map( x, a, y ) w ( x, a, y ) ∈ R given by w ( x, a, b ) = S ( x, b ) − S ( a, b ) w ( x, a, b ) = S ( x, b ) − ρ ( a, b ) S ( x, b ) − S ( a, b ) + ρ ( a, b ) S ( a, b ) w ( x, a, b ) = S y ( x, b ) − S y ( a, b ) . Then det( D w /Dx ) = det( S x ( x, b ) , S x − ρ ( a, b ) S x ( x, b ) , S xy ( x, b )) = ∆ ( x, b ) = 0 . By the implicit function theorem, there exists r > r < r such that | w | ∞ < r , a ∈ Q (2 r ), b ∈ I (2 r ), the equation w ( x, a, b ) = w is solved by a unique C ∞ function x = x ( w, a, b ) such that for | x | ∞ < (50 M ) − r , x ( w ( x, a, b ) , a, b ) = x. We also change variables in y . Define z : R × Q (2 r ) × I (2 r ) → R by z ( y, a, b ) = y − S ( a, y ) z ( y, a, b ) = y − ρ ( a, b ) y − S ( a, y ) + ρ ( a, b ) S ( a, y ) − ( y − b ) X i =1 ρ i ( y i − S i ( a, y )) z ( y, a, b ) = y − b. or z , b ∈ I ( r ), a ∈ Q (2 r ), where r < min { r , (24 M ) − } , we define the inverse z y ( z, a, b ) by y ( z, a, b ) = z + S ( a, b + z ) y ( z, a, b ) = z + z ( ρ ( a, b ) + ρ ( a, b ) z ) + (1 − z ) S ( a, b + z )1 − ρ ( a, b ) z y ( z, a, b ) = b + z . Lemma 5.2 ([20]) . The function x , y defined above have the following properties. (1) x (0 , a, b ) = a , y (0 , a, b ) = ( S ( a, b ) , S ( a, b ) , b ) , y ( z, a, b ) = b + z . (2) det (cid:16) D x ( w,a,b ) dw (cid:17) = ( x ( w,a,b ) ,y ) . (3) Let ρ = ρ ( a, b ) , and let B ( z , a, b ) = (cid:16) − ρ − ρ z − ρ z (cid:17) . Then for | z | ≤ r , a ∈ Q (2 r , q ) , | w | ≤ r B ( z , a, b ) (cid:16) S ( x ( w,a,b ) ,b + z ) − y ( z,a,b ) S ( x ( w,a,b ) ,b + z ) − y ( z,a,b ) (cid:17) = (cid:16) S ( w,z ,a,b ) − z S ( w,z ,a,b ) − z (cid:17) , where S i are C ∞ with ( S , S , S z ) (cid:12)(cid:12) ( w, ,a,b ) = w and S wz (0 , , a, b ) = 0 . (4) Let ∆ iS ( x, y ) = det( S x , S x , S ixy ) (cid:12)(cid:12) ( x,y ) ∆ i S ( x, y , a, b ) = det( S w , S w , S iwz ) (cid:12)(cid:12) ( w,z ,a,b ) . Then for ( τ , τ ) = ( µ , µ ) B ( z , a, b ) , X i =1 τ i ∆ iS ( x ( w, a, b ) , b + z ) = ∆ S ( x ( w, a, b ) , b )1 − ρ ( a, b ) z X i =1 µ i ∆ i S ( w, z , a, b ) . (5) Let κ ( a, b ) = Γ ∆ S − Γ ∆ S + ∆ S ∆ S,y − ∆ S ∆ S,y (cid:12)(cid:12)(cid:12) ( a,b ) . Then S w z z (0 , , a, b ) = κ ( a, b )∆ S ( a, b ) . The proof is found in [20]. .3. Decoupling in the General Case.Proposition 5.2.
Let < ε < , k ≫ , ℓ ≤ k/ . Let δ ∈ (2 − ℓ (1 − ε ) , − ℓε ) , and define < δ < δ such that max { − ℓ (1 − ε/ , δ − ℓε/ } < δ < δ . Define ε = ( δ /δ ) . Let J be an interval of length δ within r of b ◦ , and let I J be acollection of intervals which have disjoint interior, intersecting J . For each I ∈ I J , define f I ( y ) = f ( y ) I ( y ) . Then for ≤ p ≤ (cid:13)(cid:13)(cid:13) X I ∈I J R k,ℓ f I (cid:13)(cid:13)(cid:13) p ≤ C ε ( δ /δ ) / − /p + ε (cid:16) X I ∈I J (cid:13)(cid:13)(cid:13) R k,ℓ f I (cid:13)(cid:13)(cid:13) pp (cid:17) /p + C N,ε − kN k f k p . Proof.
Fix b ∈ J . Let σ ∈ C ∞ c supported in ( − ,
1) such that σ ≥ P n ∈ Z σ ( · − n ) = 1. For n ∈ Z define σ n ( x ) = σ ( x − n ) and for a ∈ Z let ς ℓ,ε ( x, a, b ) = σ a (2 ℓ w ( x, a, b )) σ a (2 ℓ w ( x, a, b )) σ a ( ε − w ( x, a, b )) . Then since | x | , | a | < r / | b | < r /
2, we have that ς ℓ,ε ( x ( w, a, b ) , a, b ) = σ a (2 ℓ w ) σ a (2 ℓ w ) σ a ( ε − w ) =: σ ℓ,ε ( w, a ) , and P a ∈ Z σ ℓ,ε ( w, a ) = 1 with finite overlap for all | w | < r /
2. Then by H¨older’s inequal-ity (cid:13)(cid:13)(cid:13) X I ∈I J R k,ℓ f I (cid:13)(cid:13)(cid:13) p ≤ C p (cid:16) X a ∈ Z (cid:13)(cid:13)(cid:13) X I ∈I J ς ℓ,ε ,a,b R k,ℓ f I (cid:13)(cid:13)(cid:13) pp (cid:17) /p . Note that the terms vanish for | a | > r . Fix a . Write g ( z, a, b ) = f ( y ( z, a, b )). Applychanges of variables y = y ( z, a, b ) and τ = B ⊺ − ( z , a, b ) µ , noting thatdet( D y /dz ) det B = 1so that R k,ℓ f ( x ) = 2 k Z Z e i k h µ, S ( w ( x,a,b ) − z ′ i ˜ χ k,ℓ ( x, z, µ, a, b ) g ( z, a, b ) dµdz, with ˜ χ k,ℓ ( x, z, µ, a, b ) = χ ( x, y ( z, a, b )) η ( | B ⊺ − ( z , a, b ) µ | ) η (cid:16) ℓ ∆ ( x )1 − ρ ( a,b ) z × ( µ ∆ S ( w ( x, a, b ) , z , a, b ) + µ ∆ S ( w ( x, a, b ) , z , a, b )) (cid:17) . Thus we see that ς ℓ,ε ,a,b ( x ( w, a, b )) X I ∈I J R k,ℓ f I ( x ( w, a, b )) = σ ℓ,ε ,a,b ( w ) X I ∈I J T k,ℓ,a,b g I ( w ) , where g I ( z, a, b ) = g ( z, a, b ) − b + I ( z ) and T k,ℓ,a,b ≡ T k,ℓ from the model case. Define M n ( a, b ) ≥ || S ( · , a, b ) || C n +5 ([ − r ,r ] ) + || S ( · , a, b ) || C n +5 ([ − r ,r ] ) ˜ A ( ε ) = sup a,b ∈ [ − r ,r ] max { ⌈ /ε ⌉ +2 M ⌈ /ε ⌉ ( a, b ) , κ ( a, b )(1 + 4 ⌈ /ε ⌉ +2 M ⌈ /ε ⌉ ( a, b )) } ; hese are the uniform versions of (5.1) and (5.33) respectively. We can then write (cid:13)(cid:13)(cid:13) ς ℓ,ε ,a,b X I ∈I J R k,ℓ f I (cid:13)(cid:13)(cid:13) p = (cid:16) Z (cid:12)(cid:12)(cid:12) ς ℓ,ε ,a,b ( x ( w, a, b )) X I ∈I J R k,ℓ f I ( x ( w, a, b )) k p | det( D x Dw ) | dw (cid:17) /p . (cid:13)(cid:13)(cid:13) σ ℓ,ε ,a,b X I ∈I J T k,ℓ g I (cid:13)(cid:13)(cid:13) p by the uniform upper bound on | det( D x Dw ) | . Then we can apply Proposition 5.1 with A ( ε ) = ˜ A ( ε ) to get (cid:13)(cid:13)(cid:13) σ ℓ,ε ,a,b X I ∈I J T k,ℓ g I (cid:13)(cid:13)(cid:13) p ≤ C ε ( δ /δ ) / − /p + ε (cid:16) X I ∈I J (cid:13)(cid:13) σ ℓ,ε ,a,b T k,ℓ g I (cid:13)(cid:13) pp (cid:17) /p + C ε − k − ℓ ε k g k p . Then undoing the changes of variables above (and applying the uniform lower bounds on | det( D x Dw ) | ) we may bound this by C ′ ε ( δ /δ ) / − /p + ε (cid:16) X I ∈I J (cid:13)(cid:13) ς ℓ,ε ,a,b R k,ℓ f I (cid:13)(cid:13) pp (cid:17) /p + C ε − k − ℓ ε k f k p . Finally, we recombine our partition of unity in x using the fact that there are at most C ℓ ε − many a ∈ Z for which σ ℓ,ε ,a,b is nonzero, to get (cid:13)(cid:13)(cid:13) X I ∈I J R k,ℓ f I (cid:13)(cid:13)(cid:13) p ≤ C p (cid:16) X a ∈ Z | a | ∞ Let δ = 2 − ℓε , and define δ j = δ j − − ℓε/ for j = 1 , , ... Note that this implies ε = ( δ /δ ) = 2 − ℓε/ . We will iterate the estimatein Proposition 5.2 until δ j ≤ − ℓ (1 − ε ) . Let j ∗ be the smallest j such that δ j < − ℓ (1 − ε ) .Clearly j ∗ . /ε and 2 − ℓ (1 − ε/ ≤ δ j ∗ ≤ − ℓ (1 − ε ) .For j = 0 , , , ... let I j denote an interval of length δ j inside [ b ◦ − r , b ◦ + r ], and let I I j denote the collection of intervals I j +1 of length δ j +1 intersecting I j with disjoint interior.Finally, let J = [ b ◦ − r / , b ◦ + r / 2] and let I J,j denote the collection of intervals I j oflength δ j intersecting J with disjoint interiors. Then since r < δ = 2 − ℓε , using ¨older’s and Minkowski’s inequalities we have(5.34) kR k,ℓ f k p . ℓε/p ′ (cid:16) X I ∈I J, (cid:13)(cid:13)(cid:13) R k,ℓ f I (cid:13)(cid:13)(cid:13) pp (cid:17) /p . The function and operator R k,ℓ f I now satisfy the conditions of Proposition 5.2. We claimthat for each 0 ≤ j ≤ j ∗ , kR k,ℓ f k p . C ( ε ) j ℓε/ ( p ′ ) ( δ /δ j ) / − /p + ε (cid:16) X I j ∈I J,j kR k,ℓ f I j k pp (cid:17) /p (5.35) + j ℓ C ( ε ) j − k k f k p . The case j = 0 follows immediately from (5.34). Assume (5.35) holds for some j . Thenby applying Proposition 5.2 we get (cid:16) X I j ∈I J,j kR k,ℓ f I j k pp (cid:17) /p ≤ (cid:16) X I j ∈I J,j h C ( ε ) (cid:0) δ j δ j +1 (cid:1) / − /p + ε (cid:16) X I j +1 ∈ I Ij kR k,ℓ f I j +1 k pp (cid:17) /p + C ( ε )2 − k k f k pp (cid:17) /p i p (cid:17) /p (5.36) ≤ C ( ε ) (cid:0) δ j δ j +1 (cid:1) / − /p + ε (cid:16) X I j +1 ∈I J,j +1 kR k,ℓ f I j +1 k pp (cid:17) /p + C ( ε ) δ − /pj − k k f k p . Plugging the above estimate into (5.35) gives us kR k,ℓ f k p ≤ C ( ε ) j +1 ℓε/ ( p ′ ) (cid:0) δ δ j +1 (cid:1) / − /p + ε (cid:16) X I j +1 ∈I J,j +1 kR k,ℓ f I j +1 k pp (cid:17) /p + C ( ε ) j ℓε/ ( p ′ ) (cid:0) δ δ j (cid:1) / − /p + ε C ( ε ) δ − /pj − k k f k p + j ℓ C ( ε ) j − k k f k p . Using the fact that δ = 2 − ℓε , δ j ≥ ℓ (1 − ε/ for j ≤ j ∗ , and 2 ≤ p ≤ 6, the last two termsof the above inequality are bounded by( j + 1) C ( ε ) j +1 ℓ − k k f k p , proving the claim.We apply (5.35) for j = j ∗ and use the fact that j ∗ ≤ /ε as well as the assertion εp ′ − ε εp − ε − ε ε p + ε ≤ ε o deduce kR k,ℓ f k p ≤ C ( ε ) /ε ℓε/p ′ − ℓε (1 / − /p + ε ) ℓ (1 − ε/ / − /p + ε ) (5.37) × (cid:16) X I j ∗ ∈I J,j ∗ kR k,ℓ f I j ∗ k pp (cid:17) /p + ε C ( ε ) /ε − k +2 ℓ k f k p . ε ℓ (1 / − /p +2 ε ) (cid:16) X I j ∗ ∈I J,j ∗ kR k,ℓ f I j ∗ k pp (cid:17) /p + C ( ε )2 − k k f k p . Picking ε ′ = 2 ε completes the proof.6. L Estimates We prove L estimates for a general class of oscillatory integral operators associated tofold blowdown singularities in d dimensions, given by A k f ( x ) = Z e i k φ ( x,y ) f ( y ) σ ( x, y ) dy, where x, y ∈ R d , φ ∈ C ∞ ( R d × R d ), and σ ∈ C ∞ c ( R d × R d ). The canonical relationassociated to this oscillatory integral operator is given by { ( x, φ x ) × ( y, φ y ) : x ∈ R d , y ∈ R d } We write the projections π L : ( x, y ) ( x, φ x ) and π R : ( x, y ) ( y, φ y ). The projectionsare degenerate on the variety L where the determinant of the mixed Hessian of φ vanishes.Let h ( x, y ) = det φ xy . We assume that π L is a fold and π R is a blowdown on L . We maychoose the support of σ small enough and choose coordinates x = ( x ′ , x d ), y = ( y ′ , y d ) in R d − × R vanishing at a reference point P ◦ = ( x ◦ , y ◦ ) so that φ x ′ y ′ ( P ◦ ) = I d − , φ x d y ′ ( P ◦ ) = 0 , φ x ′ y d ( P ◦ ) = 0 , φ x d y d ( P ◦ ) = 0 . Let φ x ′ y ′ = φ − x ′ y ′ , and we can define the kernel fields V R = ∂ x d − φ x d y ′ ( φ x ′ y ′ ) ⊺ ∂ x ′ V L = ∂ y d − φ x ′ y d φ x ′ y ′ ∂ y ′ . By the assumption on π L , in other words that φ xy has corank at most 1 and h ( x, y ) = 0 = ⇒ | V L h ( x, y ) | ≥ c L > . Since we assume that V R is a blowdown, i.e. that V R is tangent to the singularity surface L , we see that h ( x, y ) = 0 = ⇒ | V jR h ( x, y ) | = 0 ∀ j ≥ . Assuming small enough support of σ we may assume that for ( x, y ) ∈ supp( σ )max {| φ x ′ y d ( x, y ) | , | φ x d y ′ ( x, y ) |} < ε. Note that this implies | ( V L − ∂ y d ) h ( x, y ) | ≤ ε k φ k C . e decompose first in distance to the singularity surface L ; for ℓ ≤ k , A k,ℓ f ( x ) := Z e i k φ ( x,y ) f ( y ) σ ( x, y ) χ (2 ℓ h ( x, y )) dy. Theorem 6.1. For ℓ ≤ k and ε > kA k,ℓ f k ≤ C ε ( ℓ − dk ) / ℓε k f k . To prove Proposition 4.2 for R k,ℓ we apply a partial Fourier transform (in the y ′ vari-ables) then change variables 2 k µ = τ , which satisfies the conditions for Theorem 6.1 with d = 3.6.1. Proof of Theorem 6.1. For 2 ℓ ≤ k ≤ (2 + ε ) ℓ we use the estimate from [9], that kA k,ℓ f k . k − dk k f k . Since k ≤ (2 + ε ) ℓ , we see k ≤ ℓ + ε ℓ , hence kA k,ℓ f k . ( ℓ − dk ) / ℓε k f k , proving Theorem 6.1 for ℓ sufficiently close to k/ ℓ such that ℓ (2 + ε ) < k we decompose our operator further and use methods ofthe proof of the Calderon-Vaillancourt theorem, following the ideas of Comech in [4]. Wedecompose our operator along small boxes in y -space, by way of cutoffs χ ~m ( y ) = d Y j =1 χ (2 ℓ y j − m j )) . We fix k, ℓ for now and let A m := A k,ℓ [ χ m ( y ) · ]. Then A m A ∗ ˜ m has Schwartz kernel K AA ∗ m, ˜ m ( x, w ) = Z e i k ( φ ( x,y ) − φ ( w,y )) σ ( x, w, y, k, ℓ ) dy, where the amplitude is given by σ ( x, w, y ) = χ (2 ℓ h ( x, y )) | χ m ( y ) | χ (2 ℓ h ( w, y )) . Similarly, the Schwartz kernel for A ∗ m A ˜ m is given by K A ∗ A m, ˜ m ( y, z ) = Z e i k ( φ ( x,y ) − φ ( x,z )) ˜ σ ( x, y, z ) dx, where ˜ σ ( x, y, z ) = χ (2 ℓ h ( x, y )) χ m ( y ) χ (2 ℓ h ( x, z ) χ ˜ m ( z ) . By splitting our operator A k,ℓ into a finite number of collections of {A m } we may assumethat if m j = ˜ m j then | m j − ˜ m j | > max { c L , √ d } .We first prove the following lemmas. Lemma 6.1. There exists a constant C > such that kA m k L → L ≤ C ℓ − dk Lemma 6.2. For any N > , and ℓ (1 + ε ) < k the following estimates hold. a) If m = ˜ m then kA m A ∗ ˜ m k = 0 . (b) If m = ˜ m and | m ′ − ˜ m ′ | ≤ c L k φ k C | m d − ˜ m d | then kA ∗ m A ˜ m k = 0 . (c) If m = ˜ m and | m ′ − ˜ m ′ | ≥ c L k φ k C | m d − ˜ m d | then kA ∗ m A ˜ m k . N ℓ − dk (cid:0) k − ℓ | m − ˜ m | (cid:1) − N A few remarks. First, the estimates in Lemma 6.2 do not rely on the blowdown as-sumption and essentially reproves the results of Comech in [4], albeit through a slightlydifferent approach. Second, the separation of ℓ from k/ Proof of Lemma 6.1. Since | φ x ′ y ′ | > c > ∇ y ′ ( φ ( x, y ) − φ ( w, y )) = 0is solved uniquely by x ′ = x ′ ( w, x d , y ). By the implicit function theorem we can see that14 | x ′ − x ′ ( w, x d , y ) | ≤ | φ y ′ ( x, y ) − φ y ′ ( w, y ) | ≤ | x ′ − x ′ ( w, x d , y ) | . A further set of calculations reveal that φ y d ( x ′ ( w, x d , y ) , x d , y ) − φ y d ( w, y ) = N X j =0 V jR [det φ xy (det φ x ′ y ′ ) − ]( w, y ) ( x d − w d ) j +1 ( j + 1)!+ b ( w, y )( x d − w d ) N +2 . Since π L is a fold and V L (cid:12)(cid:12) (0 , = ∂ y d , we see that h ( x, y ) = 0 is solved uniquely by y d = y d ( x, y ′ ) near 0. From this,14 | y d − y d ( x, y ′ ) | ≤ | h ( x, y ) | ≤ | y d − y d ( x, y ′ ) | . Because π R is a blowdown and the bounds on h we see that | V jR h ( x, y ) | = | V jR h ( x, y ′ , y d ( x, y ′ )) + ( y d − y d ( x, y ′ )) ∂ y d V jR h ( x, y ′ , z d ) | ≤ C − ℓ implying by the properties of differentiation of products | φ y d ( x ′ ( w, x d , y ) , x d , y ) − φ y d ( w, y ) | ≥ c − ℓ | x d − w d | . Thus | φ y d ( x, y ) − φ y d ( w, y ) | ≥ | φ y d ( x, y ) − φ y d ( x ′ ( w, x d , y ) , x d , y ) |− | φ y d ( w, y ) − φ y d ( x ′ ( w, x d , y ) , x d , y ) | , and therefore, |∇ y ( φ ( x, y ) − φ ( w, y ) | ≥ C max { − ℓ | x d − w d | , | x ′ − x ′ ( w, x d , y ) |} . With these estimates in place we integrate by parts in the y variables, noting that for amultiindex α | D αy σ | ≤ C | α | ℓ | α | and for | α | > | D αy φ | ≤ C | α | | x − w | . hus we get the estimate | K AA ∗ ( x, w ) | . N Z k − ℓ | x ′ − x ′ ( w, x d , y ) | ) N k − ℓ | x d − w d | ) N dy. Integrating in x we see that Z | K AA ∗ ( x, w ) | dx . N Z k − ℓ | x d − w d | ) N dx d × sup x d ,y − dℓ Z k − ℓ | x ′ − x ′ ( w, x d , y ) | ) N dx ′ . ℓ − k − dℓ ( d − ℓ − k ) . ℓ − dk . (cid:3) Proof of Lemma 6.2. First, (a) follows immediately since it implies χ (2 ℓ y − m ) χ (2 ℓ y − ˜ m ) = 0 . The kernel K A ∗ A m, ˜ m vanishes under the assumption in (b) because π L is a fold, meaning that V L det dπ L is bounded away from 0 on L . Since | det φ ( x, y ) | and | det φ ( x, z ) | are bothbounded above by 2 − ℓ +1 , their sum is bounded by 2 − ℓ +3 . Expanding the difference about y = z we seedet φ xy ( x, y ) − det φ xy ( x, z ) = ( y d − z d ) ∂ y d det φ xy ( x, z )+ ( y ′ − z ′ ) · ∇ y ′ det φ xy ( x, z ) + O ( | y − z | )= ( y d − z d )[ ∂ y d − V L ] det φ xy ( x, z )+ ( y d − z d ) V L det φ xy ( x, z )+ ( y ′ − z ′ ) · ∇ y ′ det φ xy ( x, z ) + O ( | y − z | ) | det φ xy ( x, y ) − det φ xy ( x, z ) | ≥ c L | y d − z d |≥ − ℓ ) . Thus we see there are no y, z that satisfy these conditions, hence a k,ℓ, ± ( x, y ) a k,ℓ, ± ( x, z ) = 0 . To prove (c) we split into two cases: first, assume that k ≥ (2 + ε ) ℓ . Then we use thefollowing Taylor approximation of the derivative of the phase of K A ∗ A m, ˜ m .(6.1) ∇ x ′ [ φ ( x, y ) − φ ( x, z )] = φ x ′ y d ( x, z )( y d − z d ) + φ x ′ y ′ · ( y ′ − z ′ ) + O ( | y − z | ) . We know that | φ x ′ y ′ ( x, z ) · ( y ′ − z ′ ) | ≥ C d | y ′ − z ′ | , and | φ x ′ y d ( x, z )( y d − z d ) | ≤ ε | y d − z d | .By assumption | y ′ − z ′ | ≥ c L k φ k C | y d − z d | ≥ εC d | y d − z d | . Thus |∇ x ′ [ φ ( x, y ) − φ ( x, z )] | ≥ c | y − z | or some small constant c > 0. Define the operator M x ′ = i k ∇ x ′ [ φ ( x, y ) − φ ( x, z )] |∇ x ′ [ φ ( x, y ) − φ ( x, z )] | · ∇ x ′ . We apply M x ′ many times to K A ∗ A m, ˜ m , and by our lemma | K A ∗ A m, ˜ m ( y, z ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z e i k ( φ ( x,y ) − φ ( x,z )) σ dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z e i k ( ϕ Q ( x,y ) − ϕ Q ( x,z )) (cid:0) M ∗ x ′ (cid:1) N σ dx (cid:12)(cid:12)(cid:12) . N Z k − ℓ | y − z | ) N | ˜ σ | dx . N k − ℓ | y − z | ) N χ m ( y ) χ ˜ m ( z ) . Since | y − z | > − ℓ , k ≥ ℓ , and | y − z | ≃ − ℓ | m − ˜ m | ,12 k − ℓ | y − z | ≤ min n C k − ℓ | y − z | , C k − ℓ | m − ˜ m | o Integrating in y (or z ) Z | K A ∗ A m, ˜ m ( y, z ) | dy ≤ C N,d Z k − ℓ | y − z | ) d +1 × k − ℓ | m − ˜ m | ) N χ m ( y ) χ ˜ m ( z ) dy ≤ C N,d d ( ℓ − k ) (2 k − ℓ | m − ˜ m | ) − N . Since k − ℓ ≥ ℓε , if we let N = d/ε then by Schur’s Lemma kA ∗ m A ˜ m k → ≤ C ( ε, d )2 ( ℓ − dk ) | m − ˜ m | − N , proving part (c) of Lemma 6.2. (cid:3) L p -Sobolev Estimate As in [2], we prove Theorem 1.1 by applying a special case of Theorem 1.1 from [17]and a Littlewood-Paley bound adapted from [22].Let R ℓ = X k ≥ ℓ R k,ℓ . We will prove for compactly supported f kR ℓ f k F p,q /p ≤ − ℓε ( p ) k f k B p,p , < q ≤ < < p < ∞ , where F p,qs and B p,qs are respectively the Triebel-Lizorkin space and Besov spaces (see[23]). Summing in ℓ with q ≥ R : B p,ps,comp → F p,qs +1 /p , q ≤ < < p < ∞ . ince L ps = F p, s ֒ −→ B p,ps for p > F p,qs +1 /p ֒ −→ F p, s +1 /p = L ps +1 /p for q ≤ 2, this impliesthe asserted L p -Sobolev bounds for R .Let P k be standard Littlewood-Paley multipliers on R for k ∈ N and ˜Φ j ( x, y ) = S j − y j for j = 1 , . Because ∇ x ˜Φ j ( x, y ) are linearly independent, as are ∇ y ˜Φ j ( x, y ), we can find C > C − | τ | ≤ | ( τ · ˜Φ) x | , | ( τ · ˜Φ) y | ≤ C / | τ | This implies the following. Lemma 7.1. Suppose k ′ , k ′′ ∈ N , k ′ ≥ ℓ and max {| k − k ′ | , | k − k ′′ |} ≥ C , where C depends on C . Then k P k R k ′ ,ℓ P k ′′ k L p → L p ≤ C min { − kN , − k ′ N , − k ′′ N } . Proof of Lemma 7.1. This integration by parts argument is essentially due to H¨ormander[12], based on the fact that the canonical relation stays away from zero sections (cf.Lemma 2.1 in [22]). Note that the kernel of the operator P k R k ′ ,ℓ P k ′′ is given by Z Z Z Z Z e i [ h x − w,η i + τ · ˜Φ( w,z )+ h z − y,ξ i ] χ (2 − k | η | ) χ (2 − k ′ | τ | ) χ (2 − k ′′ | ξ | ) × a k,ℓ, ± ( z , τ ) χ ( | w | ) χ ( | x | ) dw dz dτ dη dξ. Our assumption on Φ implies that if max {| k − k ′ | , | k ′ − k ′′ |} > C we have ∇ ( z,w ) h h x − w, η i + τ · ˜Φ( w, z ) + h z − y, ξ i i ≥ c max { k , k ′ , k ′′ } . Thus we integrate by parts in the ( w, z ) variables to get the above bound on the kernel,implying by Minkowski the desired bound on L p . (cid:3) Using the lemma above and an argument similar to a part of the proof of Lemma 2.1in [22], we can reduce the proof of Theorem 1.1 to the estimate(7.1) (cid:13)(cid:13)(cid:13)(cid:16) X k ≥ ℓ (cid:12)(cid:12) k/p P k R k + s ,ℓ P k + s f (cid:12)(cid:12) q (cid:17) /q (cid:13)(cid:13)(cid:13) L p ≤ C − ℓε ( p ) (cid:13)(cid:13)(cid:13)(cid:16) X k> | P k + s f | p (cid:17) /p (cid:13)(cid:13)(cid:13) L p . To prove (7.1) we apply the main result from [17]. Theorem 7.1 ([17]) . Let T k be a family of operators on Schwartz functions by T k f ( x ) = Z K k ( x, y ) f ( y ) dy. Let φ ∈ S ( R ) , φ k = 2 k φ (2 k · ) , and Π k f = φ k ∗ f . Let ε > and < p < p < ∞ .Assume T k satisfies sup k> k/p k T k k L p → L p ≤ A (7.2) sup k> k/p k T k k L p → L p ≤ B . (7.3) urther let A ≥ , and assume that for each cube Q there is a measurable set E Q suchthat (7.4) | E Q | ≤ A max {| Q | / , | Q |} , and for every k ∈ N and every cube Q with k diam( Q ) ≥ , (7.5) sup x ∈ Q Z R d \ E Q | K k ( x, y ) | dy ≤ B max n(cid:0) k diam( Q ) (cid:1) − ε , − kε o . Let B = B q/p ( AA /p + B ) − q/p . Then for any q > there is a C depending on ε, p, p , q such that (7.6) (cid:13)(cid:13)(cid:13)(cid:16) X k kq/p | P k T k f k | q (cid:17) /q (cid:13)(cid:13)(cid:13) p ≤ CA (cid:2) log (cid:0) B A (cid:1)(cid:3) /q − /p (cid:16) X k k f k k pp (cid:17) /p . We apply this theorem on the family of operators T k := R k,ℓ for k ≥ ℓ (here ℓ isfixed). By Proposition 4.1 the assumptions (7.2) and (7.3) are satisfied with A . − ℓε ( p ) and B . − ℓε ( p ) . We next check assumptions (7.4) and (7.5). For a given cube Q withcenter x Q let E Q = { y : | S ( x Q , y ) − y ′ | ≤ C ℓ diam( Q ) } if diam( Q ) < 1, and a cube centered at x Q of diameter C ℓ diam( Q ) if | Q | ≥ 1. By anintegration by parts argument we derive the bound | K k ( x, y ) | . N k (1 + 2 k − ℓ | S ( x Q , y ) − y ′ | ) N . Then clearly assumptions (7.4) and (7.5) are satisfied with A . ℓ and B . ℓ respec-tively. Theorem 7.1 then implies (7.1) with Π k = P k + s and f k = P k + s f , finishing theproof of Theorem 1.1. References [1] Theresa C Anderson, Laura Cladek, Malabika Pramanik, and Andreas Seeger. Spherical means onthe Heisenberg group: Stability of a maximal function estimate. arXiv preprint arXiv:1801.06981,To appear in J. Anal. Math. , 2018.[2] Geoffrey Bentsen. 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