Mixed norm Strichartz-type estimates for hypersurfaces in three dimensions

In their work [IM16] I.A. Ikromov and D. M\"{u}ller proved the full range $L^p-L^2$ Fourier restriction estimates for a very general class of hypersurfaces in $\R^3$ which includes the class of real analytic hypersurfaces. In this article we partly extend their results to the mixed norm case where the coordinates are split in two directions, one tangential and the other normal to the surface at a fixed given point. In particular, we resolve completely the adapted case and partly the non-adapted case. In the non-adapted case the case when the linear height $h_\text{lin}(\phi)$ is below two is settled completely.


Introduction
For a given smooth hypersurface S in R n , its surface measure dσ, and a smooth compactly supported function ρ ě 0, ρ P C 8 0 pSq, the associated Fourier restriction problem asks for which p, q P r1, 8s the estimate˜ż holds true. This problem was first considered by E.M. Stein in the late 1960s. Soon thereafter the problem was essentially solved for curves in two dimensions, see [9], [5], [37]. The higher dimensional case in its most general form is still wide open. The three dimensional case, as of yet, is far from being completely understood even when S is the sphere, and there has been a lot of deep work in the direction of understanding L p´Lq estimates for surfaces with both vanishing and non-vanishing Gaussian curvature. A small sample of such works are [3], [35], [26], [36], [32], [23], [4], [14]. The case when q " 2 has proven to be more tractable since one can use the "R˚R technique". This was exploited by P.A. Tomas and E.M. Stein (see [33]) to obtain the full range of L p´L2 estimates when the hypersurface in question is the unit sphere, and later further developed by A. Greenleaf in [13] where the full range of L p´L2 estimates was obtained for surfaces with nonvanishing Gaussian curvature. In fact, Greenleaf proved that if one has a decay estimate on the Fourier transform of ρdσ (which can be interpreted as a uniform estimate for an oscillatory integral), i.e., | y ρdσpξq| À p1`|ξ|q´1 {h , ξ P R n , then the associated restriction estimate holds true for p 1 ě 2ph`1q and q " 2. However, in general, this range is not optimal. Recently I.A. Ikromov and D. Müller in their series of works (see [16], [17], [18], and also their work with M. Kempe [15]) have developed techniques for proving the full range of L p´L2 estimates for a very general class of surfaces. Their work builds upon the work of V.I. Arnold and his school (in particular, the work by Varchenko [34]) which highlighted the importance of the Newton polyhedron within problems involving oscillatory integrals, and upon the work of D.H. Phong and E.M. Stein [27] and D.H. Phong, E.M. Stein, and J.A. Sturm [28] in the real analytic case where the authors in addition to the Newton polyhedron used the Puiseux series expansions of roots to obtain results on oscillatory integral operators. For further and more detailed references we refer the reader to [18]. In [18] Ikromov and Müller proved the following theorem.
Theorem 1.1. Let S be a smooth hypersurface in R 3 and dσ its surface measure. After localisation and a change of coordinates assume that S is given as the graph of a smooth function φ : Ω Ñ R of finite type with φp0q " 0 and ∇φp0q " 0, where Ω Ď R 2 is an open neighbourhood of 0. Furthermore, assume that φ is linearly adapted in its original coordinates. Let ρ ě 0, ρ P C 8 c pSq, be a smooth compactly supported function. Then the estimate (1.1) holds true for all ρ with support contained in a sufficiently small neighbourhood of 0 when q " 2 and when either (a) φ is adapted in its original coordinates and p ě 2phpφq`1q, or (b) φ is not adapted in its original coordinates, satisfies the Condition (R), and p ě 2ph res pφq`1q.
Since linear transformations respect the Fourier transform, one can always assume linear adaptedness. The quantities hpφq and h res pφq are respectively the height and the restriction height of the function φ (the precise definitions can be found in Subsections 1.1 and 2.3 below respectively; also note that we use h res pφq to denote the restriction height of the function φ instead of h r pφq as in [18]). Condition (R) is a factorisation condition which is true for real analytic functions, but not for general smooth functions, and it remains open whether this condition can be removed in the above theorem.
In this paper we shall be interested in the mixed norm case with L p pR 3 q denoting from now on the space L p 3 x 3 pL p 2 x 2 pL p 1 x 1 qq and q " 2 in (1.1). We shall be interested in the particular case when p 1 " p 2 , i.e., we only differentiate between the tangential and the normal direction to the surface S at the point 0 P S. This means we take }f } L p pR 3 q to meañ ż˜ż ż |f | p 1 px 1 , x 2 , x 3 qdx 1 dx 2¸p Henceforth we shall denote by p the pair pp 1 , p 3 q. Our task is to determine for which pp 1 , p 3 q the inequality˜ż holds true for ρ ě 0 supported in a sufficiently small neighbourhood of 0. This question is of great interest in the theory of PDEs, as was noticed by Strichartz in [31]. Namely, one can obtain mixed norm Strichartz estimates for a wide collection of symbols φ determining the surface S since the estimate (1.2) can be reinterpreted as an a priori estimate }u} L p px,tq pR 3 q ď C}g} L 2 pR 2 q for the Cauchy problem # B t upx, tq " iφpDqupx, tq, px, tq P R 2ˆR , upx, 0q " gpxq, where g has its Fourier transform supported in a small neighbourhood of the origin and φpDq is the operator with symbol φpξq.
It turns out that we can use the same basic techniques and phase space decompositions as in [18] in proving the estimate (1.2) in the cases we consider (namely, the adapted case and the non-adapted case with h lin pφq ă 2). The main additional ingredients we shall use are some basic ideas from [11] (see also [22]) for handling mixed norms. In our case additional complications appear which were absent in the corresponding cases in [18] and some of which resemble problems appearing in some of the final chapters of [18]. For example, after making a phase space decomposition of the kernel of the convolution operator obtained by the "R˚R technique", a recurring theme will be that we will not be able to sum absolutely the operators associated to the kernel decomposition pieces whose operators were absolutely summable [18]. A further interesting feature of the mixed norm case is that estimates for the mixed norm endpoint for operators of certain kernel pieces become invariant under scalings considered in [18].
The structure of this article is as follows. In the following Subsection 1.1 we review some fundamental concepts such as the Newton polyhedron and adapted coordinates. In Subsection 1.2 we state the main results of this paper, namely Theorem 1.2 which states the necessary conditions, and Theorem 1.3 which gives us the mixed norm Fourier restriction estimates in the adapted case and the case h lin pφq ă 2. In Section 2 we derive the necessary conditions (by means of Knapp-type examples) for the exponents in (1.2). See Proposition 2.1. In Subsection 2.4 we also determine explicitly the Newton polyhedra of φ in its original and adapted coordinates in the case when the linear height of φ is strictly less than 2. Section 3 contains auxiliary results that we shall often refer to. In Subsection 3.2 we list results related to oscillatory integrals, such as the van der Corput lemma, and also some results on oscillatory sums from [18] that are useful in conjunction with complex interpolation. In Subsection 3.3 we state results which we need for handling mixed norms. In Section 4, Proposition 4.2, we deal with the adapted case, i.e., we prove that if φ is adapted in its original coordinates, then the estimate (1.2) holds for all p's determined by the necessary conditions, except occasionally for a certain endpoint. In the same section (see Proposition 4.3) we also reduce the general non-adapted case to considering the part near the principal root jet of φ. In Sections 5 and 6 we handle the case when the linear height of φ is strictly less than 2 for a class of functions φ which includes all analytic functions (see Theorem 5.1 for a precise formulation).
For reasons of consistency we use the same notational conventions as in [18]. We use the "variable constant" notation meaning that constants appearing in calculations and in the course of our arguments may have different values on different lines. Furthermore we use the symbols ", À, Á, !, " in order to avoid writing down constants. If we have two nonnegative quantities A and B, then by A ! B we mean that there is a sufficiently small positive constant c such that A ď cB, by A À B we mean that there is a (possibly large) positive constant C such that A ď CB, and by A " B we mean that there are positive constants C 1 ď C 2 such that C 1 A ď B ď C 2 A. One defines analogously A " B and A Á B. Often the constants c and C shall depend on certain parameters p in which case we occasionally write A ! p B, A À p B, etc., in order to emphasize this dependence.
A further notational convention adopted from [18] is the use of symbols χ 0 and χ 1 in denoting certain nonnegative smooth compactly supported functions on R. Namely, we require χ 0 to be supported in a neighbourhood of the origin and identically 1 near the origin, and χ 1 to be supported away from the origin and identically 1 on some open neighbourhood of 1 P R. These cutoff functions χ 0 and χ 1 may vary from line to line, and sometimes, when several χ 0 and χ 1 appear within the same formula, they may even designate different functions.
Acknowledgement. I would like to thank my supervisor Prof. Dr. Detlef Müller for numerous useful discussions we had and for his valuable comments on how to improve this paper.

Fundamental concepts and basic assumptions
Let the surface S be given as the graph S " S φ :" tpx 1 , x 2 , φpx 1 , x 2 qq : x " px 1 , x 2 q P Ω Ă R 2 u of a smooth and real-valued function φ defined on an open neighbourhood Ω of the origin. We can assume without loss of generality that φp0q " 0 and we take Ω to be a sufficiently small neighbourhood of the origin in R 2 . In the mixed norm case we cannot use the rotational invariance of the Fourier transform in order to reduce to the case ∇φp0q " 0. Instead we use a different linear transformation (for details see Subsection 3.1), and so we may and shall assume ∇φp0q " 0.
Next, we impose on φ to be a function of finite type at 0. This means that there exists a multiindex α P N 2 0 such that B α φp0q ‰ 0. By continuity, φ is of finite type on a neighbourhood of 0. We may therefore assume that φ is of finite type in each point of Ω. We define the Taylor support of φ as the set The Newton polyhedron N pφq of φ is the convex hull of the set where the union is over all α such that B α φp0q ‰ 0 (and so |α| ě 2). See Figure 1. Both edges and vertices are called faces of N pφq. We define the Newton diagram N d pφq of φ to be the union of all compact faces of N pφq.
If we are given a face e 0 of N pφq, we can define its associated (formal) series φ e 0 px 1 , x 2 q :" If e 0 is a compact face, then φ e 0 px 1 , x 2 q is a mixed homogeneous polynomial. This means that there exists a weight κ e 0 " pκ e 0 1 , κ e 0 2 q P r0, 8q 2 such that for any r ą 0 we have φ e 0 pr κ e 0 1 x 1 , r κ e 0 2 x 2 q " rφ e 0 px 1 , x 2 q, and we call φ e 0 a κ e 0 -homogeneous polynomial. κ e 0 is uniquely determined if and only if e 0 is not a vertex. In fact, in the case when e 0 is an edge, we define L κ e 0 to be the unique line containing e 0 : Then the weight κ e 0 is uniquely determined by the relation L κ e 0 " ! pt 1 , t 2 q P R 2 : κ e 0 1 t 1`κ e 0 2 t 2 " 1 ) . (1.5) When e 0 is an unbounded face, φ e 0 px 1 , x 2 q is to be taken only as a formal power series. Note that then e 0 is either a vertical or horizontal edge of N pφq, and we can also find unique κ e 0 1 and κ e 0 2 (one of them being 0 in this case) such that (1.4) holds.
Of particular interest is the principal face πpφq defined as the face of minimal dimension of N pφq which intersects the bisectrix tpt 1 , t 2 q P R 2 : t 1 " t 2 u. Its associated series (or homogeneous polynomial) we call the principal part of φ and denote by φ pr :" φ πpφq . Let κ " pκ 1 , κ 2 q determine the line L κ as in (1.5) containing the principal face if it is an edge, or when it is a vertex, let κ determine the edge of N pφq having the principal face as its left endpoint. Interchanging the x 1 and x 2 coordinates, if necessary, we may always assume that We shall denote the ratio κ 2 {κ 1 by m, and so m ě 1.
The Newton distance dpφq of φ is defined to be the coordinate d of the point pd, dq which is the intersection of the bisectrix and the principal face of N pφq. One can easily see that if κ " pκ 1 , κ 2 q determines the line containing the principal face (or any of the supporting lines to N pφq in case πpφq " tpd, dqu), then we have The Newton height hpφq of φ is defined as hpφq " suptdpφ˝ϕq : ϕ a smooth local coordinate changeu.
By a smooth local coordinate change we mean a function ϕ which is smooth and invertible in a neighbourhood of the origin, and ϕp0q " 0. We also define the linear height as h lin pφq " suptdpφ˝ϕq : ϕ a linear coordinate changeu.
For a coordinate change ϕ we shall denote the new cooridnates by y " ϕpxq. In this case we also denote d y " dpφ˝ϕq. We say that φ is adapted in the y coordinates if d y " hpφq. Analogously, we say that φ is linearly adapted in coordinates y if d y " h lin pφq. When φ is adapted in its original coordinates x we say that φ is adapted, and if φ is not adapted in its original coordinates, then we say that φ is non-adapted. Analogous expressions we shall use for linear adaptedness. We obviously always have d x " dpφq ď h lin pφq ď hpφq.
The existence of an adapted coordinate system for real analytic functions on R 2 was first proven by Varchenko in [34]. He gave an explicit algorithm on how to construct an adapted coordinate system. His result was generalised in [16] where it was shown that an adapted coordinate system exists for general smooth functions. It turns out that in the smooth case one can also essentially use Varchenko's algorithm. In this article when we refer to Varchenko's algorithm we shall always mean the variant used in [16]. In this variant one constructs an adapted coordinate system in the form of a non-linear shear transformation y 1 " x 1 , y 2 " x 2´ψ px 1 q.
The smooth real-valued function ψ can be taken in the real-analytic case to be the principal root jet of φ as defined in [18]. We denote the function φ in the new (adapted) coordinates by φ a . Then we have φ a pyq " φpy 1 , y 2`ψ py 1 qq.
We remark that when φ is not adapted, then m " κ 2 {κ 1 is a positive integer and ψpx 1 q´b 1 x m 1 " Opx m`1 1 q for some nonzero real constant b 1 . We introduce next Varchenko's exponent νpφq P t0, 1u. If hpφq ě 2 and there exists an adapted coordinate system y such that in these coordinates the principal face of φ a pyq is a vertex, we define νpφq :" 1. In all other cases we take νpφq :" 0. In particular νpφq " 0 whenever hpφq ă 2. A concrete characterisation for determining when an adapted coordinate system having the principal face as a vertex exists can be found in [17,Lemma 1.5].
Let us discuss next linear adaptedness. We assume that h lin pφq ă hpφq, i.e., that we cannot achieve adapted coordinates with a linear coordinate change. In [18,Section 1.3] it was shown that in this case we can always find a linearly adapted coordinate system, and [18,Proposition 1.7] gives an explicit characterisation of when a coordinate system is linearly adapted. It was shown in particular that if the coordinate system x is not already linearly adapted, then one just needs to apply the first step of Varchenko's algorithm in order to obtain it.
Since in our mixed norm case we consider only p 1 " p 2 , we can freely use linear coordinate changes in "tangential" variables px 1 , x 2 q in the expression (1.2). Thus we may assume without loss of generality that either the original coordinate system x is already adapted, or that it is at least linearly adapted. In particular, we may assume dpφq " h lin pφq.
The final important concept we introduce is the augmented Newton polyhedron N res pφ a q of a non-adapted φ (note the slight change in notation compared to [18], where N r pφ a q is used instead). N res pφ a q is defined as the convex hull of the set where L`is defined as follows. Let L κ be the line containing the principal face πpφq of N pφq and let P " pt P 1 , t P 2 q be the point on L κ X N pφ a q with the smallest t 2 coordinate. Such a point always exists. Then L`is the ray ) .

The main results
Let us briefly review all the conditions on the function φ which we may assume without loss of generality when considering the mixed norm restriction problem: • φp0q " 0 and ∇φp0q " 0, • φ is of finite type on Ω, • the weight κ determined by the principal face of N pφq (or by the edge containing the principal face as its left endpoint) satisfies m " κ 2 {κ 1 ě 1, and • the original coordinate system x is either adapted, or linearly adapted but not adapted. In both cases we have dpφq " h lin pφq.
Recall that S denotes the surface given as the graph of φ and dσ its surface measure. We are considering the mixed norm Fourier restriction problem (1.2) when ρ is supported in a sufficiently small neighbourhood of the origin. Figure 1: The (augmented) Newton polyhedron associated to φ a .
We begin by stating necessary conditions which will be obtained by means of Knapp-type examples. When φ is not adapted we denote by K : ro, κ 1 s Ñ r0,`8s the function defined in the following way. Consider all lines of the form whereκ P r0, 8q 2 is a weight. For each 0 ďκ 1 ď κ 1 there is a uniqueκ 2 so that (1.6) determines a supporting line Lκ to N res pφ a q. We then define Kpκ 1 q to beκ 2 forκ 1 P r0, κ 1 s (see Figure 2). Note that then the weight p0, Kp0qq determines line containing the horizontal edge of the augmented Newton polyhedron, i.e., the right most edge of N res pφ a q. The weight pκ 1 , Kpκ 1 qq " κ determines the line containing the edge associated to the principal face of N pφq which is the left most edge of N res pφ a q. Denote by L the Legendre transformation for a real-valued convex function K: Then we may state the necessary conditions in the following way: Theorem 1.2. Let φ be as above and let us assume that the estimate (1.2) holds true with ρp0q ‰ 0.
If φ is adapted, then we have the necessary condition .
If K is as above and φ is linearly adapted, but not adapted, then we necessarily have Recall that dpφq " hpφq when φ is adapted. The above theorem is a direct consequence of Proposition 2.1 in Section 2 below and the discussion in Subsection 2.2. The necessary conditions are depicted in Figure 3.
The main result of this paper is: 3. Let φ be as above and ρ supported in a sufficiently small neighbourhood of 0. If either (a) φ is adapted in its original coordinates, or (b) φ is non-adapted, h lin pφq ă 2, and φ is real analytic, then the estimate (1.2) holds true for all p1{p 1 1 , 1{p 1 3 q as determined by Theorem 1.2, except for the point p1{p 1 1 , 1{p 1 3 q " p0, 1{p2hpφqqq where it is false if ρp0q ‰ 0 and either hpφq " 1 or νpφq " 1.
In case (b) we shall actually prove the claim for a more general class of functions than is stated here.
The part (a) of the above theorem follows from Proposition 4.2, and the part (b) follows from Theorem 5.1 Let us mention that in the case h lin pφq ă 2 it turns out that we always have νpφq " 0, which will be important for the boundary point p1{p 1 1 , 1{p 1 3 q " p0, 1{p2hpφqqq. In this article we do not deal with the non-adapted case when h lin pφq ě 2 in its full generality. Let us briefly comment how one can easily get some preliminary Fourier restriction estimates. Namely, the abstract result from [22] by Keel and Tao implies that we automatically have the Fourier restriction estimate for the region labeled by KT in Figure 3 below. For details we refer to Proposition 4.1.
One can combine this result with the case p 1 " p 3 from Theorem 1.1 and get by interpolation the region labeled by IM in Figure 3.

Necessary conditions
In this section our assumptions on φ are as explained in Subsection 1.2. Our goal is to find a complete set of necessary conditions on p " pp 1 , p 3 q P r1, 8s 2 for (1.2) to hold true whenever ρp0q ‰ 0. We shall reframe the conditions in several ways: an "explicit" form in Subsection 2.1, a form as in Theorem 1.2 using the Legendre transformation of K in Subsection 2.2, and a form when we fix the ratio p 1 1 {p 1 3 in Subsection 2.3. In Subsection 2.4 we discuss the normal forms of φ when h lin pφq ă 2 and determine explicitly the necessary conditions in this case.

The explicit form
Let us first introduce some further notation. If φ is linearly adapted but not adapted, then the adapted coordinate system is obtained through where ψ is the principal root jet. The function φ is in the new coordinates y φ a py 1 , y 2 q :" φpy 1 , y 2`ψ py 1 qq, i.e., φ a represents the function φ in adapted coordinates. We denote the vertices of N pφ a q by pA l , B l q P N 2 0 , l " 0, 1, 2, . . . , n, where n ě 0 and we assume that the points are ordered from left to right, i.e., A l´1 ă A l for l " 1, 2, . . . , n. Next, we denote the compact edges of N pφ a q by γ l :" rpA l´1 , B l´1 q, pA l , B l qs, l " 1, 2, . . . , n, and also the unbounded edges by see Figure 1. Let us denote by L l , l " 0, . . . , n`1, the associated lines on which these edges lie. Each line L l is given by the equation where pκ l 1 , κ l 2 q P r0, 8q 2 is its associated weight. We also introduce the quantity which is related to the slope of L l , namely, its slope is then equal to´1{a l . We obviously have a 0 " 0 and a n`1 " 8.
Let us denote by 0 ă m ă 8 the leading exponent in the Taylor expansion of ψ. We define L κ to be the unique line κ 1 t 1`κ2 t 2 " 1 satisfying κ 2 " mκ 1 and which is a supporting line to the Newton polyhedron N pφ a q. This line coincides with the line containing the principal face of N pφq. This follows from Varchenko's algorithm. Next, let l 0 be such that Note that the point pA l 0´1 , B l 0´1 q is the right endpoint of the intersection of L κ and N pφ a q.
Varchenko's algorithm also shows that B l 0´1 ě A l 0´1 . We denote by l a the index such that κ l a is associated to the principal face of N pφ a q. If πpφ a q is a vertex, we take l a to be associated to the edge to the left of πpφ a q. Note l a ě l 0 .
We may now define the augmented Newton polyhedron N res pφ a q as the convex hull of the set where Lκ denotes the ray ) .
Before stating the necessary conditions analogous to [18,Proposition 1.16], let us recall that in the case of the principal face being a vertex, we take κ to determine the line containing the edge of N pφq which has πpφq as its left endpoint. Furthermore recall that m " κ 2 {κ 1 ě 1 and that φ is linearly adapted in its original coordinates. Proposition 2.1. Let φ be as above. Let ρ ě 0, ρ P C 8 0 pSq, be a smooth compactly supported function with ρp0q ‰ 0, and assume that the estimate (1.2) holds true. If φ is non-adapted, let us consider the nonlinear shear transformation y 1 :" x 1 , y 2 :" x 2´ψ px 1 q, and let φ a pyq :" φpy 1 , y 2`ψ py 1 qq be the function φ expressed in the adapted coordinates. Then it necessarily follows that for all weights pκ 1 ,κ 2 q such that Lκ is a supporting line to N res pφ a q we have This is equivalent to Furthermore, when φ is either adapted or non-adapted we have the conditions .
In particular when φ is non-adapted the first condition in (2.3) then coincides with the one in the second line of (2.2). Moreover in this case the conditions in (2.2) for l ą l a are redundant, and if we fix p 1 3 " 8 (resp. p 1 1 " 8) then all the conditions reduce to p 1 1 ě 2 (resp. p 1 3 ě 2hpφq).
Proof. We give only a sketch of the proof since it follows the same lines as in [18]. Let us consider any supporting line Lκ to the augmented Newton polyhedron N res pφ a q for some weight pκ 1 ,κ 2 q. This particularly implies by the definition of the augmented Newton diagram thatκ 2 ě mκ 1 . We first consider the case whenκ 1 ą 0, i.e., when the associated line Lκ is not horizontal. In this case for each sufficiently small ε ą 0 we define the region D a ε :" which in the original coordinate system has the form ) .
Using the φ ã κ part of the Taylor approximation of φ a one easily gets that for each y P D a ε we have |φ a pyq| ď Cε. Returning to the x coordinates we obtain |φpxq| ď Cε, x P D ε .
But for x P D ε one has |x 2 | ď εκ 2`| ψpx 1 q| À εκ 2`ε mκ 1 À ε mκ 1 , since |ψpx 1 q| À |x 1 | m andκ 2 ě mκ 1 . Therefore the region D ε is contained in the set where |x 1 | ď C 1 εκ 1 and |x 2 | ď C 2 ε mκ 1 . Thus we choose a Schwartz function ϕ ε which has its Fourier transform of the form for some smooth compactly supported function χ 0 which is identically 1 on the interval r´1, 1s. Then in particular we have x ϕ ε px 1 , x 2 , φpx 1 , x 2 qq ě 1 on D ε . Now on the one hand, since ρp0q ‰ 0, we havẽ and on the other Plugging these into (1.2) and letting ε Ñ 0 one obtains (2.1) for the non-horizontal edges. In the horizontal caseκ 1 " 0 one only slightly changes the argument. Namely, one defines for a sufficiently small δ ą 0 D a ε :" The associated set in the x coordinates D ε is then contained in the box determined by |x 1 | ď ε δ and |x 2 | ď ε mδ . Furthermore, using a Taylor series expansion, one can easily show that for x P D ε we have again |φpxq| ď Cε. Now one proceeds as in the non-horizontal case, the only difference is that after taking the limit ε Ñ 0, one also needs to take the limit δ Ñ 0.
Let us now briefly explain why (2.1) and (2.2) are equivalent. We obviously have that (2.1) implies (2.2). For the reverse implication we note that theκ's considered in (2.2) are by definition precisely those for which the lines Lκ contain the edges of the augmented Newton diagram. This means that all the other supporting lines touch the augmented Newton diagram at only one point. Now one just uses the fact that the associated weightκ of such a supporting line Lκ is obtained by a convex combination of weights associated to the edges which intersect at the point through which Lκ passes. Thus, all the conditions in (2.1) can be obtained as convex combinations of conditions in (2.2).
The proof of (2.3) is similar to the one for (2.1). One considers the set D ε defined by tx P R 2 : |x 1 | ď ε κ 1 , |x 2 | ď ε κ 2 u in the case when the principal face of N pφq is compact. If it is not compact, then one uses tx P R 2 : |x 1 | ď ε δ , |x 2 | ď ε κ 2 u. Using the Taylor approximation of φpxq one gets that for x P D ε we have |φpxq| À ε. The first condition in (2.3) is then obtained by plugging As in Subsection 1.2 we denote by K the function associating to eachκ 1 P r0, κ 1 s theκ 2 such that Lκ is a supporting line to the augmented Newton polyhedron of φ a , i.e., we haveκ " pκ 1 , K f pκ 1 qq. The Legendre transformation of K is given by LpKqrws :" sup uPr0,κ 1 s pwu´Kpuqq, and thus we have We have depicted the graph of K in Figure 2. Figure 4: The restriction height.

Conditions when the ratio is fixed
If we fix a ratio r " p 1 1 {p 1 3 P r0, 8s, then we are able to introduce a quantity slight more general than the restriction height h res pφq introduced in [18]. We shall not use this quantity in this article, but it may prove useful when considering the mixed norm Fourier restriction for functions φ with h lin pφq ě 2. The cases r P t0, 8u are not interesting since we shall prove the associated results in Section 4 easily, so we assume that r P p0, 8q is fixed. In this case the conditions (2.2) can be restated as i.e., where againκ is such that Lκ is a supporting line to the augmented Newton polyhedron N res pφ f q. But now we notice that the number must hold necessarily true for all r P p0, 8q, along with the inequalities p 1 1 ě 2 and p 1 3 ě 2hpφq, representing the respective cases r " 0 and r " 8.
By definition, the restriction height h res pφq from [18] coincides with h res r pφq when r " 1, and in the same way as in [18] we see from (2.6) that h res r pφq`1 can be read off as the t 2 -coordinate of the point where the line ∆ pmq r intersects the augmented Newton diagram of φ a (see Figure 4).

Necessary conditions when h lin pφq < 2
In the case when φ is non-adapted and the linear height of φ is strictly less than 2 it turns out that there are only two necessary conditions from Proposition 2.1. Namely, in this case we shall show that l 0 " l a , and therefore the only conditions are   Figure 5: The Newton polyhedra associated to A n´1 type singularity in the (linearly adapted) original and adapted coordinates respectively, and the associated necessary conditions. If we replace above the inequality signs with equality signs, we get two linear equations in p1{p 1 1 , 1{p 1 3 q. Let p1{p 1 1 , 1{p 1 3 q be the solution of this system. We shall call p " pp 1 , p 3 q the critical exponent. Then, by interpolation, it is sufficient to prove the Fourier restriction estimate (1.2) for the exponent p and the endpoint exponents associated to the points lying on the axes, i.e., p0, 1{2q and p1{p2hpφqq, 0q.
In order to obtain what precisely the critical exponent p is, we recall [18, Proposition 2.11] which gives us explicit normal forms of φ in the case when h lin pφq < 2. In the real analytic case these normal forms were derived in [29] by D. Siersma. [18,Proposition 2.11] states that there are two type of singularities, A and D.
In the case of A type singularity the form of the function φ is Here ψ, b, and b 0 are smooth functions such that ψpx 1 q " cx m 1`O px m`1 1 q (with c ‰ 0 and m ě 2), bp0, 0q ‰ 0, and b 0 px 1 q " x n 1 βpx 1 q (with either βp0q ‰ 0 and n ě 2m`1, or b 0 is flat, i.e., "n " 8"). The function ψ is the principal root jet of φ. If b 0 is flat, this is A 8 type singularity, and otherwise it is A n´1 type singularity. In adapted coordinates, the formula (2.8) turns into φ a py 1 , y 2 q " b a py 1 , y 2 qy 2 2`b 0 py 1 q, (2.9) where b a py 1 , y 2 q " bpy 1 , y 2`ψ py 1 qq, i.e., the function b in py 1 , y 2 q coordinates. From the formulas (2.8) and (2.9) one can now determine the form of the Newton polyhedron of φ and φ a (see Figure  5). Reading off the Newton polyhedra we have and so the necessary conditions (2.2) can be written as Now an easy calculation shows that p1{p 1 1 , 1{p 1 3 q " p1{p2m`2q, 1{4q, i.e., we have determined the critical exponent.
In the case of D type singularity [18, Proposition 2.11] tells us that φ a py 1 , y 2 q "´y 1 b a 1 py 1 , y 2 q`py 2`ψ py 1 qq 2 b 2 py 2`ψ py 1 qq¯y 2 2`b 0 py 1 q, i.e., the function b from (2.8) is now to be written as bpx 1 , x 2 q " x 1 b 1 px 1 , x 2 q`x 2 2 b 2 px 2 q. In this case we have the conditions b 1 p0, 0q ‰ 0 and b 2 px 2 q " c 2 x k 2`O px k`1 2 q. Again ψpx 1 q " cx m 1`O px m`1 1 q (c ‰ 0, m ě 2) and b 0 px 1 q " x n 1 βpx 1 q, but now either βp0q ‰ 0 and n ě 2m`2, or b 0 is flat. If b 0 is flat, this is D 8 type singularity, and otherwise it is D n`1 type singularity. The function b a 1 is the function b 1 in py 1 , y 2 q coordinates. Now one determines the form of the Newton polyhedra (see Figure 6) and reads off that Therefore, the necessary conditions can be written as Again, a simple calculation shows that p1{p 1 1 , 1{p 1 3 q " p1{p4m`4q, 1{4q. Note that in the A 8 and D 8 cases the necessary conditions form a right-angled trapezium in the p1{p 1 1 , 1{p 1 3 q-plane (easily seen by taking n Ñ 8; one can also do a direct calculation). As the critical exponents in the cases A n´1 and D n`1 do not depend on n, one is easily convinced that the critical exponents of A 8 and D 8 cases are equal to the respective critical exponents of A n´1 and D n`1 . Figure 6: The Newton polyhedra associated to D n`1 type singularity in the (linearly adapted) original and adapted coordinates respectively, and the associated necessary conditions.

Auxiliary results
3.1 Reduction to the case ∇φp0q " 0 In Subsection 1.1 we mentioned that one can always reduce the mixed normed Fourier restriction problem to the case when ∇φp0q " 0, despite rotational invariance not being at one's disposal. Let us justify this. Consider the linear transformation Lpx 1 , x 2 , x 3 q :" px 1 , x 2 , x 3`B1 φp0qx 1`B2 φp0qx 2 q whose inverse and transpose are Plugging in the function f˝L t into the expression of the mixed norm Fourier restriction estimate (1.2) we obtaiñ Now one just notices that }f˝L t } L p 3 since the determinant of L is 1. Thus the estimate (1.2) with the function φ is equivalent (up to a slight change in amplitude due to the Jacobian factor a 1`|∇φpξq| 2 ) to the same estimate with the function φ replaced by the function ξ Þ Ñ φpξq´ξ¨∇φp0q, which has gradient 0 at the origin.

Auxiliary results related to oscillatory sums and integrals
We shall often need the following two one-dimensional oscillatory integral results. The first one is a van der Corput-type estimate used in [18] and originating in the works of van der Corput [6], G.I. Arhipov [1], and J.E. Björk (as noted in [7]). (ii) f is of polynomial type M ě 2, that is, I is compact and there are positive constants c 1 , c 2 such that |f pjq psq| ď c 2 , for every s P I.
Then there exists a constant C which depends only on M in case (i), and on M , c 1 , c 2 , and I in case (ii), such that for every λ P R we havěˇˇż for any L 8 pIq function g with an integrable derivative on I. Furthermore, if G P L 1 pRq is a nonnegative function which is majorized by a function H P L 1 pRq such that p H P L 1 pRq, then for the same constant C as above we have We note that in the above lemma in case piiq we can use in both expressions p1`|λ|q´1 {M instead of |λ|´1 {M since the constant C depends on I anyway. 1 We also remark that we can always use G " |ϕ| for a Schwartz function ϕ since the Fourier transform of |ϕ| is integrable. The proof of this (known) fact is almost straightforward. Namely, the derivative of |ϕ| can have jumps only at the points s where ϕpsq " 0 and ϕ 1 psq ‰ 0. Denote the set of such points N and note that it is a discrete set. In order to estimate the Fourier transform of |ϕ| at ξ, one integrates by parts the expression pF |ϕ|qpξq " ż e´i xξ |ϕ|pxqdx twice and gets the additional boundary terms which can be estimated by |ξ|´2 ř sPN |ϕ 1 psq|. Using the fact that between any two neighbouring points s 1 , s 2 P N there is a point s inbetween such that ϕ 1 psq " 0 one easily gets ř sPN |ϕ 1 psq| ď ş |ϕ 2 psq|ds ă`8 and the claim follows. The second lemma (less general, but with a stronger implication than the one in [18, Section 2.2]) we need gives us an asymptotic of an oscillatory integral of Airy type. We shall also need some variants, but these we shall state and prove along the way when they are needed. where a, b are smooth and real-valued functions on an open neighbourhood of IˆK for I a compact neighbourhood of the origin in R and K a compact subset of R m . Let us assume that bpt, sq ‰ 0 on IˆK and that |t| ď ε on the support of a. If ε ą 0 is chosen sufficiently small and λ sufficiently large, then the following holds true: (a) If λ 2{3 |u| À 1, then we can write where gpv, µ, sq is a smooth function of pv, µ, sq on its natural domain.
(b) If λ 2{3 |u| " 1, then we can write where a˘are smooth functions in p|u| 1{2 , sq and classical symbols 2 of order 0 in λ|u| 3{2 , and where q˘are smooth functions such that |q˘| " 1. The function E is a smooth function satisfying for all N, α, β, γ P N 0 .
Proof. For the part (a) we only sketch the proof since it is a straightforward modification of [18, Lemma 2.2., (a)]. In the integral defining J we substitute t Þ Ñ λ´1 {3 t. Then we can write We added the smooth cutoff function χ 0 localised near 0 in order to emphasize that domain of integration. If we denote v " λ 2{3 u, then the integral can be written as ż R e ipbpµt,sqt 3´v tq apµt, sqχ 0 pµt{εqdt.
We split the integral into two parts, depending on whether the integration domain is contained in |t| À C or |t| ą C for some fixed large C, by using a smooth cutoff function. The part where |t| À C is obviously smooth in all the (bounded) parameters pv, µ, sq and hence it satisfies the conclusion of the lemma. If C is sufficiently large, ε sufficiently small, and |t| ą C, then |B t pbpµt, sqt 3´v tq| " |t| 2 , where B α is any derivative in the pv, µ, sq variables. Therefore by taking derivatives of the integral in pv, µ, sq, factors of polynomial growth in t appear. This can be controlled by using integration by parts a sufficient number of times since the phase derivative is " |t| 2 , and so we get the uniform estimate in this case too. The part (b) is also a straightforward modification of [18, Lemma 2.2, (b)], and so we sketch the proof. Here we get a stronger result for the function E compared to [18, Lemma 2.2, (b)] since we assume that there are no t 2 terms in the phase. We start by substituting t Þ Ñ |u| 1{2 t. Then one gets Jpλ, u, sq " v ż R e iµpbpvt,sqt 3´p sgn uqtq apvt, sqχ 0 pvt{εqdt, where µ denotes λ|u| 3{2 and v denotes |u| 1{2 . If |u| Á ε and if ε is sufficiently small, then the integration domain is |t| ! 1, and so we may use integration by parts and get an estimate as is required for the E term in the conclusion.
If t is away from the critical points (which only exist if u and b are of the same sign), then we can argue similarly as in the (a) part of the proof by using integration by parts and get an estimate as is required for the E term in the conclusion. If u and b have the same sign, then there are two critical points |t˘pv, sq| " 1. One now applies the stationary phase method at each of the critical points and obtains the form as in the conclusion of the theorem.
Next, we state results relating the Newton polyhedron and its associated quantities with asymptotics of oscillatory integrals. Theorem 3.3. Let φ : Ω Ñ R be a smooth function of finite type defined on an open set Ω Ă R 2 containing the origin. If Ω is a sufficiently small neighbourhood of the origin and η P C 8 c pΩq, theňˇˇˇˇż This result was proven in [17] and can be interpreted as a uniform estimate with respect to a linear pertubation of the phase. The case when hpφq ă 2 was considered earlier in [8]. The case when φ is real analytic and there is no pertubation (i.e., ξ 1 " ξ 2 " 0) the above result goes back to Varchenko [34]. In the case of a real analytic function φ one actually has a uniform estimate with respect to analytic pertubations (this was proved by Karpushkin in [20]).
We also have the following result from [17] which gives us sharpness of Theorem 3.3 in the case when ξ 1 " ξ 2 " 0. plog λq νpφq J˘pλq " c˘ηp0q, where c˘are nonzero constants depending on the phase φ only.
An analogous result was proved earlier by Greenblatt in [12] for real analytic phase functions φ. When the principal face is not compact, Theorem 3.4 may fail in general (for an example of this see [19]).
Finally, we state three lemmas which we shall often use in conjunction with Stein's complex interpolation theorem. The proofs of the first and third lemma can be found in [18, Section 2.5], while we only give a brief note on the proof of the second lemma since it is a direct modification of the first one. The proof of all of them are elementary, though the proof of the third one is quite technical.
Lemma 3.5. Let Q " ś n k"1 r´R k , R k s be a compact cube in R n for some real numbers R k ą 0, k " 1, . . . , n, and let α, β 1 , . . . , β n be some fixed nonzero real numbers. For a C 1 function H defined on an open neighbourhood of Q, nonzero real numbers a 1 , . . . , a n , and M a positive integer we define F ptq :" M ÿ l"0 2 iαlt pHχ Q qp2 β 1 l a 1 , . . . , 2 β n l a n q for t P R. Then there is a constant C which depends only on Q and the numbers α and β k 's, but not on H, a k 's, M , and t, such that We shall often use this lemma in combination with the holomorphic function when applying complex interpolation. This function has the property thaťˇˇγ p1`itqF ptqˇˇď C θ for a positive constant C θ ă`8, and γpθq " 1.
The following lemma is a slight variation of what was written in [18,Remark 2.8].
Lemma 3.6. Let Q " ś n k"1 r´R k , R k s be a compact cube in R n for some real numbers R k ą 0, k " 1, . . . , n, let α, β 1 , . . . , β n be some fixed nonzero real numbers, and let 0 ă ă 1. For a C 1 function H on a neighbourhood of Q, nonzero real numbers a 1 , . . . , a n , and M a positive integer we define F ptq :" M ÿ l"0 2 iαlt pHχ Q qp2 β 1 l a 1 , . . . , 2 β n l a n q for t P R. Then there is a constant C which depends only on Q and the numbers α, β k 's, and , but not on H, a k 's, M , and t, such that for all t P R. The constants C k are given as where the supremum goes over the set ś k j"1 r´R j , R j s. The only difference compared to the proof of [18, Lemma 2.7] is that one now writes and notes that the fractions are bounded by their respective C k 's.
In the above lemma we could have directly defined C k 's as the Hölder quotients appearing in (3.2), but the formulas used in Lemma 3.6 turn out to be more practical. One can easily construct an example though where using the Hölder quotients is more appropriate. One example is when one has an oscillatory factor such as in Hpy 1 q " y 1 e iy´1 1 , 0 ă y 1 ă 1 (cf. the Riemann singularity as in [30, Chapter VIII, Subsection 1.4.2]). This function is -Hölder continuous at 0 and satisfies the conclusion of Lemma 3.6 in the sense that |F ptq| ď C{|2 iαt´1 |, but one can show without too much effort that the integral defining C 1 in Lemma 3.6 is infinite.
The third lemma is a two parameter version of the first one.
Lemma 3.7. Let Q " ś n k"1 r´R k , R k s be a compact cube in R n for some real numbers R k ą 0, k " 1, . . . , n, and let α 1 , α 2 P Qˆ, and β k 1 , β k 2 P Q, k " 1, . . . , n, be fixed numbers such that for all k (i.e., the vector pα 1 , α 2 q is linearly independent from pβ k 1 , β k 2 q). For a C 2 function H defined on an open neighbourhood of Q, nonzero real numbers a 1 , . . . , a n , and M 1 , M 2 positive integers we define F ptq :" for t P R. Then there is a constant C which depends only on Q and the numbers α 1 , α 2 , β k 1 's, β k 2 's, but not on H, a k 's, M 1 , M 2 , and t, such that and N is a positive integer depending on the β k 1 's and β k 2 's.
For future reference, we also note the following construction from [18, Remark 2.10] of a complex function γ on the strip Σ :" tζ P C : 0 ď Re ζ ď 1u which shall be used in the context of complex interpolation together with the above two parameter lemma. If we are given 0 ă θ ă 1 and the exponents α 1 , α 2 , and β k 1 's, β k 2 's as above, we define The function γ has the following two key properties. It is an entire analytic function uniformly bounded on the strip Σ, and for the function F as in Lemma 3.7 there is a positive constant C θ ă`8 such that for all t P Rˇˇˇγ It also has the property that γpθq " 1.

Auxiliary results related to mixed L p -norms
In this subsection R shall denote the Fourier restriction operator L p pR 3 q Ñ L 2 pdµq for a positive finite Radon measure µ, and all functions and measures will have R 3 as their domain, unless stated otherwise. Recall that we assume p " pp 1 , p 3 q.
We first recall what happens in the simple case when p " p2, 1q and µ has the form where φ is any measurable function on an open set Ω and η P C 8 c pΩq is a nonnegative function. In this case the form of the adjoint of R is and it is called the extension operator. Using Plancherel for each fixed x 3 , we easily get boundedness of R˚: L 2 pdµq Ñ L 8 x 3 pL 2 px 1 ,x 2 q q. Note that the operator bound depends only on the L 8 norm of η. In particular we know that R : L 1 x 3 pL 2 px 1 ,x 2 q q Ñ L 2 pdµq is bounded. When considering the L p´L2 Fourier restriction problem for other p's, it is advantageous to reframe the problem using the so called "R˚R" method. The boundedness of the restriction operator R : L p Ñ L 2 pdµq is equivalent to the boundedness of the operator T " R˚R, which can be written as in the pair of spaces L p Ñ L p 1 , where p 1 denotes the Young conjugate exponents pp 1 1 , p 1 3 q. Note that the operator T is linear in µ and it even makes sense for a complex µ (unlike the restriction operator R). This enables us to decompose the measure µ into a sum of complex measures, each having an associated operator of the same form as in (3.4).
The following few lemmas give us information on the boundedness of convolution operators such as in (3.4).
Lemma 3.8. Let us consider the convolution operator T : f Þ Ñ f˚p µ for a tempered Radon measure µ (i.e., a Radon measure which is a tempered distribution).
(i) If p µ is a measurable function which satisfies for someσ P r0, 1q, then the operator norm of T : (ii) If µ is a bounded function such that }µ} L 8 À B, then the operator norm of T : L 2 Ñ L 2 is bounded (up to a multiplicative constant) by B.
Proof. One can easily show by integrating (3.4) in px 1 , x 2 q variables that and therefore we can now apply the (one-dimensional) Hardy-Littlewood-Sobolev inequality and obtain the claim in the first case. The second case when p 1 " p 3 " 2 is a well known classical result for multipliers.
For a more abstract approach to the above lemma see [11] and [22]. There one also obtains an appropriate result forσ " 1 when 1{p 1 1 ą 0, but shall not need this. A particular useful application of the above lemma is the following. Lemma 3.9. Let us consider T : f Þ Ñ f˚p µ for a tempered Radon measure µ which is now localised in the frequency space: Let us assume that µ and p µ are measurable functions satisfying Then T is a bounded operator for p 1 q " p0,σ 2 q for allσ P r0, 1q, with the associated operator norm being at most (up to a multiplicative constant) A λσ 3 . The operator norm of T : L 2 Ñ L 2 is bounded (up to a multiplicative constant) by B.
Proof. We only need to obtain the decay estimate (3.5). We note that since p µ has x 3 support bounded by λ 3 , it follows At the end of this subsection we note the following simple result which tells us that the conclusion of Lemma 3.8 is in a sense quite sharp. We remark that the last conclusion in the lemma below is consistent with the conditionσ ă 1 in (3.5).
Lemma 3.10. Consider the convolution operator T : f Þ Ñ f˚p µ for a tempered Radon measure µ whose Fourier transform p µ is continuous. Let ϕ : r0,`8q Ñ p0,`8q be an increasing and unbounded continuous function and assume that at least one of the limits exists for someσ P p0, 1q, with the limiting value being a nonzero number. Then T : L p Ñ L p 1 is not a bounded operator for p1{p 1 1 , 1{p 1 3 q " p0,σ{2q. The conclusion also holds in the case when ϕ is the constant function 1,σ " 1, and if we additionally assume that p µ is an L 8 pR 3 q function and that both of the above limits exist and are equal, with the limiting value being a nonzero number.
Proof. Let us begin the proof by assuming that the operator is bounded. Since p µ is continuous, without loss of generality we can assume that for all x in the open set U of the form where K ą 0 and U is a continuous and strictly positive function on R. Now consider the function where χ 0 is smooth, identically 1 in the interval r´1, 1s, and supported within the interval r´2, 2s.
and if we assume ε to be sufficiently small and M sufficiently large, one obtains by a simple calculation that and the lower bound on the norm is But now by the boundedness assumption we obtain i.e., ϕp|M |q À 1. This is impossible in general since we can take M Ñ 8.
In the case when the limits are equal,σ " 1, and ϕ is the constant function 1, we can take (3.7) to be true for x P U too. If we use the same f as above, then for any x 3 P r´M {2, M {2s we easily obtain from the definition of T that for an M sufficiently large and ε sufficiently small. Thus the norm }T f } L 2 In the caseσ " 1 and when ϕ is identically equal to a nonzero constant the above proof does not work if the limits have the same absolute value but opposite signs. This is related to the fact that an operator given as a convolution against x Þ Ñ x{p1`x 2 q is bounded L 2 pRq Ñ L 2 pRq since the Fourier transform of x Þ Ñ x{p1`x 2 q is up to a constant ξ Þ Ñ e´| ξ| sgn ξ.

The adapted case and reduction to restriction estimates near the principal root jet
Here we mimic [18,Chapter 3] and the last section of [17], where the adapted case for p 1 " p 3 was considered. In this section we shall be concerned with measures of the form where φp0q " 0, ∇φp0q " 0, and η is a smooth nonnegative function with support contained in a sufficiently small neighbourhood of 0. We assume that φ is of finite type on the support of η. The associated Fourier restriction problem is for any η with support contained in a sufficiently small neighbourhood of 0.
The following proposition will be useful in this section.
Let us now recall what happens in the non-degenerate case, i.e., when the determinant of the Hessian det H φ p0, 0q ‰ 0. This is equivalent to hpφq " 1 and in this case φ is adapted in any coordinate system. Here we have the bound (4.2) for all of the p1{p 1 1 , 1{p 1 3 q given in the necessary condition (2.5), except for the point p0, 1{2q, for which it does not hold. This fact is actually true globally, i.e., the Strichartz estimates hold (see [11,22] and references therein) in the same range, and one can easily convince oneself that the same proof as in say [22] goes through in our local case. For the negative results at the point p0, 1{2q in the case of Strichartz estimates see [21] and [25]. We can also get a negative result at the point p0, 1{2q directly in our case by applying Lemma 3.10 for the caseσ " 1 and ϕ is identically equal to 1. The limits in Lemma 3.10 are obtained by a simple application of the two dimensional stationary phase method. Furthermore, since the Hessian does not change its sign when changing the phase φ Þ Ñ´φ, the limits in both directions are equal.
The claims for the case when hpφq ą 1 follow easily by applying Theorems 3.3 and 3.4 to Lemmas 3.8 and 3.10 respectively. In Lemma 3.10 we take ϕ to be the logarithmic function x Þ Ñ logp2`xq.

The adapted case
The following proposition tells us precisely when the Fourier restriction estimate holds in the adapted case.
Proposition 4.2. Let us assume that µ, φ, and η are as explained at the beginning of this section, and let us assume that φ is adapted.
(i) If hpφq " 1 or νpφq " 1, then the full range Fourier restriction estimate given by the necessary condition (2.5) holds true, except for the point p1{p 1 (ii) If hpφq ą 1 and νpφq " 0, then the full range Fourier restriction estimate given by the necessary condition (2.5) holds true, including the point p1{p 1 Proof. The case when hpφq " 1 is the classical known case and it was already discussed in the proof of Proposition 4.1. The case when hpφq ą 1 and νpφq " 0 follows from Proposition 4.1 by interpolation.
Let us now consider the remaining case when hpφq ą 1 and νpφq " 1. Then if we would use Proposition 4.1 and interpolation as in the previous case, we would miss all the boundary points determined by the line of the necessary condition (2.5) except the point p1{2, 0q where we know that the estimate always holds. Recall that this is essentially because we have the logarithmic factor in the decay of the Fourier transform of µ. Instead, one can use the strategy from [17, Section 4] to avoid this problem. We only briefly sketch the argument. One decomposes µ " where µ k are supported within ellipsoid annuli centered at 0 and closing in to 0. This is done by considering the partition of unity ηpxq " where χ is an appropriate C 8 c pR 2 q function supported away from the origin and where κ " pκ 1 , κ 2 q is the weight associated to the principal face of N pφq. Next, one rescales the measures µ k and obtains measures µ 0,pkq having the form (4.1). These new measures have uniformly bounded total variation and Fourier decay estimate with constants uniform in k:ˇz Note that there is no logarithmic factor anymore. Now we can use Proposition 4.1 and interpolation to obtain the mixed norm Fourier restriction estimate within the range (2.5) for each µ 0,pkq . As in [17,Section 4], one now easily obtains the bound 3 where δ e r px 1 , x 2 , x 3 q " pr κ 1 x 1 , r κ 2 x 2 , rx 3 q. The scaling in our mixed norm case is by the necessary condition 1 and the equalities dpφq|κ| " hpφq|κ| " 1. The rest of the proof is the same as in [17] if we assume p 1 ą 1, since then one can use the Littlewood-Paley theorem 4 and the Minkowski inequality (which we can apply since p 1 " p 2 ď 2 and p 3 ď 2) to sum the above inequality in k. The proof of Proposition 4.2 is done.

Reduction to the principal root jet
In this subsection we make some preliminary reductions for the case when φ is not adapted. Recall that we may assume that φ is linearly adapted and that we denote by ψ the principal root of φ.
Then we can obtain the adapted coordinates y (after possibly interchanging the coordinates x 1 and x 2 ) through Before stating the last proposition of this section (analogous to [18, Proposition 3.1]) let us recall some notation from [18]. We write where b 1 ‰ 0 and m ě 2 by linear adaptedness (see [18,Proposition 1.7]). If F is an integrable function on the domain of η, say Ω Ď R 2 , then we denote If χ 0 denotes a C 8 c pRq function equal to 1 in a neighbourhood of the origin, we may define where ε is an arbitrarily small parameter. The domain of ρ 1 is a κ-homogeneous subset of Ω which contains the principal root jet x 2 " ψpx 1 q of φ when Ω is contained in a sufficiently small neighbourhood of 0.
Assume φ is of finite type on Ω, non-adapted, and linearly adapted (i.e., dpφq " h lin pφq). Let ε ą 0 be sufficiently small and let µ 1´ρ 1 have support contained in a sufficiently small neighbourhood of 0. Then the mixed norm Fourier restriction estimate (4.2) with respect to the measure µ 1´ρ 1 holds true for all p1{p 1 1 , 1{p 1 3 q which satisfy 1 dpφq , i.e., within the range determined by the necessary condition associated to the principal face of N pφq, except maybe the boundary points of the form p0, 1{p 1 3 q. In particular, it also holds true within the narrower range determined by all of the necessary conditions, excluding maybe the boundary points of the form p0, 1{p 1 3 q.
We just briefly mention that the proof of the Proposition 4.3 is trivial as soon as one uses the results from [18,Chapter 3]. Analogously to the previous subsection, one decomposes the measure µ 1´ρ 1 by using the κ dilations associated to the principal face of N pφq. The measures ν k obtained by rescaling are of the form (4.1), have uniformly bounded total variation, and have the Fourier transform decay (with constants uniform in k) |p ν k pξq| À p1`|ξ|q´d pφq .
Note that the estimates for the boundary points of the form p0, 1{p 1 3 q can be directly solved for the original measure µ through Proposition 4.1.

The case h lin pφq < 2
In the remainder of this article we shall we concerned with the proof of: Theorem 5.1. Let φ : R 2 Ñ R be a smooth function of finite type defined on a sufficiently small neighbourhood Ω of the origin, satisfying φp0q " 0 and ∇φp0q " 0. Let us assume that φ is linearly adapted, but not adapted, and that h lin pφq ă 2. We additionally assume that the following holds: Whenever the function b 0 appearing in (2.8), (2.9), (2.10) is flat (i.e., when φ is A 8 or D 8 type singularity), then it is necessarily identically equal to 0. In this case, for all smooth η ě 0 with support in a sufficiently small neighbourhood of the origin the Fourier restriction estimate (4.2) holds for all p given by the necessary conditions determined in Subsection 2.4.
The above condition on the function b 0 is implied by the Condition (R) from [18] (see [18,Remark 2.12 We begin with some preliminaries. As one can see from the Newton diagrams in Subsection 2.4, the assumption in our case h lin pφq < 2 implies that hpφq ď 2. Additionally, we see that hpφq " 2 implies that we either have A 8 or D 8 type singularity. As mentioned in Subsection 1.1, the Varchenko exponent is 0, i.e., νpφq " 0, if hpφq ă 2. When hpφq " 2 the equality νpφq " 0 also holds true in our case since the principal faces are non-compact. We conclude that if h lin pφq < 2, then by Proposition 4.1 we have the mixed norm Fourier restriction estimate (4.2) for both of the points p1{p 1 1 , 1{p 1 3 q " p1{2, 0q and p1{p 1 1 , 1{p 1 3 q " p0, 1{p2hpφqqq. Therefore, according to Subsection 2.4, by interpolation it remains to prove the estimate (4.2) for the respective critical exponents given by´1 n case of A type singularity, n case of D type singularity, where m ě 2 is the principal exponent of ψ from Subsection 2.4.
Recall that according to Proposition 4.3 we may concentrate on the piece of the measure µ located near the principal root jet: for an arbitrarily small ε and ωp0qx m 1 the first term in the Taylor expansion of where ω is a smooth function such that ωp0q ‰ 0.
As we use the same decompositions of the measure µ ρ 1 as in [18], we shall only briefly outline the decomposition procedure.

Basic estimates
Before we outline the further decompositions and rescalings of µ ρ 1 , we first describe here the general strategy for proving the Fourier restriction estimates for the pieces obtained through these decompositions. All of the pieces ν of the measure µ ρ 1 will essentially be of the form where Φ is a phase function and a ě 0 an amplitude. The amplitude will usually be compactly supported with support away from the origin. Both Φ and a will depend on various decomposition related parameters. We shall need to prove the Fourier restriction estimate with respect to these measures with estimates being uniform in a certain sense with respect to the appearing decomposition parameters. At this point one uses the "R˚R" method applied to the measure ν. The resulting operator is T ν which acts by convolution against the Fourier transform of ν. Now one considers the spectral decomposition pν λ q λ of the measure ν so that each functions ν λ is localised in the frequency space at λ " pλ 1 , λ 2 , λ 3 q, where λ i ě 1 are dyadic numbers for i " 1, 2, 3. For such functions ν λ we shall obtain bounds of the form (3.6). By Lemma 3.9 then we have the bounds on their associated convolution operators T λ ν : Now it remains to sum over λ.
When θ ă 1{4, we shall be able to always sum absolutely. In the cases when θ " 1{4 and particularly θ " 1{3 (note that both appear only in A type singularity with m " 3 and m " 2 respectively) we shall need the complex interpolation method developed in [18].

First decompositions and rescalings of µ ρ 1
As in Section 4, we use the κ dilatations associated to the principal face of N pφq, and subsequently a Littlewood-Paley argument. Then it remains to prove the Fourier restriction estimate for the renormalised measures ν k of the form xν k , f y " ż f px, φpx, δqq apx, δq dx, uniformly in k. As was shown in [18, Section 4.1], the function φpx, δq has the form φpx, δq :"bpx 1 , in case of A type singularity, Above the functions b, b 1 , b 2 , β, and the quantity n are as in Subsection 2.4. Recall that m " κ 2 {κ 1 ě 2 and so δ 2 " δ m 1 . The amplitude apx, δq ě 0 is a smooth function of px, δq supported at Furthermore, due to the ρ 1 cutoff function which has a κ-homogeneous domain, we may assume |x 2´x m 1 ωp0q| ! 1. Since we can take k arbitrarily large, the parameter δ approaches 0. This implies that on the domain of integration of a we have thatbpx 1 , x 2 , δ 1 , δ 2 q converges as a function of px 1 , x 2 q to bp0, 0q (resp. b 1 p0, 0qx 1 ) in C 8 when k Ñ 8 and φ has A type singularity (resp. D type singularity). The amplitude apx, δq converges in C 8 c to apx, 0q. We also recall that according to the assumption in Theorem 5.1, we may assume that δ 3 " 0 if "n " 8", i.e., if b 0 is flat in the normal form of φ.
The next step is to decompose the (compactly) supported amplitude a into finitely many parts, each localised near a point v " pv 1 , v 2 q for which we may assume that it satisfies v 2 " v m 1 ωp0q (by compactness and since in (5.2) we can take ε arbitrarily small). The newly obtained measures we denote by ν δ and their new amplitudes by the same symbol apx, δq ě 0: where now the support of ap¨, δq is contained in the set |x´v| ! 1.
Since we can use Littlewood-Paley decompositions in the mixed norm case (see [24,Theorem 2], and also [2,10]), we can now decompose the measure ν δ in the x 3 direction in the same way as in [18,Section 4.1]. This is achieved by using the cutoff functions χ 1 p2 2j φpx, δqq in order to localise near the part where |φpx, δq| " 2´2 j . Then it remains to prove the mixed norm estimate (4.2) for measures ν δ,j with bounds uniform in paramteres j P N and δ " pδ 1 where j can be taken sufficiently large and δ sufficiently small. The function 2 2j φpx, δq can be written as Following [18], we distinguish three cases: 2 2j δ 3 ! 1, 2 2j δ 3 " 1, and the most involed 2 2j δ 3 " 1.

5.3
The case 2 2j δ 3 " 1 As was done in [18, Subsection 4.1.1], we change coordinates from px 1 , x 2 q to px 1 , 2 2j φpx, δqq and subsequently perform a rescaling (which we adjust to our mixed norm case). Then one obtains that the mixed norm Fourier restriction for ν δ,j is equivalent to the estimate The function apx, δ, jq has in δ and j uniformly bounded C l norms for an arbitrarily large l ě 0, and the phase function is given by where x 1 " 1, x 2 " 1, and without loss of generality we may assumeb 1 px 1 , x 2 , 0, 0q " 1 and βp0q " 1; for details see [18,Subsection 4.1.1]. There the phase function φpx, δ, jq was obtained by solving the equation 2 2j φpy, δq " 2 2jb py 1 , y 2 , δ 1 , δ 2 qpy 2´y m 1 ωpδ 1 y 1 qq 2´22j δ 3 y n 1β pδ 1 y 1 q in y 2 after substituting x 1 " y 1 and x 2 " 2 2j φpy, δq. By using the implicit function theorem one can show that when δ Ñ 0, then we have the following C 8 convergence in the px 1 , x 2 q variables: In both the A and D type singularity cases we see thatb 1 does not depend on x 2 in an essential way. Now we proceed to perform a spectral decomposition ofν δ,j , i.e., for pλ 1 , λ 2 , λ 3 q dyadic numbers with λ i ě 1, i " 1, 2, 3, we define the spectrally localised measures ν λ j through (5.11) We slightly abuse notation in the following way. Whenever λ i " 1, then the appropriate factor χ 1 p ξ i λ i q in the above expression should be considered as a localisation to |ξ i | À 1, instead of |ξ i | " 1.
If we define the operatorsT δ,j f :" f˚p ν δ,j , T λ j f :" f˚x ν λ j , then we formally haveT and according to (5.8) and by applying the "R˚R" technique we need to prove In case when we are able to obtain this estimate by summing absolutely the operator pieces T λ j we shall proceed as explained in Subsection 5.1. In this case in order to obtain the (5.3) estimates we need an L 8 bound for x ν λ j , which we shall get from the expression (5.11), and an L 8 bound for ν λ j , which we shall derive next.
Using the equation (5.11) we get by Fourier inversion ν λ j px 1 , x 2 , x 3 q "λ 1 λ 2 λ 3 ż q χ 1 pλ 1 px 1´y1 qq q χ 1 pλ 2 px 2´φ py, δ, jqqq q χ 1 pλ 3 px 3´y2 qq apy, δ, jq χ 1 py 1 q χ 1 py 2 q dy. (5.13) Here we immediately obtain that the L 8 bound on ν λ j is up to a multiplicative constant λ 2 using the first and the third factor within the integral by substituting λ 1 y 1 and λ 3 y 2 . On the other hand, one can easily verify that B y 2 φpy, δ, jq " δ´1 and hence by substituting z 1 " λ 1 y 1 , z 2 " λ 2 φpy, δ, jq, and utilising the first two factors within the integral, we obtain }ν λ j } L 8 À δ 1{2 3 2 2j λ 3 , and therefore combining these two estimates we get It remains to estimate the Fourier side; for this we shall need to consider several cases depending on the relation between λ 1 , λ 2 , and λ 3 . Let us mention that as in [18], here we shall have no problems when absolutely summing the "diagonal" pieces where λ 1 " λ 2 " δ 1{2 3 2 2j λ 3 . However, unlike in [18], a case appears which is not absolutely summable. This will be a recurring theme in this article. It will also indicate that we should take care even when estimates are obtained by integration by parts.
Case 1. λ 1 ! λ 2 or λ 1 " λ 2 , and λ 3 " λ 2 . In this case we can use integration by parts in both x 1 and x 2 in (5.11) to obtain for any nonnegative integer N . Therefore, after plugging this estimate and the estimate (5.14) into (5.3) and (5.6), we may sum in all three parameters λ 1 , λ 2 , and λ 3 , after which one obtains an admissible estimate for (5.12). Case 2. λ 1 ! λ 2 or λ 1 " λ 2 , and λ 3 À λ 2 . Here it is sufficient to use integration by parts in x 1 . Therefore, we have for any nonnegative integer N . Again, after interpolating summation of operators T λ j is possible in all three parameters.
In this case we see that necessarily λ 1 Á δ 1 2 3 2 2j . Also we note that if we fix say λ 1 , then there are only finitely many dyadic numbers λ 2 such that λ 1 " λ 2 , and therefore we essentially need to sum in only two parameters in this case. By stationary phase (and integration by parts when away from the critical point) in x 1 and integration by parts in The better bound in (5.14) is δ 1{2 3 2 2j λ 3 . Therefore (5.6) becomes in our case and hence by summation in λ 3 and taking N " 1 we get Now we obviously get the desired result by summation over λ 1 Á δ 1 2 Here we essentially sum in only one parameter. Let us first determine the estimate in (5.6).
Here we have by the stationary phase method in x 1 and integration by parts in x 2 } x ν λ j } L 8 À λ´1 2 1 pλ 3 q´N , and the bound in (5.14) is λ 1 " λ 2 . Interpolating, we obtain (with a different N ) Now if θ ă 1{3, then we can easily sum in both λ 1 and λ 3 . Therefore, we assume in the following that θ " 1{3.
Now we may sum in λ 3 to get the desired result.
Note that here we sum λ 3 over all the dyadic numbers greater than or equal to 1. We can also assume that λ 1 " δ 3 2 j . This is admissible for (5.12).
In order to obtain the required bound in the remaining range: we need to use the complex interpolation technique developed in [18]. For simplicity we assume that λ 1 " λ 2 (we can do this without losing much on generality since for a fixed λ 1 there are only finitely many dyadic numbers λ 2 such that λ 1 " λ 2 ). We need to consider the following function parametrised by the complex number ζ and the dyadic number λ 3 : where γpζq " 2´3 pζ´1q{2´1 .
The associated convolution operator (given by convolution against the Fourier transform of the function µ λ 3 ζ ) we denote by T λ 3 ζ . At this point let us mention that whenever we use complex interpolation we shall generically denote by µ ζ the considered measure parametrised by the complex number ζ, sometimes with an additional superscript, as is in the current case. Similarily, the associated operator shall be denoted by T ζ , up to possible appearing superscripts.
The first estimate is trivial in (5.17). Namely, since x ν λ j have essentially disjoint supports, it follows from the formula (5.16) and the estimate on the Fourier transform of ν λ j that } y µ λ 3 it } L 8 À λ´N 3 , for any N P N, the implicit constant depending of course on N . Now one just uses the results from Subsection 3.3.

5.4
The setting when 2 2j δ 3 À 1 As explained in Section [18,Subsection 4.2], in this case we use the change of coordinates px 1 , x 2 q Þ Ñ px 1 , 2´jpx 2`x m 1 ωpδ 1 x 1 qqq in the expression (5.7) for ν δ,j . After renormalising the measure ν δ,j we obtain that the mixed norm Fourier restriction estimate for ν δ,j is equivalent to whereν δ,j is the rescaled measure xν δ,j , f y :" The function apx, δ, jq has the form apx, δ, jq :" χ 1 pφ a px, δ, jqqapx 1 , 2´jx 2`x m 1 ωpδ 1 x 1 q, δq and the phase function is given by where |bpx 1 , x 2 , 0, 0q| " 1 and |βp0q| " 1. Also, we recall that when δ Ñ 0, thenbpx 1 , x 2 , δ 1 , δ 2 q converges in C 8 to a nonzero constant if φ has A type singularity, and that it converges up to a multiplicative constant to x 1 if φ has D type singularity. We shall assume without loss of generality thatbpx 1 , x 2 , δ 1 , δ 2 q ą 0 since one can just reflect the third coordinate of f in the expression for the measureν δ,j . Support assumptions on ap¨, δq from Subsection 5.2 (namely, that the support is contained in a small neighbourhood of the point pv 1 , v m 1 ωp0qq for some v 1 ą 0) imply that ap¨, δ, jq is supported in a set where x 1 " 1 and |x 2 | À 1.
We also introduce the operatorsT δ,j f :" f˚p ν δ,j and T λ j f :" f˚x ν λ j . Then we need to prove: In most of the cases this will be done in a similar manner as in the previous subsection. In the case when 2 2j δ 3 " 1, θ " 1{3, and λ 1 " λ 2 " λ 3 , with which we shall deal in the next Section, we shall need to perform a finer analysis.

(5.22)
Similarily as in the case 2 2j δ 3 " 1, we can consider either the substitution pz 1 , z 2 q " pλ 1 y 1 , λ 2 2´jy 2 q, or the substitution pz 1 , z 2 q " pλ 1 y 1 , λ 3 φ a py, δ, jqq (in order to carry this out one needs to consider the cases y 2 " 1 and y 2 "´1 separately). Then one can easily obtain Next we calculate the L 8 bounds on the Fourier transform by using the expression (5.20). Case 1. λ 1 ! λ 2 or λ 1 " λ 2 , and λ 3 ! maxtλ 1 , λ 2 u. By integration by parts in x 1 one has The operators T λ j are now summable which can be seen by using the estimate in (5.6) obtained by interpolation.
Case 2. λ 1 ! λ 2 or λ 1 " λ 2 , and λ 3 Á maxtλ 1 , λ 2 u. Here we use integration by parts in x 2 only and so we have the bound After interpolating we can again sum operators T λ j in all three paramteres. Case 3. λ 1 " λ 2 and λ 3 ! 2´jλ 2 . Note that necessarily λ 2 ě 2 j . Here we use stationary phase in x 1 and integration by parts in x 2 . Then one gets the estimate The better bound in (5.23) is 2 j λ 3 . Therefore (5.6) becomes If θ ă 1{3, then we can rewrite we note that one can now easily sum in both λ 1 and λ 3 . If θ " 1{3, then the first inequality for T λ j can be rewritten as for some different N . Now we first sum in λ 3 up to 2´jλ 1 , and then we sum in λ 1 ě 2 j . Case 4. λ 1 " λ 2 and λ 3 " 2´jλ 2 . Again necessarily λ 2 Á 2 j . One uses in both x 1 and x 2 the stationary phase method and gets The estimate for }ν λ j } L 8 from (5.23) is À λ 2 . Hence, we get the estimate By summation in λ 1 Á 2 j we obtain the bound 2 3jθ{2´j{2 . Now since θ ď 1{3, we get the desired result. Case 5. λ 1 " λ 2 and λ 3 Á λ 2 . Here it suffices to use integration by parts in x 2 only. One easily gets and one can now sum in both λ 1 and λ 3 . Case 6. λ 1 " λ 2 and 2´jλ 2 ! λ 3 ! λ 2 . By the stationary phase method in x 1 and integration by parts in x 2 } x ν λ j } L 8 À λ´1 2 1 pλ 3 q´N , and the better bound in (5.23) is λ 2 .
Similarily as in the case 2 2j δ 3 " 1 one easily sees that, unless θ " 1{3, one can sum in both parameters. Henceforth we shall assume θ " 1{3 and use complex interpolation in order to deal with this case. Here we know that φ has A type singularity andσ " 1{4. For simplicity we shall again assume that λ 1 " λ 2 .
We consider the following function parametrised by the complex number ζ and the dyadic number λ 3 : We denote the associated convolution operator by T λ 3 ζ . For ζ " 1{3 we see that Hence, by interpolation it suffices to prove 1`it } L 2 ÑL 2 À 1, for some N ą 0, with constants uniform in t P R.
The first estimate follows right away since x ν λ j have essentially disjoint supports, and so the L 8 estimate for x ν λ j implies } y µ λ 3 it } L 8 À λ´N 3 , for any N P N. We prove the second estimate using Lemma 3.5. We need to prove uniformly in t.
Within the second factor in the integral we can use a Taylor approximation at x 1 and obtain where |B N 1 r| " 1 for N ě 0 since the term ψ δ is dominant, and Q is a smooth function with uniform bounds. Now we notice that this form is the same as in the case 2 2j δ 3 " 1 in the part where we used complex interpolation, and hence the same proof using the oscillatory sum lemma can be applied, up to obvious changes such as changing the summation bounds.

(5.26)
Recall also that in this case we have the weaker conditions x 1 " 1 and |x 2 | À 1 for the domain of integration in the integral in (5.20). We furthermore slightly modify the notation in this case, as it was done in [18]. Namely, δ shall denote in this subsection pδ 1 , δ 2 q since δ 3 appears only in σ. We also note that in this case there is no A 8 nor D 8 type singularity. Let us introduce the notation ψ ω py 1 q " y m 1 ωpδ 1 y 1 q, ψ β py 1 q " σy n 1 βpδ 1 y 1 q.
Then, after applying the inverse Fourier transform to (5.20), we may write ν λ j pxq " λ 1 λ 2 λ 3 ż q χ 1 pλ 1 px 1´y1 qq q χ 1 pλ 2 px 2´2´j y 2´ψω py 1 qqq q χ 1 pλ 3 px 3´b # py, δ, jqy 2 2´ψβ py 1 qqq apy, δ, jq χ 1 py 1 q χ 0 py 2 q dy. Namely, in the first factor within the integral in (5.27) we can substitute λ 1 y 1 Þ Ñ y 1 , and afterwards either substitute λ 2 2´jy 2 Þ Ñ y 2 in the second factor, or use the van der Corput lemma (i.e., Lemma 3.1, piq) in the third factor with respect to the y 2 variable. As can easily be seen from (5.26) by using integration by parts in x 1 , if one of λ 1 , λ 2 is considerably larger than any other λ i , i " 1, 2, 3, then we can easily gain a sufficiently strong estimate with which one can sum absolutely in all three parameters λ i , i " 1, 2, 3, the operators T λ j . If λ 3 is significantly larger than both λ 1 and λ 2 and φ is of type A, we can also use integration by parts in x 1 in order to get a sufficiently strong estimate. In the case when λ 3 is the largest and φ is of type D, then b # px, δ, jq is approximately x 1 in the C 8 sense, and so in this case and when |x 2 | " 1, we use integration by parts in x 2 , and when |x 2 | ! 1 integration by parts in x 1 . In both parts we get the bound λ´N 3 with which we can obtain a summable estimate for T λ j in all three parameters.
As it turns out, in almost all the other possible relations between λ i , i " 1, 2, 3, we shall need complex interpolation if θ " 1{3, or if θ " 1{4 and it is the "diagonal" case, i.e., all the λ i , i " 1, 2, 3, are of approximately the same size. If θ " 1{3 and λ i , i " 1, 2, 3, are of approximately the same size we shall actually need a finer analysis where estimates on Airy integrals are needed. This will be done in the next section.
Case 1.1. λ 1 " λ 3 , λ 2 ! λ 1 , and λ 2 ď 2 j λ 1{2 1 . On the part where |x 2 | " 1 we can use integration by parts in x 2 and obtain much stroger estimates sufficient for absolute summation. When |x 2 | ! 1 we use stationary phase in both variables, and so from which one can calculate that Let us denote by T I δ,j the sum of the operator pieces T λ j in this case. We need to separate the sum in λ 1 into two subcases λ 1 ď 2 2j and λ 1 ą 2 2j : Therefore if θ ă 1{3, then we obtain the desired result, and if θ " 1{3, we need to use complex interpolation for the first sum where λ 1 ď 2 2j . For θ " 1{3, we have and one is easily convinced that we may restrict ourselves to the case The bound on the operator norm motivates us to define k through 2 k :" λ 1 λ´1 2 " 2 k 1´k2 , where 2 k 1 " λ 1 and 2 k 2 " λ 2 . Our goal is to prove that for each k within the range 1 ! 2 k ! 2 2j we have since then we obtain the desired estimate by summation in k.
We shall slightly simplify the proof by assuming that λ 1 " λ 3 . Let us consider the following function parametrised by the complex number ζ and the integer k: .
The associated convolution operator (convolution again the Fourier transform of µ k ζ ) we denote by T k ζ . For ζ " 1{3 we see that Therefore, it is sufficient to prove 1`it } L 2 ÑL 2 À 1, with constants uniform in t P R. Recall thatσ " 1{4 since m " 2 and θ " 1{3.
Using the first three factors we can reduce the problem to the case |x| ď C for some large constant C. Now, as we have done in previous instances of complex interpolation, we use the substitution λ 1 x 1´y1 Þ Ñ y 1 , conclude that it is sufficient to consider the part of the integration domain where |y 1 | ď λ ε 1 . In particular then x 1 " 1 and we can use Taylor approximation for ψ ω and ψ β at x 1 . Then one gets where |B N 1 r ω | " 1 and |B N 1 r β | " 1 for any N ě 0. Also note that 2´jλ 1{2 1 ! 1. We may now conclude that it is sufficient to consider the cases when either |A| " 1 or |B| " 1, where since otherwise, when both |A| and |B| are bounded, we could apply Lemma 3.5, similarily as in the case 2 2j δ 3 " 1, to the function Hpz 1 , z 2 , z 3 , z 4 , z 5 ; x, δ, σq :" where we would plug in pz 1 , z 2 , z 3 , z 4 , z 5 q " p2´kλ 1 Q ω px 1 , x 2 , δ 1 q, λ 1 Q β px 1 , x 3 , δ 1 q, 2´jλ 1{2 1 , λ´ε 1 , 2 k λ´1 1 q.
Note that the upper bounds on z 4 and z 5 are given by the summation bounds for the parameter λ 1 , and that the function H does not depend on z 5 . Furthermore, the C 1 norm of H in pz 1 , z 2 , z 3 , z 4 , z 5 q is bounded since derivatives of Schwartz functions are Schwartz and only factors of polynomial growth in y 1 and y 2 appear when taking the derivatives. The polynomial growth in y 1 can be dealt with by using the first factor. For the polynomial growth in y 2 one has to consider the cases |y 2 | À |y 1 | N and |y 2 | " |y 1 | N separately. In the first case we can obviously again use the first factor, and in the second case we use the third factor inside which the term b # y 2 2 is now dominant. Let us now first assume |B| " 1. The first three factors within the integral are behaving essentially like q χ 1 py 1 qq χ 1 pA´2´ky 1´2´j´k λ 1{2 1 y 2 qq χ 1 pB´y 1´y 2 2 q. We may reduce ourselves to the discussion of the part of the integration domain where |y 1 | ! |B| ε B since otherwise, when |y 1 | Á |B| ε B , we could use the first factor, obtain the estimate |B|´N ε B for the integral, and then sum this geometric series in λ 1 . Then |B´y 1 r β | " |B|, and the integral we need to estimate is bounded by ż |q χ 1 pB´y 1 r β´y for some constant C. Now one can again sum in λ 1 .
Let us now assume |B| ď C B for some large, but fixed constant C B , and let |A| " C B . Again, we can reduce ourselves to the part where |y 1 | ! |A| ε A , and so |A´2´ky 1 r ω | " |A|. Therefore if |y 2 | ď |A| 1{2 , then using the second factor we get that the integral is bounded (up to a constant) by |A|´N . If |y 2 | ą |A| 1{2 , then |B´y 1 r β´y 2 2 b # | Á |A| and so we can use the third factor, and sum in λ 1 .
Case 1.2. λ 1 " λ 3 , λ 2 ! λ 1 , and λ 2 ą 2 j λ 1{2 1 . In this case we have the same bound for the Fourier transform. Hence from which one can calculate that If we denote by T II δ,j the sum of the operator pieces in this case, then we have: Case 2.1. λ 2 " λ 3 , λ 1 ! λ 2 , and λ 2 ď 2 2j . Here again we may use stationary phase in both variables (and when |x 2 | " 1 even integration by parts in x 2 ). The estimates are and therefore independent of λ 1 . As in [18] we define and note that then we can write where χ 0 is a smooth cutoff function supported in a sufficiently small neighbourhood of 0. Therefore, one easily sees that using the same argumentation as for ν λ j we have The operator norm bound is Hence, if θ ă 1{3, then we obtain the desired result by summing the geometric series, and if θ " 1{3, we need to use complex interpolation. As usual, we consider only the case λ 2 " λ 3 . Also note that we may reduce ourselves to the summation over λ 2 ! 2 2j instead of λ 2 ď 2 2j . We define the following function parametrised by the complex number ζ: .
The associated convolution operator we denote by T ζ . For ζ " 1{3 we see that and so it is sufficient to prove with constants uniform in t P R.
Now we can apply Lemma 3.8. We prove the second estimate by using the oscillatory sum lemma (Lemma 3.5). We need to prove First note that since we obtain the function σ λ 2 ,λ 2 j by summation in λ 1 , the expression (5.27) has to be replaced by q χ 0 pλ 2 px 1´y1 qq q χ 1 pλ 2 px 2´2´j y 2´ψω py 1 qqq q χ 1 pλ 2 px 3´b # py, δ, jqy 2 2´ψβ py 1 qqq apy, δ, jq χ 1 py 1 q χ 0 py 2 q dy. Recall that the function χ 0 of the first factor within the integral has support contained in r´ , s where the small constant depends on the implicit constant in the relation λ 1 ! λ 2 .
After substituting λ 2 y 1 Þ Ñ y 1 and λ 1{2 2 y 2 Þ Ñ y 2 in the expression (5.31), we get that the sum on the left hand side of (5.30) is Since otherwise we could use the first three factors within the integral to gain a factor of λ´N 2 , we may assume that |x| ď C for some large constant C. Now again we use the substitution λ 2 x 1´y1 Þ Ñ y 1 , conclude that it is sufficient to consider the part of the integration domain where |y 1 | ď λ ε 2 , which implies x 1 " 1, and so we may use Taylor approximation for ψ ω and ψ β at x 1 . Then one gets where |B N 1 r ω | " 1 and |B N 1 r β | " 1 for any N ě 0. Note that 2´jλ 1{2 2 ! 1. Now we may restrict ourselves to cases when either |A| " 1 or |B| " 1, where A :" λ 2 Q ω px 1 , x 2 , δ 1 q, B :" λ 2 Q β px 1 , x 3 , δ 1 q, since otherwise we could apply the oscillatory sum lemma similarily as in Case 1.1. The first three factors within the integral are behaving essentially like q χ 1 py 1 qq χ 1 pA´y 1´2´j λ 1{2 2 y 2 qq χ 1 pB´y 1´y 2 2 q.
Let us first consider |B| " 1, as in Case 1.1. As usual, we may restrict ourselves to the part of the integration domain where |y 1 | ! |B| ε B . Therefore there we have |B´y 1 | " |B|, and the integral is bounded by ż |q χ 1 pB´y 1 r β´y for some constant C. Now one can sum in λ 2 .
Let us now assume |B| ď C B for some large, but fixed constant, and |A| " C B . Again, we may consider only the part of the integration domain where |y 1 | ! |A| ε A , and so here we have |A´y 1 | " |A|. Therefore, if |y 2 | ď |A| 1{2 , then using the second factor we get that the integral is bounded (up to a constant) by |A|´N . If |y 2 | ą |A| 1{2 , then |B´y 1 r β´y 2 2 b # | Á |A| and so we can use the third factor to gain |A|´N , and sum in λ 2 .
Case 2.2. λ 2 " λ 3 , λ 1 ! λ 2 , and λ 2 ą 2 2j . As in the previous case we use and note that in this case the bounds are The operator norm bound is This is summable over λ 2 ą 2 2j for all θ ď 1{3. Case 3.1. λ 1 " λ 2 , λ 3 ! λ 1 , and λ 1{2 3 Á 2´jλ 1 . In this case, by stationary phase in both variables, the estimates are from which one can calculate that The sum of the operator pieces in this case we denote by T V δ,j . Then This is summable if and only if θ ă 1{3. For θ " 1{3 we see Therefore, in this case we shall need the oscillatory sum lemma with two parameters (Lemma 3.7) when applying complex interpolation. As usual we assume λ 1 " λ 2 . We consider the following function parametrised by the complex number ζ: where γpζq is to be defined later as appropriate. The summation is over all λ 1 and λ 3 satisfying the conditions of this case (Case 3.1). Notice that we necessarily have λ 1 " 1.
We denote by T ζ the associated convolution operator against the Fourier transform of µ ζ . For ζ " 1{3 we require that i.e., γp1{3q " 1. Then by interpolation it suffices to prove with constants uniform in t P R.
In order to prove the first estimate, we need the decay bound (3.5), i.e., But this follows automatically by (5.32), the definition of µ ζ , and the fact that each x ν λ j has its support located at λ.
It remains to prove the L 2 Ñ L 2 estimate by showing uniformly in t.
After substituting λ 1 y 1 Þ Ñ y 1 and λ 1{2 3 y 2 Þ Ñ y 2 in the expression (5.27), we get that the sum on the left hand side of (5.33) is Using the first two factors we can restrict ourselves to the case when |px 1 , x 2 q| ď C for some large constant C.

If we define
A :" λ 1 Q ω px 1 , x 2 , δ 1 q, B :" λ 3 Q β px 1 , x 3 , δ 1 q, then we need to see what happens when either |A| " 1 or |B| " 1. Let us assume that C B is a sufficiently large positive constant.
Subcase |B| ą C B and |A| À 1. In this case we shall use the Hölder variant of the one parameter oscillatory sum lemma (Lemma 3.6) for each fixed λ 3 . We definẽ Hpz 1 , z 2 , z 3 , z 4 ;λ 3 , x 1 , x 3 , δ, 2´jq (5.35) :" where we shall plug in Note that the parameters λ 3 and x 3 are not bounded. Applying Lemma 3.6 we get if we add an appropriate factor to γ (i.e., our γ needs to contain a factor equal to the expression (3.1)). It remains to prove that one can estimate }H} L 8 and the constants C k , k " 1, 2, 3, 4, by |B|´ε B since then we can sum in λ 3 . First let us consider the expression forHpzq. The first three factors within the integral are behaving essentially like q χ 1 py 1 qq χ 1 pz 1´y1´z2 y 2 qq χ 1 pB´z 4 y 1´y 2 2 q. Since we could otherwise use the first factor and estimate by |B|´ε B , we may restrict our discussion to the part of the integration domain where |y 1 | ! |B| ε B . Then we have |B´z 4 y 1 r β | " |B|, and therefore ż |q χ 1 pB´z 4 y 1 r β´y for a constant C. Hence, we have the required bound for }H} L 8 . Next, we see that taking derivatives in z 1 and z 4 , doesn't change in an essential way the actual form ofH since we only obtain polynomial growth in y 1 which can be absorbed by p χ 1 py 1 q, and since derivatives of Schwartz functions are again Schwartz. Therefore, we may estimate C k , k " 1, 4, in the same way as we estimated the original integral.
Permuting the order of the variables z k , k " 1, 2, 3, 4 appropriately, we see from the expressions for C k in Lemma 3.6 that we may now assume z 1 " z 4 " 0. Taking the derivative in z 3 we obtain several terms. We deal with the terms where a y 1 factor appears in the same way as we have dealt with in the previous cases. It remains to deal with the term where y 2 2 factor appears, that iś This integral can be estimated by The key is now to notice that if we fix λ 3 , then λ 1 goes over the set where λ 1 " λ 3 . In particular, since we shall plug in z 3 " λ´ε 1 , we have |z 3 |´1`1 {ε À λ´1`ε 3 . Therefore using the first factor in (5.36) we obtain the bound for (5.36) to be for some different ε. Now one subsitutes t " y 2 2 b # and easily obtains an admissible bound of the form |B|´ε B .
We shall now consider only the part where y 2 ě 0 and z 2 ě 0, as other cases can be treated in the same way. Then substituting t " y 2 2 one gets that the estimate for B z 2H is ĳˇˇˇq χ 1 py 1 q pq χ 1 q 1 p´y 1 r ω´z2 t 1{2 qq χ 1 pB´tb # qˇˇdy 1 dt.
From this form it is obvious that we may now restrict ourselves to the part of the integration domain where |y 1 | ! |B| ε B and |t| " |B| by using the first and the third factor respectively. If we denote this integration domain by U B , then the bound for the C 2 constant in Lemma 3.6 reduces to estimating |z 2 | 1´ϑ ż 1 0 ĳ U Bˇq χ 1 py 1 q pq χ 1 q 1 p´y 1 r ω´s z 2 t 1{2 qq χ 1 pB´tb # qˇˇdy 1 dtds "|z 2 |´ϑ ż z 2 0 ĳ U Bˇq χ 1 py 1 q pq χ 1 q 1 p´y 1 r ω´s t 1{2 qq χ 1 pB´tb # qˇˇdy 1 dtds, where ϑ represents the Hölder exponent. If |z 2 | ď |B|´1 {4 , then we obviously have the required estimate. Therefore, let us assume |z 2 | ą |B|´1 {4 . Then |z 2 |´ϑ ă |B| ϑ{4 and so integration on the domain |s| ď |B|´1 {4 is not a problem. On the other hand, if |s| ą |B|´1 {4 , then |st 1{2 | Á |B| 1{4 by our assumption on the size of t. Thus we may use the Schwartz property of the second factor in the integral and obtain the required estimate. This finishes the proof of the case where |B| " 1 and |A| À 1.
Subcase |B| ą C B and |A| " 1. The preceding argumentation for the estimate of }H} L 8 is also valid in this case since we have not used the second factor, and so we see that we can always estimate the integral appearing in (5.34) by |B|´1 {2 . It remains to gain a decay in |A|.
If we furthermore assume |A| ď |B|, then |B|´1 {2 ď |B|´1 {4 |A|´1 {4 , and so we can sum in both λ 1 and λ 3 . Therefore we may consider |A| ą |B| next, and reduce our problem using the first factor in the integral in (5.34) to the part where |y 1 | ! |A| ε A . Then |z 1´y1 r ω | " |A´y 1 r ω | " |A|, and so we can gain an |A|´ε A using the second factor in the integral, unless |z 2 y 2 | " |A|. But since |z 2 | À 1, we see that |z 2 y 2 | " |A| implies |y 2 | Á |A|, and so we can use finally the third factor where then the y 2 2 term is dominant.
Subcase |B| ď C B and |A| ą C 2 B . We can reduce ourselves to the integration over |y 1 | ! |A| ε A , and so |A´y 1 r ω | " |A|. Therefore, if |y 2 | ď |A| 1{2 , then using the second factor we get that the integral is bounded (up to a constant) by |A|´1. If |y 2 | ą |A| 1{2 , then |B´z 4 y 1 r β´y 2 2 b # | Á |A|, and so we can use the third factor, and sum in both λ 1 and λ 3 (since |B| ă |A|).
Case 3.2. λ 1 " λ 2 , λ 3 ! λ 1 , and λ 1{2 3 ! 2´jλ 1 . Here we have the same bound for the Fourier transform as in the previous case. Therefore from which one can get by interpolation We first consider the case λ 1 ą 2 2j and denote its sum of the operator pieces by T V I,1 δ,j . Then The other case is when 2 j ! λ 1 ď 2 2j and we denote the sum of these operator pieces by T V I,2 δ,j . Then Again, this is summable if and only if θ ă 1{3. For θ " 1{3, we have This operator norm estimate motivates us to define k through 2 k :" λ 2 1 λ´1 3 " 2 2k 1´k3 , where 2 k 1 " λ 1 and 2 k 3 " λ 3 . Our goal is to prove for each k that for some 0 ď ε ă 1{2. Since k ě 2j, we then obtain the desired result by summation in k.
We shall slightly simplify the proof by assuming that λ 1 " λ 2 . Let us consider the following function parametrised by the complex number ζ and k: Let T k ζ denote the associated convolution operator. For ζ " 1{3 we have and so, by interpolation, we need to prove for some 0 ď ε ă 1{2, and with constants uniform in t P R. The first estimate follows right away since x ν λ j have supports located at λ, and therefore by the L 8 estimate for the Fourier transform of ν λ j we have p1`|ξ 3 |q 1{4 . We prove the second estimate by using the oscillatory sum lemma. We need to prove Let us discuss first the index ranges for λ 1 , λ 3 , and 2 k " λ 2 1 λ´1 3 . Recall that we are in the case where 2 j ! λ 1 ď 2 2j and 1 ď λ 3 ! λ 2 1 2´2 j , which implies λ 3 ! λ 1 and 2 2j ! 2 k ď 2 4j . Let us now fix any k satisfying 2 2j ! 2 k ď 2 4j , and let us consider all pλ 1 , λ 3 q such that 2 k " λ 2 1 λ´1 3 . We shall use the oscillatory sum lemma by summing in λ 1 and consider λ 3 " λ 2 1 2´k as a function of λ 1 and k. The conditions for λ 1 are then , which determine an interval of integers I j,k for k 1 (recall λ 1 " 2 k 1 ).
Since using the first two factors we can get a decay in λ 1 , we can restrict ourselves to the case |px 1 , x 2 q| À 1. When |x 3 | " 1, then by using the third factor we can gain a factor λ´1 3 " pλ 2 1 2´kq´1, which sums up to a number of size " 1, by definition of I j,k . Therefore we may and shall assume |x| À 1.
We concentrate on the first three factors within the integral: where r ω , r β , and b # are all converging in C 8 to constant functions of magnitude " 1 when λ 1 Ñ 8, δ Ñ 0, and j Ñ 8. Let us denote by M a large enough positive number. Subcase |B| ą M 3 and |A| ď M . Then because of the first factor we may restrict our discussion to the integration domain where |y 1 | ă |B| 1{3 . There |A´r ω y 1 | ď C|B| 1{3 for some C. We may then furthermore assume |y 2 | ď 2C|B| 1{3 , since otherwise we could use the second factor. Now, if we take M sufficiently large, we have and so we can now use the third factor's Schwartz property to obtain a factor |B|´1, which gives summability in λ 1 .
Subcase |A| ą M . Here we shall need a slightly finer analysis. Note that using the first factor within the integral we can actually reduce ourselves to the integration within the slightly narrower range |y 1 | ă |A| ε 2 10εp2j´kq for some small ε (see (5.37)), and therefore we can also assume using the second factor that y 2 P rA´C|A| ε 2 10εp2j´kq , A`C|A| ε 2 10εp2j´kq s, for some C. Now if |A| ε 2 10εp2j´kq ď 1, we obtain that the bound on the integral is |A| 2ε 2 20εp2j´kq (the area of the surface over which we integrate), and this is summable in λ 1 over the set |A| ε 2 10εp2j´kq ď 1.
Therefore, we assume |A| ε 2 10εp2j´kq ą 1, that is |A| 1{10 ą 2 k´2j . Now, if M is sufficiently large, we then have by the restraint on y 2 that |A| 2 {2 ă y 2 2 ă 2|A| 2 , and hence Therefore if either |B| ! C 1 |A| 2´1{10 or |B| " C 2 |A| 2 , we can simply use the Schwartz property of the third factor within the integral. Let us now assume that B is within the range |B| P rC 1 |A| 2´1{10 , C 2 |A| 2 s. We denote δ A :" |A| ε 2 10εp2j´kq and recall δ A ą 1 and |y 1 | ă δ A ď |A| ε . Using the third factor within the integral we can reduce our problem to when The implicit function theorem implies that for some C 1 . Since δ A ď |A| 1{10 and |B| ą |A| 3{2 , we can conclude that is, y 2 goes over a set with length at most C 1 |A|´1 {2 . This implies that our integral is bounded by C 1 |A|´1 {2`ε , which is summable in λ 1 . Case 4.1. λ 1 " λ 2 " λ 3 and λ 1 ą 2 2j . Here one first applies stationary phase in x 2 . Afterwards, as easily seen and explained in a bit more detail at the end of [18,Chapter 4] (and also in the next section of this article), one gets a phase function in x 1 which has a singularity of Airy-type. By using Lemma 3.1, with condition piiq and M " 3, one gets that the Fourier transform estimate is from which one gets by interpolation The bound on the operator norm is whereT V II δ,j denotes the sum of the associated operator pieces. This is uniformly bounded if and only if θ ď 1{4. For θ " 1{3, we can only sum in the range λ 1 ą 2 6j and so it remains to see what happens when 2 2j ă λ 1 ď 2 6j . We denote the sum of the associated operator pieces for this remaining range by T V II δ,j . We shall deal with this case in the following section. Case 4.2. λ 1 " λ 2 " λ 3 and λ 1 ď 2 2j . Here only the space-side estimate changes and we have By interpolation one can obtain We denote the sum of the associated operator pieces by T V III δ,j . The above estimate is obviously summable if and only if θ ă 1{4. For θ " 1{4 we shall now use complex interpolation, and we deal with θ " 1{3 in the next section. We obviously may assume in this case λ i " 1 for all i " 1, 2, 3.
For simplicity, we assume that λ 1 " λ 2 " λ 3 . We consider the following function parametrised by the complex number ζ: The associated operator is denoted by T ζ . For ζ " 1{4 it holds and so by Stein's interpolation theorem it suffices to prove with constants uniform in t P R. Hereσ " 1{3 since θ " 1{4.
In order to prove the first estimate, we need the decay bound (3.5), i.e., But this follows automatically by (5.39), the definition of µ ζ , and the fact that each x ν λ j has its support located around λ.
We prove the second L 2 Ñ L 2 estimate by using the oscillatory sum lemma [18, Lemma 2.7]. We need to prove uniformly in t. After substituting λ 1 y 1 Þ Ñ y 1 and λ 1{2 1 y 2 Þ Ñ y 2 in the expression (5.27), we get that the sum on the left hand side of (5.41) is We may assume that |x| ď C for some large constant C, since otherwise we could use the first three factors to gain a decay in λ 1 .
If |B| " 1, then since we could otherwise use the first factor, we can assume |y 1 | ! |B| ε B . Then |B´y 1 r β | " |B|, and we can estimate the integral by ż |q χ 1 pB´y 1 r β´y Now one can sum in λ 1 .
Let us now assume |B| ď C B for some large, but fixed constant C B , and |A| " C B . Again, we can assume |y 1 | ! |A| ε A , and so |A´y 1 r ω | " |A|. Therefore if |y 2 | ď |A| 1{2 , then using the second factor we get that the integral is bounded (up to a constant) by |A|´N . If |y 2 | ą |A| 1{2 , then |B´y 1 r β´y 2 2 b # | Á |A| and so we can use the third factor, and sum in λ 1 .
Recall that according to Case 4.2 from the last subsection of the previous section we have and we can assume λ " 1. Furthermore, recall that σ " 1, and that b # py, δ 1 , δ 2 , jq " b 0 py, δq :" b a pδ 1 y 1 , δ 0 δ 2 y 2 q, where b a is the same function as in Subsection 2.4. It is the function b from Subsection 2.4 expressed in adapted coordinates. Recall that βp0q ‰ 0, ωp0q ‰ 0, and b 0 py, 0q " b a p0, 0q " bp0, 0q ‰ 0 for all y.
In terms of s the expression for the Fourier transform of ν λ δ :" ν λ j becomes χ 1 ps 1 s 3 qχ 1 ps 2 s 3 qχ 1 ps 3 q ż e´i λs 3Φ py,δ,σ,s 1 ,s 2 qã py, δqdy, where the amplitudeãpy, δq :" apy, δqχ 1 py 1 qχ 0 py 2 q is a smooth function supported in the sets where x 1 " 1 and |x 2 | À 1 and whose derivatives are uniformly bounded with respect to δ. If we denote then the estimate we need to prove is This estimate corresponds to the estimate of the sum T V II δ,j`T V III δ,j considered in the last subsection of the previous section (Case 4.1 and Case 4.2).
Recall that as a 0 is a classical symbol we can express it as a 0 py 1 , s, δ; λq " a 0 0 py 1 , s, δq`λ´1a 1 0 py 1 , s, δ; λq, where a 0 0 does not depend on λ and a 1 0 has the same properties as a 0 . This induces the decomposition The function ν λ δ,a 1 0 associated to the amplitude a 1 0 has Fourier transform bounded by λ´3 {2 and the L 8 norm on the space side is bounded by λ 3{2 (by the same reasoning as used to obtain (5.39)). From these two bounds we can easily get the required estimate for the operator associated to ν λ δ,a 1 0 . Therefore from now on, by an abuse of notation, we may and shall assume that ν λ δ has an amplitude which does not depend on λ, i.e., x ν λ δ pξq " λ´1 {2 χ 1 ps 1 s 3 qχ 1 ps 2 s 3 qχ 1 ps 3 q ż e´i λs 3 Ψpy 1 ,δ,σ,s 1 ,s 2 q a 0 py 1 , s, δqdy 1 .
The next step is to localise the integration in the above integral to a small neighbourhood of the point where the second derivative vanishes. For δ " 0 this point is Away from this point the estimate for the integral is at worst λ´1, by stationary phase or integration by parts. We now briefly explain how to deal with the part away from x c 1 . Recall from Case 4 in the last subsection of the previous section that the space bound on ν λ δ is 2 j λ " δ´1 0 λ if λ ą 2 2j " δ´2 0 . Now using the results from Subsection 3.3 one can easily see that we can sum absolutely in λ ą δ´2 0 . The case when λ ď δ´2 0 has to be dealt with complex interpolation as in the Case 4.2. from the last subsection of the previous section. In fact, the proof is completely the same, except that one needs to appropriately change γ and the exponent over λ 1 in the expression for µ ζ in (5.40) since θ " 1{3 in this case, and there it was θ " 1{4. One also has a different amplitude a localising near x c 2 in y 2 integration and away from x c 1 in y 1 integration. Hence we may now consider only the part near the critical point x c 1 . Abusing the notation again, we shall denote the part near the critical point x c 1 by ν λ δ too. Following [18] we shall furthermore assume without loss of generality´2 and that in (6.1) we are integrating over an arbitrarily small neighbourhood of x c 1 . Therefore, we now have x c 1 p0, σ, s 2 q " s 1{pn´2q 2 , |Ψ 3 px c 1 pδ, σ, s 2 q, δ, σ, s 1 , s 2 q| " 1, (by implicit function theorem) x c 1 " x c 1 pδ, σ, s 2 q depends smoothly in all of its variables, and Ψ 2 px c 1 pδ, σ, s 2 q, δ, σ, s 1 , s 2 q " 0.
We restate [18,Lemma 5.2.] how to locally develop Ψ at the critical point of Ψ 1 , i.e., the point x c 1 . Its proof is straightforward.
We denote the associated convolution operators, convolving against the Fourier transform of ν λ δ,Ai and ν λ δ,l , by T λ δ,Ai and T λ δ,l . Note that the size of the number M 0 is related to how large of a neighbourhood of 0 the cutoff function χ 0 covers in the first equation of (6.5), and the size of the number M 1 is related to how small of a neighbourhood of 0 we take in (6.4) for the y 1 variable.

Estimates near the Airy cone
From Lemma 3.2, (a), we get that the bound on the Fourier transform of ν λ δ,Ai is λ´5 {6 . Unlike in [18] we shall need to use complex interpolation to be able to estimate the part T λ δ,Ai . The proof here is actually similar to certain cases when h lin pφq ě 2 in [18,Subsection 8.7.1].
We consider the following function parametrised by ζ P C: .
The associated operator acting by convolution against the Fourier transform of µ ζ is denoted by T ζ .
For ζ " 1{3 we see that which means, by interpolation, that it is sufficient to prove with constants uniform in t P R.
In order to prove the first estimate, we need the decay bound (3.5), i.e., This follows right away by using the estimate on the Fourier transform of ν λ δ,Ai , the definition of µ ζ , and the fact that each z ν λ δ,Ai has its support located at pλ, λ, λq. We prove the second L 2 Ñ L 2 estimate by using Lemma 3.5. We need to prove uniformly in t.
We may now also restrict ourselves to the situation where |x| À 1, since otherwise we can get a factor λ´N by integrating by parts. Finally, we change coordinates from s 1 " ps 1 , s 2 q to pz, s 2 q, where z :" λ 2{3 B 1 ps 1 , δ, σq, and so by Lemma 6.1 we have Thus we obtain where by using the expressions for B 0 ps 1 , δ, σq and G 5 ps 2 , δ, σq from Lemma 6.1 one gets whereg is smooth with uniformly bounded derivatives and localising the integration domain to |z| À 1, s 2 " |s 3 | " 1.
We are interested in localising the integration in (6.9) to the place where B 2 s 0Φ " 0 and B 3 s 0Φ ‰ 0. In order to carry out this reduction we need another simple lemma. It will be applied to the first three terms ofΦ pz, s 0 , x, 0, σq "s n 0 G 5 ps n´2 which constitute a polynomial in s 0 whose derivatives have at most two zeros not located at the origin. Note that the last term in the above expression is arbitrarily small. Lemma 6.2. Assume n ě 5 and consider a number x 0 " 1. Let us define a polynomial of the form whose second derivative can be written as If |ε| ď c 1 for a sufficiently small constant c 1 , then |P 1 pxq| " 1 on a neighbourhood of x 0 , which depends on c 1 , but not on ε. On the other hand, if |ε| ą c 2 for some c 2 ą 0 and x 0´ε " 1 (resp. x 0`ε " 1), then |P 3 px 0´ε q| " c 2 1 (resp. |P 3 px 0`ε q| " c 2 1).
Proof. One needs to express b and c in terms of x 0 and ε, after which it is easy to prove the lemma by a straightforward calculation.
From the first conclusion of Lemma 6.2 we see that if the zeros of B 2 s 0Φ which are away from the origin are too close to each other, then we may use stationary phase or integration by parts to obtain a factor of λ´1 {2 (or better) and so the left hand side of (6.6) is absolutely summable. Therefore we may assume that there is at least some distance between the zeros of B 2 s 0Φ . From the second conclusion of Lemma 6.2 we obtain |B 3 s 0Φ | " 1 in a neighbourhood of those zeros within the integration domain (i.e., for those located at " 1).
Therefore, we may now use the implicit function theorem and obtain a parametrisation of a zero of the first three terms of B 2 s 0Φ : n´2 0 x 2 q, which we shall denote by s c 0 px, δ, σq, and assume it is located away from the origin. All such zeros can be treated the same way.
From this expression one sees that we can get an integrable factor of size p1`|s 0 | 2 q´N {2 in the amplitude ofν λ δ,Ai by using integration by parts in s 0 , i.e., we can assuměˇˇB as the unbounded terms in the expression for the s 0 derivative of λΦ 1 pz, λ´1 {3 s 0 , x, δ, σq vanish. Let us denote by E :" λB 0 px, δ, σq, F :" λ 1{3G 1 p0, x, δ, σq, the unbounded terms of the phase. We need to reduce our problem to the case when |E| À 1 and |F | À 1 since then we can simply apply the oscillatory sum lemma.
We begin with the case |F | " 1. Let us consider the z integration. The factor tied with z in the phase is We may therefore assume we are integrating over the area in s 0 where |F´G 3 s 0 | À |F | ε , since otherwise we can use integration by parts in z and gain a factor |F |´ε. In particular, in this case we have |s 0 | " |F |. But then the integrable factor p1`|s 0 | 2 q´N {2 is of size |F |´N and so we obtain the required bound. It remains to consider the case |F | À 1 and |E| " 1. The idea in this case is to use integration by parts in s 3 , which enables us to localise the integration to the set where |λΦ 1 | À |E| ε . If we now take |E| sufficiently large compared to both |A| and |F |, then we see that |λΦ 1 | À |E| ε forces |s 0 | " |E| 1{3 . But this implies that the integrable factor p1`|s 0 | 2 q´N {2 is of size |E|´N {3 , which is what we wanted. We are done with the part near the Airy cone.
From this we easily see that We plan to use complex interpolation and the two parameter oscillatory sum lemma (Lemma 3.7). We consider the following function parametrised by ζ P C: for an appropriate γpζq to be chosen later as in (3.3). We shall also use the one parameter oscillatory sum lemma for certain subcases, and therefore we shall need to add appropriate factors to γ of the form 3.1. The operator associated to µ ζ we denote by T ζ . For ζ " 1{3 we see that which means, by Stein's interpolation theorem, that it is sufficient to prove with constants uniform in t P R.
In order to prove the first estimate we need the decay bound (3.5), i.e., This bound follows easily by the L 8 bound on the Fourier transform of ν λ δ,l , the definition of µ ζ , and the fact that each x ν λ δ,l has its support located at pλ, λ, λq. It remains to prove the L 2 Ñ L 2 estimate p1`itqˇˇ, (6.10) uniformly in t.
6.4 Estimates away from the Airy cone -the estimate for ν E λ,l The function ν E λ,l can be treated similarily as the function ν λ δ,Ai in the case near the Airy cone. We first apply the inverse of the Fourier transform to y ν E λ,l , and then substitute s " ps 1 , s 2 , s 3 q for ξ " pξ 1 , ξ 2 , ξ 3 q. Recall that z " p2´lλq 2{3 B 1 ps 1 , δ, σq and so by Lemma 6.1 one has We plug in this expression for s 1 and also substitute s 0 for s 1{pn´2q 2 . In the end one gets ν E λ,l pxq "λ 3{2 2´N l ż e´i λs 3 Φ 2 pz,s 0 ,x,δ,σq g 2´2 l , p2 l λ´1q 1{3 , z, s 0 , s 3 , δ, σ¯dzds 0 ds 3 , where g 2 is smooth and has all of its derivatives Schwartz in the first variable, and where Φ 2 pz, s 0 , x, δ, σq :"s n 0 G 5 ps n´2 0 , δ, σq´s n´1 The only difference compared to the phase in (6.8) is that there |z| À 1, while here |z| " 1, and instead of the λ´2 {3 factor in front of z in the phase in (6.8), here we have the much larger factor p2 l λ´1q 2{3 . We can now reduce to the situation where |x| À 1. Namely, if |x 1 | " 1 then we integrate by parts in z to gain a factor pλp2 l λ´1q 2{3 q´N . Otherwise if |x 1 | À 1 and |x 2 | " 1, then we integrate by parts in s 0 to obtain a factor λ´N , and if |px 1 , x 2 q| À 1 and |x 3 | " 1, we integrate by parts in s 3 to again gain a factor of λ´N .
Next, recall that p2 l λ´1q 2{3 ! 1. Therefore, we may use again Lemma 6.2 and argue similarily as we did in the case near the Airy cone to reduce ourselves to a small neighbourhood of a point where the second derivative in s 0 of the first three terms of Φ 2 vanishes and |B 3 s 0 Φ 2 | " 1. By the implicit function theorem we may parametrise this point as s c " s c px, δ, σq: The point s c depends smoothly on px, δ, σq.
If we shorten ρ " p2 l λ´1q 2{3 z, then the expression for the first derivative ofΦ 2 at the point s c 0 has the form B s 0Φ 2 pz, s c 0 , x, δ, σq " ps c 0 q 2 bps c 0 , x, δ, σq´ρhps c 0 , x, δ, σq´B 1 px, δ, σq " ρ 2 ps c 0 q 2 bps c 0 , x, δ, σq´ρhps c 0 , x, δ, σq´B 1 px, δ, σq, where hps c 0 , x, δ, σq " 1 and |bps c 0 , x, δ, σq| " 1 for some smooth functions h and b. One can easily check that |B 3 s 0Φ 2 pz, s 0 , x, δ, σq| " 1. Therefore, developing the phaseΦ 2 at the point s c 0 , we may writeΦ 3 pz, s 0 , x, δ, σq "b 0 pρq´"b 1`ρb1 pρq ı s 0`b3 ps 0 , ρqs 3 0 , (6.11) where we suppressed the dependence of b 0 , b 1 ,b 1 , and b 3 on the bounded parameters px, δ, σq. Here we know thatb 1 " 1 and |b 3 | " 1. We may again assume |s 0 | ! 1 as on the other part where |s 0 | Á 1 we could use integration by parts or stationary phase and obtain an expression which when plugged into (6.10) would be absolutely summable in both λ and 2 l . Finally, we develop the term b 0 at 0 and substitute s 0 Þ Ñ λ´1 {3 s 0 . Then and the remaining part of the functionν E λ,l is of the form ν E λ,l pxq "λ 7{6 2´N l ż e´i λs 3Φ3 pz,λ´1 {3 s 0 ,x,δ,σq (6.12) g 3´2 l , p2 l λ´1q 1{3 , z, λ´1 {3 s 0 , s 3 , δ, σ¯dzds 0 ds 3 , where again g 3 has the same properties asg 2 and in the area of integration we have |s 0 | ! λ 1{3 . Now, we first note that we can assume λ´1 {3 2 4l{3 ! 1 since otherwise we can easily sum in both λ and l using the factor 2´N l for a sufficiently large N . Next, we introduce We need to reduce our problem to the situation when A, B, and D are bounded since then we can simply apply the (one parameter) oscillatory sum lemma. When this is the case, the size of the integration domain in (6.12) is not a problem since, if we split the integration domain to the areas where |s 0 | À 2 l{3 and |s 0 | " 2 l{3 , the first part has domain size 2 l{3 , which is admissible, and in the second part the amplitude is integrable in s 0 after using integration by parts.
Then necessarily again |D| " |2 2l{3 |, and this can happen only for Op1q λ's. By (6.12) we have for maybe some different N . The factor λ 7{6 is retained since in this case we can get an integrable factor in s 0 by using integration by parts. After plugging into (6.10) we may sum over the Op1q λ's and then in l.
In order to simplify the situation a bit, we develop the amplitude function g 3 into a sum of tensor products, separating the s 3 variable from the others. It is sufficient to consider each of these tensor product terms separately, and so we can assume without loss of generality that g 3´2 l , p2 l λ´1q 1{3 , z, λ´1 {3 s 0 , s 3 , δ, σ¯"g 3´2 l , p2 l λ´1q 1{3 , z, λ´1 {3 s 0 , δ, σ¯χ 1 ps 3 q, whereg 3 has the same properties as g 3 , except it does not depend on s 3 . Then, after using the Fourier transform in s 3 , the integral in s 0 for the functionν E λ,l is of the form l , p2 l λ´1q 1{3 , z, λ´1 {3 s 0 , δ, σ¯ds 0 , (6.13) where we have suppressed the variables of B 0 and B 1 . One can easily check that this integral is bounded by 2 l{3 by considering the situations where |s 0 | À 2 l{3 and |s 0 | " 2 l{3 separately. This is in fact true if we use any L 1 X L 8 function instead of q χ 1 . If now |B 0´B1 s 0`b3 pλ´1 {3 s 0 , ρqs 3 0 | Á |A| ε , by using the Schwartz property we obtain the bound with a different N , which after plugging into (6.10) is summable. Next, if |B 0´B1 s 0`b3 pλ´1 {3 s 0 , ρqs 3 0 | ! |A| ε , then for some small c ą 0. In particular, the fact |B 0 | " A gives us |B 1 s 0´b3 pλ´1 {3 s 0 , ρqs 3 0 | " |A|.
First we consider integration over the domain |s 0 | À 2 l{3 . In this case we get which in turn implies that |A| À 2 l . But this means we can trade a 2´l factor for a |A|´1 and so we are done. The second part of the integral is where |s 0 | " 2 l{3 , which implies |B 1 s 0´b3 pλ´1 {3 s 0 , ρqs 3 0 | " |s 0 | 3 , i.e., |s 0 | " |A| 1{3 . But as the derivative of is of size |s 0 | 2 " |A| 2{3 , then if we substitute t " B 1 s 0´b3 pλ´1 {3 s 0 , ρqs 3 0 in the integral (6.13), the Jacobian is of size |A|´2 {3 and so the same |A|´2 {3 bound holds for the integral. We are done with the estimate for the function ν E λ,l .
We assume z " 1 since the case z "´1 can be treated in the same way. We can restrict ourselves to the case |x| À 1 arguing in the same way as in the previous case. In fact, we can restrict ourselves to the case |x 1´s0 G 1 ps n´2 0 , δ, σq| ! 1, since otherwise we can use integration by parts in z. From this it follows |x 1 | " 1. Since G 1 ps n´2 0 , 0, σq " 1, we can also localise the integration in s 0 to an arbitrarily small interval containing x 1 . Lemma 6.4. Define the polynomial P ps 0 ; x 1 , x 2 , σq :" pn´1qpn´2q 2 σβp0qs n 0´n pn´2qσβp0qx 1 s n´1 0´x 2 s n´2 0 .
The coefficients of the polynomial in the above lemma come from the first three terms of Φ 4 pz, s 0 , x, 0, σq and from Lemma 6.1. Hence, the above lemma relates the first and the second s 0 derivative of Φ 4 at x 1 .
We need to reduce our problem to the case when A, B, and D are bounded. As here the integral itself is bounded by À 1, we can assume that it is not the case that |A| " |B|, nor |B| " |C|, nor |A| " |C|, since otherwise λ's would go over a finite set, and we could sum in l. Furthermore, as soon as |A| (resp. |B|, or |C|) is greater than 1, then we can automatically assume that |A| " 2 4l (resp. |B| " 2 4l , or |C| " 2 4l ), since otherwise we could trade some factors 2´l N to obtain a factor |A|´ε (resp. |B|´ε, or |D|´ε) giving summability in λ in the expression (6.10).
If |A| " maxt2 4l , |B|, |D|u, then we can easily gain a factor |A|´1 using the Schwartz property of q χ 1 . If |B| " maxt2 4l , |A|, |D|u, then the size of the derivative in v of the function within q χ 1 is B and so we get the bound |B|´1 by substitution. Finally, if |D| " maxt2 4l , |A|, |B|u, we use the van der Corput lemma and obtain the bound |D|´1 {2 .
Subcase maxt|B|, |D|u ě 1. As mentioned before, this actually implies that we can assume maxt|B|, |D|u ě 2 4l . If now |D| " |B|, then since we could otherwise use the factor p1`|v| 2 q´N {2 in (6.15), we can restrain the integration to the domain |v| ! |D| ε . Here the derivative in v of the expression A`Bv`Dv 2`2lf pv, z, 2 l λ´1qv 3 (6.16) inside the Schwartz function q χ 1 in (6.15) is of size |B`cDv| for some |c| " |cpvq| " 1. But recall that |v| " 1 and so |B`cDv| " |Dv| " |D|. This means that substituting the above expression would give a Jacobian of size at most |D|´1.
Next let us consider the case |D| À |B|. If have the slightly stronger estimate |D| À |B| 1´ε , and if we assume |v| ! |B| ε (which we can because of the factor p1`|v| 2 q´N {2 ), then in this case the derivative of (6.16) is of size |B|, which means substituting this expression yields an admissible bound.
Applying the van der Corput lemma we obtain the estimate p|B||D|´1q p|B| 2 |D|´1q´1 {2 " |D|´1 {2 , and so we are done with the case maxt|B|, |D|u ě 1. Subcase maxt|B|, |D|u ď 1 and |A| " 1. Again, we may actually assume |A| " 2 4l . We may also then reduce ourselves to the discussion of the case |v| ! |A| ε , since in the other part of the integration domain we can gain a factor |A|´ε. But then the expression (6.16) is of size " |A| and we can get a factor |A|´1, and hence we are also done with the function ν a I . Estimates for ν a II . Here we have a non-degenerate critical point in z which would give us a factor 2´l {2 . We shall not apply directly the stationary phase method here since in this case some crucial information has been lost while we were deriving the form of the phase in this and the previous subsections. It seems that one cannot prove the required bound for complex interpolation using the information from the form of the phase (6.14). One needs to go back to the phase form in the original coordinates (the one before taking the inverse Fourier transform is (6.1)) and find the critical point in the variables py 1 , s 1 q. This was carried out in [18] (see the discussion before [18,Lemma 5.6.]). Here we only sketch the steps.
The phase in (6.1) is Ψpy 1 , δ, σ, s 1 , s 2 q " s 1 y 1`s2 y 2 1 ωpδ 1 y 1 q`σy n 1 βpδ 1 y 1 q`pδ 0 s 2 q 2 Y 3 pδ 1 y 1 , δ 2 , δ 0 s 2 q, and one integrates in the y 1 variable. The phase function after one applies the Fourier transform is Φ 0 py 1 , s 1 , s 2 , x, δ, σq " Ψpy 1 , δ, σ, s 1 , s 2 q´s 1 x 1´s2 x 2´x3 , (6.17) and one now integrates in the s and y 1 variables, after substituting s for ξ. Recall that s 0 " s Therefore fixing ps 2 , s 3 q is equivalent to fixing pv, s 3 q, and in this case, finding the critical point in py 1 , s 1 q is equivalent to finding the critical point in the py 1 , zq coordinates. Recall that the phase form in (6.14) was derived by using the stationary phase method in y 1 (implicitly done as a part of Lemma 3.2) and changing variables from s " ps 1 , s 2 , s 3 q to pz, v, s 3 q.
In particular, there is no significant difference between v andṽ.
We define A :" λb 0 px, δ, σq, B :" λ 2{3 2 l{3b 1 px, δ, σq, D :" δ 2 0 2 2l{3 λ 1{3 , suppress the variables ofb 2 , and shorten ρ " δ 0 p2 l λ´1q 1{3 . Then λΦ 6 pṽ, x, δ, σq " A`Bṽ`Db 2 pρṽqṽ 2 , and in order to use the oscillatory sum lemma for two parameters we need to reduce the problem to the situation where |A|, |B|, and |D| are of size À 1. In the following we define k through λ " 2 k . First we treat the case when at least two of |A|, |B|, and |D| are comparable. When this is the case, λ can go over only a finite set of indices (the index sets depending on l and other constants), and it remains to sum only in l. This is done in the following way. If |D| Á 1, then we can use van der Corput lemma and obtain a factor |D|´1 {2 , which is summable in l. If |D| ! 1, then the only case remaining is |A| " |B|, and here we can use integration by parts inṽ and obtain a factor |B|´1 which we use to sum in l.
Next, we assume that we have a "strict order" between |A|, |B|, and |D|. First we shall consider the cases when at least two of |A|, |B|, and |D| are greater than 1. If |A| " maxt|B|, |D|u Á 1, we use integration by parts in s 3 and obtain which is summable. Similarly, if |B| " maxt|A|, |D|u Á 1, we can integrate by parts inṽ and obtain the estimate which is summable. And if now |D| " maxt|A|, |B|u Á 1, we use the van der Corput lemma and obtain which is again summable. We are thus reduced to the case where one of |A|, |B|, or |D| are greater than 1, and the other two much smaller.
Case |A| ě 1 and maxt|B|, |D|u ! 1. In this case by using integration by parts in s 3 we can get a factor |A|´1. We use the one dimensional oscillatory sum lemma in l, and afterwards, we can sum in λ using the factor |A|´1 which can be obtained as the bound on the C 1 norm of the function to which we applied the oscillatory sum lemma.
Case |B| ě 1 and maxt|A|, |D|u ! 1. Here we change the summation variables 2 k 1 :" λ 2 2 l , 2 k 2 :" λ, so that we now sum over pk 1 , k 2 q. This change of variables corresponds to the system k 1 " 2k`l, k 2 " k, which has determinant equal to 1, and so the associated linear mapping is a bijection on Z 2 .
Since the summation bounds (without the constraints set by A, B, or D) are 1 ! λ ď δ´6 0 and 1 ! 2 l ! λ, for each fixed k 1 the summation in k 2 is now within the range 2 k 1 {3 ! 2 k 2 ! 2 k 1 {2 , and the summation in k 1 is for 1 ! 2 k 1 ! δ´1 8 0 . The quantities B and D can be rewritten as Now for a fixed k 1 we can apply the one-dimensional oscillatory sum lemma to sum in 2 k 2 since all the terms coupled with 2 k 2 are now within a bounded range. In order to sum in k 1 , one needs to estimate the C 1 norm of the function to which we have applied the oscillatory sum lemma. One can easily see that integrating by parts in s 0 we obtain a factor |B|´1 which in the new indices depends only on 2 k 1 .
Therefore when we fix k 1 , the summation in k 2 goes over an interval of even or uneven integers, depending on the parity of k 1 . Since the summation bounds (without the constraints set by A, B, or D) are 1 ! λ ď δ´6 0 and 1 ! 2 l ! λ, for each k 1 the summation in k 2 is now within the range 2 k 1 {3 ! 2 k 2 ! 2 k 1 , and the summation in k 1 is for 1 ! 2 k 1 ! δ´1 8 0 . The quantities B and D can be rewritten as 1 px, δ, σq, D " δ 2 0 2 k 1 {3 .
For a fixed k 1 we want to apply the oscillatory sum lemma to the summation in k 2 . We remark that formally one should write k 2 as either 2r`1 or 2r (depending on the parity of k 1 ), and then apply the oscillatory sum lemma to the summation in r instead of k 2 .
Formally, one should also add further dummy z i 's for controlling the range of the summation indices.
Since we are in the case where |D| ě 1, |z 1 | ! 1, and |z 2 | ! 1, integrating by parts in s 3 we get that the L 8 estimate is |D|´1. Taking derivatives in z 1 and z 2 does not change the form of the integral in an essential way, and so we can also estimate the L 8 norm of the these derivatives by |D|´1.
Taking the derivative in z 3 a factor of size at most |D| appears, but now we just apply integration by parts in s 3 two times and get that we can estimate the C 1 norm of H by |D|´1.