Sharp $L^p$ estimates for oscillatory integral operators of arbitrary signature

The sharp range of $L^p$-estimates for the class of H\"ormander-type oscillatory integral operators is established in all dimensions under a general signature assumption on the phase. This simultaneously generalises earlier work of the authors and Guth, which treats the maximal signature case, and also work of Stein and Bourgain--Guth, which treats the minimal signature case.


Main results
This article concerns L p bounds for oscillatory integral operators that are natural variable coefficient generalisations of the Fourier extension operator associated to surfaces of nonvanishing Gaussian curvature. To describe the basic setup, for d ≥ 1 let B d denote the unit ball in R d and fix a dimension n ≥ 2. Suppose a ∈ C ∞ c (R n × R n−1 ) is supported in B n × B n−1 and consider a smooth function φ : B n × B n−1 → R which satisfies the following conditions: (H1) rank ∂ 2 ωx φ(x; ω) = n − 1 for all (x; ω) ∈ B n × B n−1 .
A prototypical example is given by the choice of phase in this case (1.2) is the well-known extension operator E ell associated to the elliptic paraboloid (with the additional cutoff function a λ localising the operator to a spatial ball of radius λ): see Example 1.4 below. Operators of the form (1.2) were introduced by Hörmander [24] as a simultaneous generalisation of Fourier extension operators and operators which arise in the Carleson-Sjölin approach to the study of Bochner-Riesz means [17]. The L p theory of Hörmander-type operators has been investigated in a number of articles over the last few decades: see, for instance, [5][6][7]9,12,21,24,26,27,32,40] and references therein. A recent survey of the history of the problem can be found in the introductory section of [21].
It has been observed that, in general, the L p mapping properties of T λ are determined by finer geometric conditions on the phase than (H1) and (H2) above [7,9,27,40]. In particular, in addition to the Hessian in (1.1) having full rank, the behaviour of the operator can often depend on the signature of the matrix. Definition 1.1 Suppose φ is a phase which satisfies (H1) and (H2) above. The eigenvalues of the symmetric matrix ∂ 2 ωω ∂ x φ(x; ω), G(x; ω 0 ) | ω=ω 0 can be defined as continuous functions on B n × B n−1 which are bounded away from 0. The signature of φ is defined to be the quantity sgn(φ) := |σ + − σ − | where σ + and σ − are, respectively, the number of positive and the number of negative eigenvalue functions.
The aim of this article is to prove L p estimates for general Hörmander-type operators, with a range of p determined by the signature of the phase. 2 · sgn(φ) + 2(n + 1) sgn(φ) + 2(n − 1) if n is odd 2 · sgn(φ) + 2n + 3 sgn(φ) + 2n − 1 if n is even The 'extreme' cases of this result already appear in the literature: Minimal σ Stein [32] and Bourgain-Guth [12] showed that all Hörmander-type operators satisfy (1.3) for 2 This yields Theorem 1.2 in the special case where the signature is minimal (so that sgn(φ) = 0 if n is odd and sgn(φ) = 1 if n is even).
Maximal σ. On the other hand, if sgn(φ) = n − 1, then it was shown by Lee [26] for n = 3 (see also [12]) and by Guth and the authors [21] for n ≥ 4 that (1.3) holds for agreeing with the range of exponents in (1.4). Theorem 1.2 gives new bounds away from these extremes. In particular, in all other cases the previous best known range of exponents is (1.5), arising from the work of Stein [32] and Bourgain-Guth [12]. If 0 < sgn(φ) < n − 1 for n odd or 1 < sgn(φ) < n − 1 for n even, then (1.4) provides a strictly larger range than (1.5).

Sharpness
An interesting feature of the result is that it is sharp for specific choices of operator, in the following sense.
These examples are given by essentially taking tensor products of existent examples for the σ = 0 and σ = n − 1 cases, which are due to Bourgain [7,9] and Bourgain-Guth [12] (see also [27,40]). The details are discussed in Sect. 2 below.

Non-sharpness
It is also important to note that there exist examples of operators for which (1.3) is known to hold for a wider range of exponents than (1.4). For instance, the extension operator E ell associated to the elliptic paraboloid, which is a prototypical example in the maximal signature case, has been shown to satisfy a wider range of L p estimates than (1.4) in all but a finite number of dimensions (see [12,19,23,39]). More generally, one may consider extension operators associated to arbitrary paraboloids. Example 1. 4 Given a non-degenerate quadratic form Q : R n−1 → R, define the associated extension operator (1.6) Let 0 ≤ σ ≤ n − 1 be such that n − 1 − σ is even. Affine invariance typically reduces the study of these operators to that of the prototypical examples where Here, writing I d for a d × d identity matrix, the (n − 1) × (n − 1) matrix I n−1,σ is given in block form by In this case, the corresponding phase in (1.6) has signature σ and E σ := E Q σ is the extension operator associated to (a compact piece of) the hyperbolic paraboloid H n−1,σ := {(ω, Q σ (ω)) : ω ∈ R n−1 }.
As discussed in Sect. 4.3 below, at a local level all Hörmander-type operators are smooth perturbations of the prototypical operators E σ .
It is conjectured [33] that the operators E Q (and, in fact, extension operators associated to any surface of non-vanishing Gaussian curvature) are L p (B n−1 ) → L p (R n ) bounded for p > 2 · n n−1 , regardless of the signature. Restriction theory for hyperbolic parabolae involves a number of novel considerations compared with that of the elliptic case, and has been investigated in a variety of works [1,12,18,25,35,38]. There has also been a recent programme [13][14][15][16] to investigate L p -boundedness of extension operators associated to negatively-curved surfaces given by smooth perturbations of the hyperbolic paraboloid H 2,0 from Example 1.4; this turns out to be a rather subtle problem for p < 4.

Relation to other problems
It is well-known that L p estimates for the Fourier extension operators are related to many central questions in harmonic analysis such as the Kakeya conjecture, the Bochner-Riesz conjecture and the local smoothing conjecture for the wave equation (see, for instance, [37]). In the maximal signature case, L p estimates for Hörmander-type operators imply Bochner-Riesz estimates and are further connected to curved variants of the above problems defined over manifolds (see, for instance, [4,30,31]), although some of the implications are not as strong as in the Euclidean setting (see 3 [21, §1.2] for results and further details). For operators with general signature, Theorem 1.2 relates to further generalisations of the Kakeya and local smoothing problems, the latter now defined with respect to a class of Fourier integral operators. The connections with FIO theory are discussed in detail in [3,4]; see [40] and [12] for further details of the underlying Kakeya-type problems.

The rôle of the signature
The proof of Theorem 1.2 follows the argument used to establish the sgn(φ) = n − 1 case from [21], with a number of modifications to take account of the relaxed signature hypothesis. There are two significant points of departure from [21], where the signature plays a critical rôle in the argument (also reflected in the sharp examples in Sects. 2 and 3). In both cases, to illustrate the underlying ideas it suffices only to consider the prototypical operators E Q introduced in Example 1.4.
Partial transverse equidistribution. Transverse equidistribution estimates were introduced in [20] in relation to the elliptic extension operator E ell and play a significant rôle here. In order to describe the setup, it is necessary to briefly review the notion of wave packet decomposition (see Sect. 4.4 for further details). Decompose B n−1 into a family of finitely-overlapping R −1/2 discs θ = B(ω θ , R −1/2 ). By means of a partition of unity, for f : Forming a Fourier series decomposition, one may further decompose f θ = v f θ,v where the frequencies v lie in the lattice R 1/2 Z n−1 and thef θ,v are essentially supported in disjoint balls of radius R 1/2 . The functions E Q f θ,v satisfy the following key properties: (i) On B(0, R), each E Q f θ,v is essentially supported in a tube T θ,v of length R and diameter R 1/2 which is parallel to the normal direction G(ω θ ) := (−∂ ω Q(ω θ ), 1) and has position dictated by v. (ii) The Fourier transform E Q f θ,v has (distributional) support on the cap For general Hörmander-type operators T λ a similar setup holds, with the exception that the tubes T θ,v carrying the functions T λ f θ,v may be curved (see Sect. 4.4).
The incidence geometry of the tubes T θ,v is a major consideration in the L p -theory of Hörmander-type operators. A critical case occurs when f is chosen so that the T θ,v for which E f θ,v / ≡ 0 4 are aligned along a lower dimensional manifold Z (or, more precisely, a lower dimensional algebraic variety) inside B(0, R); indeed, analogous situations appear when considering extremal configurations in classical incidence geometry (see, for instance, [22]), and in fact the (variable coefficient) sharp examples in Sect. 2 exhibit similar structure. Under this hypothesis, by property i) above, E Q f is essentially supported in N R 1/2 Z , the R 1/2 -neighbourhood of Z . It is important to note, however, that the E f θ,v each carry some oscillation. If there is sufficient constructive/destructive interference between the wave packets, then it could be the case that the mass of E Q f is concentrated in a much thinner subset of N R 1/2 Z .
Transverse equidistribution in the elliptic case. On the spatial side (right-hand figure) the wave packets are aligned along a plane V . On the frequency side (left-hand figure), the frequency support is aligned along The signature influences the way in which the wave packets E Q f θ,v can interfere with each other. The reason behind this, as explained below, is that the signature largely determines the relationship between the direction G(ω θ ) of each tube T θ,v on the spatial side and the position of the cap (θ) on the frequency side. In the maximal signature case this relationship, together with the uncertainty principle, ensures that the mass of E Q f cannot concentrate in a thinner neighbourhood of the variety, but must be evenly spread across N R 1/2 Z . For general maximal signature Hörmander-type operators, this property can be formally realised via transverse equidistribution estimates, which roughly take the form 5 (1.7) These estimates play an important rôle in the proof of the maximal signature case of Theorem 1.2 by efficiently relating the wave packet geometry at different scales (see [20,21]). If the maximal signature hypothesis is dropped, however, then (1.7) no longer holds in general. Nevertheless, there is a spectrum of weaker variants of (1.7), involving additional powers of (R/ρ), which do hold in the general case. The relevant strength of these partial transverse equidistribution estimates depends on the signature of the underlying operator. The precise form of these inequalities is discussed in Sect. 5 below. It remains to explain how the signature affects the localisation properties of E Q f . Here an elliptic case is contrasted with a hyperbolic case in R 3 , for wave packets aligned along the subspace V := e 1 ⊥ , the 2-dimensional plane orthogonal to e 1 . In particular, consider the elliptic extension operator E ell in R 3 given by the signature 2 form Q ell (ω) := 1 2 ω 2 1 + ω 2 2 . The situation is depicted in Fig. 1. The directions G(ω θ ) all lie inside V , thus the ω θ lie along the line A ell = {ω 1 = 0} in R 2 . The Fourier support of E ell f thus lies in a union of caps (θ) over θ centred along A ell , so supp E ell f ⊆ N R −1/2 (A ell × R). Owing to this localisation, the uncertainty principle implies that E ell f is essentially constant at scale R 1/2 in the direction transverse (that is, normal) to A ell × R. Crucially, A ell × R = V , thus the mass of E ell f must be equidistributed across the slab 5 Here Failure of transverse equidistribution in the hyperbolic case. On the spatial side (right-hand figure) the wave packets are aligned along the same plane V as in the elliptic case. However, on the frequency side (left-hand figure), the frequency support is aligned along This observation can be used to prove (a suitably rigorous formulation of) the transverse equidistribution estimate (1.7) in this case: see [20]. The above case is somewhat special since V equals A ell × R, the plane along which the Fourier support of E ell f is aligned. For general 2-planes V , the Fourier support is aligned along a (possibly) different 2-plane V . However, a key observation is that, in the elliptic case, V and V only ever differ by a small angle, so again equidistribution of E ell f holds at scale R 1/2 in the direction transverse to V . Moreover, the argument generalises to higher dimensions: if the tubes T θ,v lie along a k-plane V in R n , then E ell f is equidistributed in directions belonging to V ⊥ . Variants also hold when V is replaced by a more general algebraic variety Z (see [20]).
For contrast, now consider the case of the hyperbolic extension operator E hyp in R 3 given by the signature 0 form Q(ω) := ω 1 ω 2 . This situation is depicted in Fig. 2. The ω θ must lie along A hyp = {ω 2 = 0}, so supp E hyp f is contained in N R −1/2 (A hyp × R). This localisation of the Fourier support guarantees that E hyp f is equidistributed at scale R 1/2 in directions transverse to A hyp × R. However, this time, these directions are not transverse to V ; instead, they lie along V . Indeed, not only are A hyp × R and V different, but in fact V ⊥ ⊆ A hyp × R. Consequently, the transverse equidistribution estimate (1.7) no longer holds, and the constructive/destructive interference patterns between the T θ,v can in fact lead to the concentration of the mass of E Q f in a tiny O(1)-neighbourhood of V . The variable coefficient counterexamples of Bourgain [7,9] for Hörmander-type operators of signature 0 exhibit destructive interference of this kind (see [21] for further details).
In the mixed signature case in R n , in general only partial equidistribution occurs as a fusion of the above two situations. Specifically, consider an operator E Q associated to some Q with signature σ and let V be a k-dimensional subspace of R n . In general, if the T θ,v are aligned along V , then the Fourier support of E Q f will be aligned along a k-dimensional The problem is to understand the relationship between V and V . In particular, if V and V are close to one another (that is, the angle between them is small), then this mirrors the situation in the elliptic case and transverse equidistribution holds. If V and V are far from one another (that is, the angle between them is large), then this mirrors the above hyperbolic case and transverse equidistribution can fail. It transpires that, in general, a hybrid of these two situations occurs: a partial transverse equidistribution holds for E Q f inside N R 1/2 V , where the equidistribution property holds only for directions lying in a certain subspace W of V ⊥ . The dimension of W can be bounded as a function of n, k and, importantly, σ . If σ is large then W has large dimension and one is close to guaranteeing the full transverse equidistribution property (1.7) enjoyed by the elliptic case. If σ is small, then the dimension of W is small and only a weak version of (1.7) holds. For instance, if σ ≤ 2k − n − 1, then the subspace W can be zero dimensional, in which case no non-trivial transverse equidistribution estimates hold: see Sect. 5 for details.
Decoupling. Although both elliptic and hyperbolic paraboloids have non-vanishing Gaussian curvature, hyperbolic paraboloids contain linear subspaces. The existence of such subspaces precludes certain bilinear estimates for extension operators associated to hyperbolic paraboloids [25,38] and means only weak 2 -decoupling inequalities hold for such operators [11]. In the present paper, the norm T λ f L p (R n ) is studied via a broad/narrow analysis, as introduced in [12] (see also [20,21]). This analysis involves certain p -decoupling estimates, the strength of which also depends on the signature. Similar observations have appeared previously in [11] and the recent paper [1].
In particular, the broad/narrow analysis requires analysing the so-called "narrow" contributions to T λ f L p (R n ) , which arise when the support of f is localised close to a submanifold of R n−1 . Consequently, one is led to consider certain slices of the (variable) hypersurfaces defined with respect to the phase φ. These contributions are dealt with using a combination of a decoupling inequality and a rescaling argument. The efficiency of the decoupling inequality depends on how curved these slices are, which in turn depends on the signature.
More concretely, for the extension operator E σ f from Example 1.4, the narrow contributions occur when the support of f is localised close to an affine subspace A of R n−1 . In this case, as in the earlier discussion on transverse equidistribution, the Fourier transform E σ f is supported in a neighbourhood of the slice A of H n−1,σ formed by intersecting H n−1,σ with the plane A × R. The favourable situation occurs when A is well-curved, in the sense that the principal curvatures of this surface (viewed as a hypersurface lying in A ×R) are all bounded away from zero. This is always the case for the elliptic paraboloid. For well-curved A one may use the strong decoupling inequalities from [11] (or [8,10] in the elliptic case) to study the narrow contribution. For hyperbolic paraboloids, however, it can happen that a given slice coincides with a linear subspace of H n−1,σ : for instance, H n−1,σ contains the n−1−σ 2 -dimensional linear subspace of all (ξ 1 , . . . , ξ n ) ∈R n satisfying In this case, owing to the lack of curvature, no non-trivial decoupling inequalities exist to control the narrow contribution and, consequently, much poorer estimates hold. In general, to obtain the best possible decoupling inequalities for a slice A , one needs to rely on the principal curvatures of A which are bounded away from zero. The number of these curvatures can be estimated in terms of the signature σ . If σ is large, then typically there will be many large principal curvatures and strong decoupling estimates will hold. If σ is small, then for certain slices there will be few large principal curvatures and only weak decoupling estimates are available. This discussion is made precise in Proposition 7.3 and Corollary 7.7 below.

Methodology: k-broad estimates
As in [20,21], the main ingredient in the proof of Theorem 1.2 is a k-broad estimate.
Theorem 1.5 Let T λ be a Hörmander-type operator of reduced phase φ. For all 2 ≤ k ≤ n and ε > 0 the k-broad estimate For the definition of the k-broad norm, see [20,21]. For technical reasons, the theorem is stated for the slightly restrictive class of reduced phases, which are defined in Sect. 4.3. Once Theorem 1.5 is established, Theorem 1.2 follows by a now-standard argument originating in [12]: see Sect. 8 for further details.
As with Theorem 1.2, certain 'extreme' cases of Theorem 1.5 can be deduced from existent results: the result follows from Stein's oscillatory integral estimate [32].
In all other cases Theorem 1.5 is new. It is also sharp in the sense that the range of p cannot be extended. This can be shown by considering extension operators of the type discussed in Example 1.4 above. The range of L p is then given by testing the estimate against functions formed by tensor products of the standard test functions appearing in, for instance, [38]. The sharpness of Theorem 1.5 is discussed in detail in Sect. 3 below. Theorem 1.5 has a multilinear flavour, and serves as a substitute for the stronger k-linear Conjecture 1.7 below. Definition 1.6 Let 1 ≤ k ≤ n and T = (T 1 , . . . , T k ) be a k-tuple of Hörmander-type operators of the same signature, where T j has associated phase φ j , amplitude a j and generalised Gauss map G j for 1 ≤ j ≤ k. Then T λ is said to be ν-transverse for some 0 < ν ≤ 1 (and |T λ holds whenever p satisfies p ≥p(n, σ, k).
This conjecture is a natural generalisation of a conjecture of Bennett [5] concerning the elliptic case. It formally implies Theorem 1.5 (see [21, §6.2]).

Structure of the article
The layout of the article is as follows: • In Sect. 2 the sharpness of Theorem 1.2 is demonstrated and, in particular, the proof of Proposition 1.3 is presented. • In Sect. 3 the sharpness of Theorem 1.5 and Conjecture 1.7 is discussed.
The remainder of the article deals with the proofs of Theorems 1.2 and 1.5. The presentation is not self-contained. In particular, the sister paper [21], which treats the maximal signature case, is heavily referenced. The argument in [21] is fairly modular in nature and, as discussed in Sect. 1.5, the signature hypothesis plays a crucial rôle only in two places in the argument: (i) The transverse equidistribution estimates, which are used to prove the bounds for the k-broad norms. (ii) The decoupling estimates, used in the passage from k-broad to linear estimates as part of the Bourgain-Guth method [12].
These two isolated steps are treated in detail in the present paper. Many other parts of the proof are merely sketched or even omitted entirely, since they are either minor modifications of or identical to corresponding arguments in [21]. Indeed, once the transverse equidistribution and decoupling theory is established in the general signature setting, the rest of the argument from [21] carries through with only changes to the numerology. In particular, the remainder of the article proceeds as follows: • In Sect. 4 various preliminaries for the proofs of Theorems 1.2 and 1.5 are recalled from the literature. This includes the definition of the k-broad norms and operators of reduced phase. • In Sect. 5 the crucial transverse equidistribution estimates are stated and proved.
• In Sect. 6 there is a brief description of how to adapt the argument from [20,21], using the transverse equidistribution results from the previous section, to prove Theorem 1.5. • In Sect. 7 the relevant decoupling theory is discussed.
• In Sect. 8 Theorem 1.5 is combined with the decoupling estimates from Sect. 7 to complete the proof Theorem 1.2.

Overview
In this section sharp examples for Theorem 1.2 are obtained, thereby proving Proposition 1.3. They arise simply by tensoring existing examples for the extremal cases of minimal and maximal signatures. All of the phases considered below are of the following basic form: given a smooth 1parameter family of symmetric matrices A : In order for this phase function to satisfy the conditions (H1) and (H2) from the introduction, the component-wise derivative A of A must be invertible on a neighbourhood of the origin. In this case, the signature of the phase function φ corresponds to the common signature of the matrices A (x n ) for x n near 0.
In the forthcoming examples T λ is taken to be a Hörmander-type operator defined with respect to the phase φ λ for some φ as in (2.1), and an amplitude with sufficiently small support so that the conditions (H1) and (H2) are satisfied. The analysis pivots on finding suitable choices of A and test functions f so that T λ f is highly concentrated near a low degree algebraic variety. In particular, the varieties in question will be hyperbolic paraboloids of the form (2.2) Note that each Z d is of dimension m d := d+2 2 . This corresponds to the minimal dimension for 'Kakeya sets of curves' in R d : see [7,12,40]. For further details on the rôle of algebraic varieties in the study of oscillatory integral operators see, for instance, the introductory discussions in [20] or [21].

Hyperbolic example
The first example is due to Bourgain [7] (see also [9]) and corresponds to the minimal signature case.
Near the origin the derivative matrix H d (t) is a perturbation of and is therefore invertible with signature 0. Note that (2.3) corresponds to the matrix I d−1,σ from Example 1.4 after a coordinate rotation.
Taking A = H d , let T λ hyp be a Hörmander-type operator with phase φ λ for φ as defined in (2.1). A key observation of Bourgain [7] is that there exists 7 a smooth function h : R d−1 → C satisfying: where the variety Z d is as in (2.2).
This bound follows from a simple stationary phase computation. In addition to [7,9], see the expositions in [21,31,40] for further details.

Elliptic example
The second example is due to Bourgain-Guth [12] and corresponds to the maximal signature case.
where the * indicates that the final (t) block appears if and only if d is even. Near the origin the derivative matrix E n is a perturbation of the identity and is therefore invertible with maximal signature d − 1.
Taking A = E d , let T λ ell be a Hörmander-type operator with phase φ λ for φ as defined in (2.1). Roughly speaking, in [12] it is shown that there exists a smooth function g : R d−1 → C satisfying: where the variety Z d is as in (2.2) and m d = dim Z d = d+2 2 . The estimate (2.5) is not quite precise since the example in [12] is randomised and the pointwise bound (2.5) holds only in expectation. However, there exists a function g for which the weaker substitute does hold, and this suffices for the present purpose. In addition to [12], see the exposition in [21] for further details.

Tensored examples
To prove Proposition 1.3, the linear estimates are tested against examples formed by tensoring the hyperbolic and elliptic examples described above. To this end, fix 1 ≤ σ ≤ n − 1 with n − 1 − σ even and let Taking A = A n,σ , let T λ be a Hörmander-type operator with phase φ λ for φ as defined in (2.1). Let f denote the tensor product f : If the amplitudes are suitably defined, then it follows that where T λ hyp is defined with respect to H n−σ and T λ ell is defined with respect to E σ +1 . Suppose that for all ε > 0 the estimate holds for T λ and f as above, uniformly in λ. The construction ensures that f L p (B n−1 ) ∼ 1 and so Thus, to obtain the desired p constraints, the problem is to bound the left-hand side of (2.7) from below. Before proceeding, it is helpful to make a few simple geometric observations regarding the varieties Z d . Given By (2.6) and Fubini's theorem, At the expense of an inequality, one may restrict the ell has suitably small x d -support, then the right-hand norm coincides with the global L p -norm and one may apply (2.5 ) to conclude that In order for (2.7) to hold uniformly in λ, the exponent on the right-hand side of (2.9) must be non-positive. Note that the parities of n and σ + 1 agree and so Thus, a little algebra shows that the non-positivity of the right-hand exponent in (2.9) is equivalent to which yields the desired condition (1.4) after rearranging.

Necessary conditions: multilinear bounds
Here examples of Hörmander-type operators are constructed which demonstrate that the range of exponents in Conjecture 1.7 cannot be extended. The proof of Proposition 3.1 can be slightly modified to demonstrate the sharpness of Theorem 1.5, up to ε-loss. The details of this simple modification are omitted; see [20] for a discussion of the elliptic case.
Similarly to the examples for Theorem 1.2 discussed in the previous section, the sharpness of the multilinear estimates may be deduced by tensoring appropriate examples from extremal signature regimes. In the multilinear case, however, one may simply work with the prototypical extension operators associated to hyperbolic parabolae from Example 1.4.

Hyperbolic example
The first example exploits the fact that hyperbolic parabolae contain affine subspaces and is a direct generalisation of the bilinear example from [38]. The example is applied in the extreme case where the signature of the underlying quadratic form is zero. In particular, let d ∈ N be odd and consider the zero signature quadratic form Note that this agrees with the form Q 0 from Example 1.4 after an orthogonal coordinate transformation.

Elliptic example
The second example corresponds to the sharp example for L 2 -based multilinear restriction for the elliptic paraboloid. It is a direct generalisation of the bilinear example described, for instance, in [36]. This example will be applied in both elliptic and hyperbolic cases, but nevertheless is referred to as the elliptic example to distinguish it from the hyperbolic example described above.
be a quadratic form in d − 1 variables of signature σ . In contrast with the hyperbolic case, here the choice of σ is not relevant to the numerology arising from the elliptic example. Note that this form agrees with the form Q σ from Example 1.4 after an orthogonal coordinate transformation.
For C ≥ 1 a suitably large dimensional constant and given λ ≥ 1, define V to be a maximal is given by a fixed function which is non-negative, supported in the unit ball and equal to 1 in a neighbourhood of the origin; see Fig. 3. For any such G(d, ), using Plancherel, one may bound On the other hand, (non)-stationary phase shows that, on B(0, λ), the function E Q g j,v is rapidly decaying away from the 'tube' where c > 0 is a suitable choice of small dimensional constant, and satisfies In particular, provided C is chosen appropriately in the definition of V , it follows that The tubes in each family (T j,v ) v∈V are pairwise disjoint and their union can be thought of as the intersection of a fixed (that is, independent of j) -plane slab formed around V of thickness λ 1/2 with B(0, λ). More precisely, using the transversality condition in particular, the left-hand set contains a union of roughly λ /2 disjoint balls in R d of radius roughly λ 1/2 .

Tensored examples
To prove Proposition 3.1, the multilinear estimates are tested against examples formed by tensoring the hyperbolic and elliptic examples described above. To this end, fix 1 ≤ σ ≤ n−1 with n − 1 − σ even and let be a quadratic form in n − 1 variables of signature σ . The multilinear examples subsequently constructed will prove the sharpness of Conjecture 1.7 when tested against the extension operator E Q , irrespective of the level k of multilinearity. Fix d satisfying and split the variables ω and x by writing The quadratic form is decomposed accordingly by writing The condition (3.9) implies that Q has zero signature, and therefore it makes sense to consider the hyperbolic examples h(d, ) defined in Sect. 3.1 applied to this form. Note that, where E Q , E Q and E Q are the extension operators associated to the respective quadratic forms, as defined in Example 1.4. Fix 1 ≤ k ≤ n and for 1 ≤ ≤ k satisfying and λ ≥ 1 a large parameter let be hyperbolic and elliptic examples as defined above. For every level of multilinearity k, appropriate d and will be chosen so that tensor products of functions from H( , d) and G(k − +1, n −d +1) demonstrate the sharpness of Conjecture 1.7 for this k. The constraints on the parameters in (3.10) are important: • The first constraint is required in order to carry out the construction of the hyperbolic example H( , d) from Sect. 3.1. Combined with (3.9), it implies that ≤ (n−σ −1)/2+1, which corresponds to the fact that maximal linear subspaces contained in the graph Q of the form (3.8) have dimension (n −σ −1)/2. Furthermore, this constraint will account for the transition in the numerology of Proposition 3.1 at k = (n − σ + 1)/2. • The second constraint is required in order to carry out the construction of the elliptic example G(k − + 1, n − d + 1) from Sect. 3.2. This constraint will account for the transition in the numerology of Proposition 3.1 at k = (n + σ + 1)/2.
Define k functions In order to apply these examples in the proof of Proposition 3.1, the supports of the h i and g j functions must satisfy the transversality hypothesis. Since the supports of these functions are well-separated, it suffices to check the transversality condition at the centres of the supports only. Given ω = (ω 1 , . . . , The value of q(n, k, ) is obtained by substituting the corresponding value into the formula in (3.12). In all cases, and d are chosen so as to satisfy (3.9) and (3.10) and note that Computing the values of the Gauss map applied to these vectors, forming the relevant matrix and rearranging the rows, it suffices to show that the n × k matrix 8 are the matrices whose columns are formed by the vectors (−a 1 , . . . , −a l−1 ) and (−b 1 , . . . , −b k− ), respectively. The desired rank condition is immediate from the choices of a i and b j and, in particular, (3.1) and (3.4).
For now, suppose that the k-linear inequality holds uniformly in λ. Presently, it is shown that, for appropriately chosen d, this forces Plugging the optimal values of into the formula for q(n, k, ) yields the desired range of p described in Proposition 3.1. In particular, to maximise q(n, k, ) one should choose as large as possible, under the condition that (3.9) and (3.10) should hold for some d. The correct choices of and d, which depend on the k regime, are tabulated in Fig. 4.
The first step is to obtain a lower bound for the expression on the left-hand side of (3.11). One may write the function appearing in the p-norm as a product of two functions Apply (3.3) at multilinearity and dimension d to each factor in H to deduce that (3.13) On the other hand, apply (3.6) at multilinearity k − + 1 and dimension n − d + 1 to each factor in G to deduce that using the fact that the tubes T j,v are pairwise disjoint as v varies over V . Combining these observations, where: • the λ −(n−1)/2 factor is the product of the coefficients from (3.13) and (3.14), • the λ (d−1)/2 p factor corresponds to the L p x -norm of the characteristic function in (3.13), • the λ (n+k−d− +2)/2 p factor arises from (3.14) owing to (3.7).
The right-hand side of (3.11) is now bounded from above. In particular, by exploiting the tensor structure and applying the bounds (3.2) and (3.5), Since the inequality is assumed to hold for all large λ, this forces the condition described in (3.12).

Overview
The remainder of the article deals with the proof of the k-broad estimates from Theorem 1.5 and the passage from k-broad to linear estimates used to establish Theorem 1.2. In this section a variety of definitions and basic results are recalled from the literature (primarily [20] and [21]), which will be used throughout the remainder of the paper. In particular: • In Sect. 4.2 the underlying geometry of Hörmander-type operators is discussed.
• In Sect. 4.3 the notation of a reduced phase is introduced, and various technical reductions are described. • In Sect. 4.4 the wave packet decomposition for Hörmander-type operators is recounted.
The treatment here is rather brief and readers new to these concepts are encouraged to consult [20] or [21] for further details.

Variable coefficient operators: basic geometry
Consider a smooth phase function φ : B n × B n−1 → R satisfying H1) and H2) from the introduction. Fixingx ∈ B n , the condition H1) implies that the mapping is a (compact piece of) a smooth hypersurface in R n . Furthermore, the condition H2) implies that for eachx the corresponding hypersurface has non-vanishing Gaussian curvature. After further localisation and a suitable coordinate transformation, the condition H1) ensures the existence of a local diffeomorphism x on R n−1 such that In particular, the map x corresponds to a graph reparametrisation of the hypersurface x , with graphing function Throughout the remainder of the paper, it is always assumed that any Hörmander-type operator with phase φ is suitably localised and that coordinates are chosen so that the above functions are defined globally on the support of the amplitude.
In view of the rescaled phase and amplitude functions appearing in the definition of T λ , given λ ≥ 1 andx ∈ B(0, λ) define λ x := x/λ , λ x := x/λ and h λ x := hx /λ . Similarly, define the rescaled generalised Gauss map taking G to be as defined in condition H2) from the introduction. Since the mapping λ x corresponds only to a change of coordinates, it follows that G λ (x; ω) is parallel to the vector

Reductions
To prove Theorem 1.2 for all Hörmander-type operators with phases of a given signature σ , one needs only to consider operators which are perturbations of the prototypical extension operators E σ from Example 1.4. In particular, recall that the Hörmander-type operators under consideration are those of the form where the phase φ satisfies the general conditions H1) and H2). For any 0 ≤ σ ≤ n − 1 with n − 1 − σ even, let I n−1,σ denote the (n − 1) × (n − 1) matrix of signature σ from Example 1.4.
Here h and E are smooth functions, h is quadratic in ω and E is quadratic in x and ω. 9 Furthermore, letting c ex > 0 be a small constant, which may depend on the admissible parameters n, p and ε, one may assume that the phase function φ satisfies for all (x; ω) ∈ X × and 1 ≤ k ≤ n. In addition, for some large integer N ex ∈ N, which can be chosen to depend on n, p and ε. If |α| ≥ 2, then the lower bound on |β| can be relaxed to 0 in (4.1). Finally, it may be assumed that the amplitude a satisfies The proof of Lemma 4.1 is a simple adaptation of the proofs of Lemma 4.1 and Lemma 4.3 in [21] (which describe the case σ = n − 1) and is thus omitted here.

Definition 4.2
Henceforth c ex > 0 and N ex ∈ N are assumed to be fixed constants (which are allowed to depend only on admissible parameters), chosen to satisfy the requirements of the forthcoming arguments. A phase of signature σ satisfying the properties of Lemma 4.1 for this choice of σ , c ex and N ex is said to be reduced. 9 Explicitly, if (α, β) ∈ N 0 × N n−1 0 is a pair of multi-indices, then: x E(0; ω) = 0 whenever ω ∈ and |α| ≤ 1.

Wave packet decomposition
The wave packet decomposition from [21] is now reviewed and some notation is established. All statements in this subsection are proved in [21]. Throughout the following sections ε > 0 is a fixed small parameter and δ > 0 is a tiny number satisfying 10 δ ε and δ ∼ ε 1. For any spatial parameter satisfying 1 R λ, a wave packet decomposition at scale R is carried out as follows. Cover B n−1 by finitelyoverlapping balls θ of radius R −1/2 and let ψ θ be a smooth partition of unity adapted to this cover. These θ are referred to as R −1/2 -caps. Cover R n−1 by finitely-overlapping balls of radius C R (1+δ)/2 centred on points belonging to the lattice R (1+δ)/2 Z n−1 . By Poisson summation one may find a bump function adapted to B(0, R (1+δ)/2 ) so that the functions  (θ, v). Thus, for f : R n−1 → C with support in B n−1 and belonging to some suitable a priori class one has For each R −1/2 -cap θ let ω θ ∈ B n−1 denote its centre. Choose a real-valued smooth functioñ ψ so that the functionψ θ (ω) :=ψ(R 1/2 (ω−ω θ )) is supported in θ andψ θ (ω) = 1 whenever ω belongs to a cR −1/2 neighbourhood of the support of ψ θ for some small constant c > 0.
Finally, define whilst the functions f θ,v are also almost orthogonal: if S ⊆ T, then A precise description of the rapidly decaying term RapDec(R), frequently used in forthcoming sections, is inserted here.

Definition 4.3
The notation RapDec(R) is used to denote any quantity C R which is rapidly decaying in R. More precisely, where N ex is the large integer appearing in the definition of reduced phase from Sect. 4.3. Note that N ex may be chosen as large as desired, under the condition that it depends only on n and ε. 10 For A, B ≥ 0 the notation A B or B A is used to denote that A is 'much smaller' than B; a more precise interpretation of this is that A ≤ C −1 ε B for some constant C ε ≥ 1 which can be chosen to be large depending on n and ε.
Let T λ be an operator with reduced phase φ and amplitude a supported in X × as in Lemma 4.1. For (θ, v) ∈ T, within B(0, R) the function T λ f θ,v is essentially supported inside a curved R 1/2+δ -tube T θ,v determined by φ, θ and v. More precisely, there exists a curve This curve λ θ,v forms the core of the tube T θ,v . In particular, for the following concentration estimate holds.
The geometry of the core curve of T θ,v is related to the generalised Gauss map G λ associated to the operator T λ : the tangent line T λ , giving rise to an extension operator, then the T θ,v are straight tubes.

Overview
In this section the key tool required for the proof of Theorem 1.5 is introduced and proved. This is a 'partial' transverse equidistribution estimate, which bounds the L 2 norm of T λ g under certain geometric hypotheses on the wave packets of g: see Lemma 5.4 below. This lemma generalises the transverse equidistribution estimates for the elliptic case in [20] and [21]. It is a key step in the argument where the signature sgn(φ) plays a rôle. Indeed, once Lemma 5.4 is in place, the remainder of the proof of Theorem 1.5 follows as in the elliptic case, with only minor numerological changes, as discussed in the following section.

Tangential wave packets and transverse equidistribution
Throughout this section let T λ be a Hörmander-type operator with reduced phase φ of signature σ and for some 1 R λ define the (curved) tubes T θ,v as in Sect. 4.4. Here a special situation is considered where T λ g is made up of a sum of wave packets which are tangential to some algebraic variety, in a sense described below. To begin, the relevant algebraic preliminaries are recounted. Definition 5.1 Given any collection of polynomials P 1 , . . . , P n−m : R n → R, the common zero set Z (P 1 , . . . , P n−m ) := x ∈ R n : P 1 (x) = · · · = P n−m (x) = 0 will be referred to as a variety. 11 Given a variety Z = Z (P 1 , . . . , P n−m ), define its (maximum) degree to be the number deg Z := max{deg P 1 , . . . , deg P n−m }.
It will often be convenient to work with varieties which satisfy the additional property that n−m j=1 ∇ P j (z) = 0 for all z ∈ Z = Z (P 1 , . . . , P n−m ). (5.1) In this case the zero set forms a smooth m-dimensional submanifold of R n with a (classical) tangent space T z Z at every point z ∈ Z . A variety Z which satisfies (5.1) is said to be an m-dimensional transverse complete intersection.
Let δ m denote a small parameter satisfying 0 < δ δ m 1 (here δ is the same parameter as that which appears in the definition of the wave packets).

Definition 5.2 Suppose
Herec tang > 0 (respectively,C tang ≥ 1) is a dimensional constant, chosen to be sufficiently small (respectively, large) for the purposes of the following arguments.

Definition 5.3 If S ⊆ T, then f is said to be concentrated on wave packets from
One wishes to study functions concentrated on wave packets from the collection Let B ⊆ R n be a fixed ball of radius R 1/2+δ m with centrex ∈ B(0, R). Throughout this section the analysis will be essentially confined to a spatially localised operator η B · T λ g where η B is a suitable choice of Schwartz function concentrated on B. It is remarked that, for any (θ, v) ∈ T, a stationary phase argument shows that the Fourier transform of η B · T λ g θ,v is concentrated near the surface Now consider the refined set of wave packets 11 The ideal generated by the P j is not required to be irreducible.
Let R 1/2 < ρ R and throughout this subsection let τ ⊂ R n−1 be a fixed cap of radius O(ρ −1/2+δ m ) centred at a point in B n−1 . Now define With these definitions, the key partial transverse equidistribution result is as follows.

Lemma 5.4
With the above setup, if dim Z = m and deg Z ε 1 and g is concentrated on wave packets from T Z ,B,τ , then The remainder of the section is dedicated to the proof of this lemma. For a discussion of the philosophy and heuristics behind estimates of this kind, see [20, §6] or [21, §8], as well as Sect. 1.5. It is noted that in the maximum signature case μ(n, n − 1, m) = n − m for all 1 ≤ m ≤ n, so this lemma recovers the previous elliptic case result in [21,Lemma 8.4] (see also [20,Lemma 6.2]). On the other hand, in the range n+σ +1 2 ≤ m ≤ n where μ(n, σ, m) = 0 the result follows from a classical L 2 bound of Hörmander and does not depend on any geometric considerations regarding the wave packets.

Wave packets tangential to linear subspaces
Here, as a step towards Lemma 5.4, transverse equidistribution estimates are proven for functions concentrated on wave packets tangential to some fixed linear subspace V ⊆ R n . As before, let B be a ball of radius R 1/2+δ m with centrex ∈ R n and define is the cap concentric to τ but with 1/10th of the radius. The key estimate is the following. (3) If g is concentrated on wave packets from T V ,B,τ , is any plane parallel to V and x 0 ∈ ∩ B, then the inequalitŷ

Lemma 5.5 If V ⊆ R n is a linear subspace, then there exists a linear subspace V with the following properties:
holds up to the inclusion of a RapDec(R) g L 2 (B n−1 ) term on the right-hand side.
Proof of Lemma 5.5 Many of the steps of the proof are similar to the proof of Lemma 8.7 from [21], although the construction of V itself is different from that used in the positive-definite case.

Constructing the subspace V aux
The first step in the argument is to construct an auxiliary space V aux ; the desired subspace V is then obtained by rotating V aux . One may assume without loss of generality that since otherwise the family of tubes T V ,B is empty and there is nothing to prove. Consider the horizontal slice where the orthogonal complements are taken inside R n−1 . The following example partially motivates the above definition.

Example 5.6
Consider the prototypical case of the extension operator E σ from Example 1.4.
Here the unnormalised Gauss map G 0 is an affine map, and so is an affine subspace. A simple computation shows that A ω is parallel to V sl .

Dimension bounds for V aux
The next step of the proof is to show that the auxiliary space satisfies the dimension bounds described in part 1) of the lemma. It is clear that dim V aux ≤ n − dim V since V aux ⊆ V ⊥ sl and the latter subspace has dimension equal to from which the estimate dim V aux ≥ n − 2 dim V + 1 directly follows. It thus suffices to prove that dim V aux ≥ (n + σ + 1)/2 − dim V , or equivalently Fix an orthonormal basis The angle condition (5.3) implies that {N 1 , . . . , N n−dim V } is a linearly independent set of vectors, where N k = (N k , N k,n ) ∈ R n−1 × R and, clearly, 12 On the other hand, Combining the observations of the previous paragraph, and, consequently, Note that and, since matrix rank is preserved under elementary column operations, For the right-hand block, the number of linearly independent columns can be at most the number of non-zero rows, which is equal to (n − 1 − σ )/2. Altogether, this bounds the matrix rank above by as desired. 12 To establish the desired dimensional bounds, the only required property of the vectors N k is that they form a basis of V ⊥ sl , not that they arise from a basis for V ⊥ in the above manner. However, the vectors N k are introduced as they will be used in subsequent parts of the proof.

Constructing the subspace V
One may assume without loss of generality that S ω ∩ τ = ∅ where since otherwise the family of tubes T V ,B,τ is empty and there is nothing to prove. Recalling (5.3), it follows that S ω is a smooth surface in R n−1 of dimension dim V − 1; indeed, this can be verified as a simple calculus exercise, but it is also treated explicitly as Claim 1 in the proof of Lemma 8.7 from [21] (the claim is stated in the positive-definite case, but the argument does not depend on the signature). For notational convenience, write for the functions as defined in Sect. 4.2. Consider the surface given by the diffeomorphic image of S ω under the map . Fix some u 0 ∈ S u ∩ −1 (τ ) and let A u denote the tangent plane to S u at u 0 . Here, the tangent plane is interpreted as a (dim V −1)dimensional affine subspace of R n−1 through u 0 . Now define A ξ := A u × R ⊆ R n , so that dim A ξ = dim V , and let V u and V ξ be the linear subspaces parallel to A u and A ξ , respectively. The spaces V sl ⊂ R n−1 and V u ⊂ R n−1 both have dimension dim V − 1. Moreover, the localisation to the cap τ and ball B implies that V sl and V u are close to one another in the following sense.
Claim Let c ex be the constant defined in Sect. 4.3. Then The proof of the claim is temporarily postponed. Assuming its validity, it follows that there exists a choice of O V ∈ SO(n − 1, R) mapping V sl to V u which satisfies Applying the Gram-Schmidt process, one may further assume {v 1 , . . . , v dim V −1 } is orthonormal, at the expense of a larger implied constant. A rotation O V with the desired properties is given by stipulating that it maps v * k to v k for 1 ≤ k ≤ n. Fixing a rotation O V which satisfies the above property, In particular, the space V inherits the dimension bounds from V aux and therefore the dimension condition (1) from the lemma is immediately verified.
It remains to prove the claim. The argument is almost identical to that used to prove Claim

Proof of Claim
Fixing v * ∈ V sl ∩ S n−2 , elementary linear geometry considerations reduce the problem to showing Forh as in (5.4), recall that u → (u,h(u)) is a graph parametrisation of the surface λ x from Sect. 4.2 and u → G λ 0 (x; (u)) is the unnormalised Gauss map associated to this parametrisation. It follows that Differentiating the defining equations in the above expression and recalling that u 0 is a fixed point featured in the definition of A u , one deduces that a basis for V ⊥ u is given by Lemma 4.1 together with some calculus (see [21,Lemma 4.5] for a similar computation) imply that Since

Verifying the transversality condition in (2)
Provided c ex is chosen to be sufficiently small, the transversality condition holds for the subspace V . To see this, first consider the auxiliary space V aux . By elementary geometric considerations, where the latter inequality is by (5.3); this computation is discussed in detail in [20,Sublemma 6.6] and is represented diagrammatically in Fig. 5. The above inequality implies that V and V aux are quantitatively transverse, since V aux is a subspace of V ⊥ sl . It remains to pass from the auxiliary space V aux to V .

Verifying the transverse equidistribution estimate in (3)
The remaining steps of the proof closely follow the argument used to prove Lemma 8.7 of [21]. The localisation to τ implies that the tangent space A u is a good approximation for the surface S u . In particular, the key observation is that As in Sect. 4.4, here ω θ ∈ B n−1 denotes the centre of the cap θ whilst is the parametrisation of the smooth hypersurface from (5.2). The inequality (5.6) follows from the proof of Claim 3 in the proof of Lemma 8.7 of [21]. Since V ξ is the linear subspace parallel to the affine subspace A ξ , the above inequality implies that proj V ⊥ ξ ξ θ lies in some fixed ball of radius O(R −1/2+δ m ) whenever (θ, v) ∈ T V ,B,τ .
As in [21] and [20], the desired transverse equidistribution estimate (3) follows as a consequence of the localisation of the proj V ⊥ ξ ξ θ described above. Indeed, since each η B · T λ g θ,v is essentially Fourier supported in a small ball around ξ θ , this implies the projection of the Fourier support of η B · T λ g θ,v onto V ⊥ ξ is also localised to a O(R −1/2+δ m )-ball. The transverse equidistribution property now follows as a manifestation of the uncertainty principle (see, in particular, [21,Lemma 8.5]). The reader is referred to [21] for the full details.

The proof of the transverse equidistribution estimate
Using ideas from [20,21], one may easily pass from Lemma 5.5 to Lemma 5.4. Much of the proof is essentially identical to the proof of [21,Lemma 8.4] therefore only a sketch of the argument is provided.
Consider Z , B, τ and g as in the statement of Lemma 5.4. It may be assumed that g is concentrated on those wave packets (θ, v) from T Z ,B,τ for which T θ,v intersects N R 1/2+δm (Z )∩ B, as for all other (θ, v) the function |T λ θ,v g| is very small on N ρ 1/2+δm (Z ) ∩ B. By the R 1/2+δ m -tangent condition, it follows that there exists z ∈ Z ∩ 2B such that for all wave packets (θ, v) upon which g is concentrated. This implies that g is concentrated on wave packets T V ,B,τ , as defined in Sect. 5.2. By Lemma 5.5 there exists a linear subspace V ⊆ R n satisfying 7) (v, v ) ≥ 2c trans for all non-zero vectors v ∈ V and v ∈ V and the transverse equidistribution estimatê for every affine subspace parallel to V and x 0 ∈ B. In contrast to the positive-definite case in [21], where one may ensure that dim V + dim V = n, only the generally weaker dimension bounds (5.7) hold here. However, the subspace V := V ⊕(V +V ) ⊥ satisfies dim V +dim V = n and the quantitative transversality condition (v, v) ≥ 2c trans for all non-zero vectors v ∈ V and v ∈ V , as well the transverse equidistribution estimatê for every affine subspace parallel to V and x 0 ∈ ∩ B, which follows from (5.8) by Fubini and Hölder's inequality (as well as the fact that δ δ m ). Following closely the proof of Lemma 8.4 in [21], one may further prove that for each z ∈ Z ∩ 2B the pair T z Z , V satisfies the quantitative transversality condition Lemma 8.13 in [21] implies that for every plane parallel to V . As ∩ Z is a complete transverse intersection of dimension dim Z + dim V − n = m − dim V , it follows by Wongkew's theorem [41] that ∩ N ρ 1/2+δm (Z ) ∩ B can be covered by balls of radius ρ 1/2+δ m . Applying the estimate (5.9) in each of these balls and summing, one obtainsˆ for all planes parallel to V . Integrating over all such planes and applying Hölder's inequality, one deduces that . (5.10) By Hörmander's L 2 bound [24] (see also [34,Chapter IX] or [21, Lemma 5.5]), . (5.11) Substituting this into (5.10), the desired estimate in Lemma 5.4 follows provided It remains to show (5.12) holds. In view of (5.7), this would follow from By the initial reduction at the beginning of the subsection, dim V ≤ m ≤ (n + σ + 1)/2. If 0 ≤ dim V ≤ (n − σ + 1)/2, then μ(n, σ, dim V ) = n − 2 dim V + 1 and On the other hand, if (n − σ This concludes the proof of Lemma 5.4.
holds for all translates T λ of Hörmander-type operators with reduced phase of signature σ , whenever f is concentrated on wave packets from T Z and Here, T Z is defined as in Sect. 5; that is, and the parameters D m,ε , θ m ,Ā ε , δ, δ 1 , . . . , δ n−1 , as well as translates of Hörmander-type operators, are defined as in [21].

Proof
The proof is the same as that of Proposition 10.1 in [21], with the exception that the exponent n −m in inequality (10.30) of [21], which is due to equidistribution under a positive definite assumption on the phase, is here replaced by μ(n, σ, m), the exponent appearing in the equidistribution Lemma 5.4. This exponent is carried through to the end of the inductive proof, and the induction closes due to the above definition of e k,n,σ ( p).

Overview
It remains to pass from the k-broad estimates of Theorem 1.5 to linear estimates for the oscillatory integral operators T λ . As in [20,21], this is achieved via the Bourgain-Guth method from [12], which recursively partitions the norm T λ f L p (B R ) into two pieces: Broad part. This is the part of the norm which can be estimated using the k-broad inequalities from Theorem 1.5.
Narrow part. This consists of the remaining contributions to the norm, which cannot be controlled using the k-broad estimates.
In this section the tools for analysing the narrow part are reviewed. The main ingredient is a Wolff-type p -decoupling inequality: see Proposition 7.3 below. In the next section, a sketch of the Bourgain-Guth argument is provided which combines Theorem 1.5 and Proposition 7.3 (or, more precisely, Corollary 7.7) in order to deduce Theorem 1.2.

Decoupling regions
Let h : B n−1 → R be a smooth function such that h(0) = 0 and ∂ u h(0) = 0 and such that the Hessian ∂ 2 uu h(u) is non-degenerate for all u ∈ B n−1 with fixed signature 0 ≤ σ ≤ n − 1. In such cases h is said to be of signature σ . Consider the surface which is of non-vanishing Gaussian curvature and has second fundamental form of constant signature σ . Note that the Gauss map G h : B n−1 → S n−1 associated to this surface is given by 1 .
In particular, G h (0) = e n and the image set G h (B n−1 ) is contained in a spherical cap in the northern hemisphere, centred around the north pole. Givenū ∈ B n−1 and δ > 0 define the matrices where D δ = diag[δ 1/2 , . . . , δ 1/2 , δ] corresponds to an anisotropic (parabolic) scaling of the coordinates. This definition may be partially motivated by considering a quadratic form Q(u) := 1 2 Lu, u for L : R n−1 → R n−1 an invertible, self-adjoint linear mapping. By forming the Taylor expansion of Q , it follows that In particular, the above identity shows that the surface [Q] can be diffeomorphically mapped to a δ 1/2 -cap 13  If θ = θ(ū; δ) is a δ 1/2 -slab, thenū is referred to as the centre of the slab and in such cases the notationū = u θ is used. It will also be convenient to write [h] θ for [h] u θ ,δ whenever θ = θ(u θ ; δ).
These regions are defined in view of the scaling considerations discussed above. In particular, in the quadratic case, where h = Q as above, the slabs inherit a scaling structure from (7.1), as described in the proof of Lemma 7.5 below. As in Sect. 5 (see (5.3)), to avoid degenerate situations it is assumed that

Constant coefficient decoupling: quadratic case
For n ≥ d ≥ 2 and 0 ≤ σ ≤ n − 1 such that n − 1 − σ is even, define the exponents With this and the definitions from the previous subsection, the main decoupling inequality reads as follows.
For the 2 ≤ d ≤ n−σ +1 2 range the decoupling is elementary, but for the remaining d values Proposition 7.3 relies on the Bourgain-Demeter decoupling theorem for surfaces of non-vanishing Gaussian curvature [11].
In this subsection the proof of Proposition 7.3 (or, more precisely, the reduction of this proposition to the main theorem in [11]) is described in the special case where the surface under consideration is quadratic. In particular, here h := Q for some quadratic form where L : R n−1 → R n−1 is an invertible, self-adjoint linear mapping of signature σ . This prototypical case is essentially treated in [2] (see also [11]) but, for completeness, the details are given.
Slice geometry. Fix Q as in (7.5) and a d-dimensional subspace V satisfying (7.2). The first step is to understand the basic geometry of [Q; V ]. This is a quadratic surface, associated to some potentially degenerate quadratic form. The key is to determine the possible degree of degeneracy, which depends on the signature σ of the original matrix L.
For Q as in (7.5), the unnormalised Gauss map G Q,0 is an affine function. Thus, the preimage is an affine subspace of dimension d − 1 (see also Example 5.6 above) and In The following lemma is a minor modification of [2,Lemma 3.3], which in turn is adapted from the proof of Proposition 3.2 in [11].
Lemma 7.4 [2,11] Let 2 ≤ d ≤ n, 0 ≤ σ ≤ n −1 with n −1−σ even and L : R n−1 → R n−1 be an invertible, self-adjoint linear mapping of signature σ . If U is a vector space of dimension d − 1, then where L U is the linear mapping obtained by restricting to U the quadratic form associated to L, as described above.
Applying Lemma 7.4 to the subspace U := V u , it follows that the slice [Q; V ] has at least d − 1 − ν(n, σ, d) principal curvatures bounded away from zero.

Proof of Lemma 7.4 The desired inequality is equivalent to showing
The bound d−1 is obvious, since the total number of eigenvalues cannot exceed the dimension of U . In order to prove the remaining bounds, form the following orthogonal decompositions of R n−1 and U : • Let X − and X + denote the subspaces of R n−1 spanned by the eigenvectors of L with negative and positive eigenvalues, respectively. In this notation, The key observation is that which is a simple consequence of the definitions. Thus, and these inequalities together with (7.8) immediately imply the n−σ −1 2 bound in (7.7). The remaining bound in (7.7) follows by summing together (7.9a) and (7.9b), using the fact that dim Trivial decoupling. Recall, V u is the (d − 1)-dimensional linear subspace parallel to the affine subspace A u defined in (7.6). Consider the eigenspace decomposition V u = E − ⊕ E 0 ⊕ E + defined with respect to L V u as in the proof of Lemma 7.4. The eigenvectors generating E 0 have eigenvalues of small modulus and therefore correspond to the (relatively) flat directions of [Q; V ]. Note that E 0 has dimension ν := N L V u ; ρ(L) , which is bounded by Lemma 7.4. In these flat directions one applies a trivial decoupling inequality, based on Plancherel's theorem.
To make the above discussion precise, note that where A ξ = A u × R is the affine subspace introduced above. Since A u is parallel to V u , one may write A u = V u + b for some b ∈ R n−1 . Thus, W Fix a smooth partition of unity (ζ α ) α∈A(V ;δ) . Thus, given any g ∈ L 1 (R n−1 ) with Fourier support in N δ 1/2 A ξ ∩ B(0, 1), one may write g = α∈A(V ;δ) g α where each g α is defined via the Fourier transform byĝ α :=ĝ · ζ α . In particular, For all 2 ≤ p ≤ ∞, an elementary argument shows that (7.10) Indeed, this follows by interpolation between the p = 2 and p = ∞ cases (first setting up the estimate in a suitably general formulation, amenable to interpolation), which follow from Plancherel's theorem and the triangle inequality, respectively.
Applying the Bourgain-Demeter theorem. Now consider the (d − 1 − ν)-dimensional eigenspace E := E − ⊕ E + . The eigenvectors generating E have eigenvalues of large modulus and correspond to 'curved' directions. In particular, the restriction of Q to E is a non-degenerate form. Owing to this, one may take advantage of the Bourgain-Demeter theorem [11].
ξ ∈ A(V ; δ) and consider the linear subspace Choose coordinates x = (x , x ) for x ∈ V (a) and x ∈ (V (a) ) ⊥ . Fix x ∈ (V (a) ) ⊥ and define By elementary properties of the Fourier transform, it follows that Since the eigenvalues associated to eigenvectors in E are bounded away from zero, it follows that Q restricts to a nondenegerate form on W For a proof of these observations see, for instance, [3,Lemma 3.4].
In light of the above, for each fixed x ∈ (V (a) ) ⊥ , the function g x ,θ satisfies the hypotheses of the decoupling theorem for negatively-curved surfaces from [11]. Thus, for all 2 ≤ p ≤ 2 · d−ν+1 d−ν−1 and ε > 0 the inequality holds uniformly in x . Taking p powers, integrating over all x ∈ (V (a) ) ⊥ and then taking the p roots, one concludes that This efficiently decouples the L p -norms on the right-hand side of (7.10).

Constant coefficient decoupling: general case
To complete the proof of Proposition 7.3, it remains to extend the result from quadratic surfaces to graphs of arbitrary smooth h of signature σ . This is achieved via a now standard iteration argument originating in the work of Pramanik-Seeger [28]. The argument relies on the fact that, locally, each such h is a small perturbation of a quadratic surface of the same signature, and also on special scaling properties of the decoupling inequalities which manifest in the proof of Lemma 7.5 below.
Consider the slight generalisation of the setup from the previous subsection where h : R n−1 → R is a quadratic of signature σ defined by where L : R n−1 → R n−1 is an invertible, self-adjoint linear mapping of signature σ , whilst b ∈ R n−1 and a ∈ R. Fix V a d-dimensional subspace satisfying (7.2), a pair of scales 0 < δ < ρ < 1 and a ρ 1/2 -slab α on [h] with (G(u α ), V ) ≤ ρ 1/2 .
Proof Define functionsF θ via the Fourier transform by and note that it suffices to prove the same inequality but with each F θ replaced withF θ . Indeed, this follows by applying an affine rescaling and modulation to the functions appearing in both sides of the inequality in (7.17).
Let (N j ) n−d j=1 be an orthonormal basis for V ⊥ and write N j = (N j , N j,n ) where N j ∈ R n−1 is the vector formed by the first n − 1 components of N j . The angle condition (7.2) implies that the vectors N j are quantitatively transverse in the sense that | n−1 j=1 N j | h 1. Definẽ The condition (G h (u θ ), V ) ≤ δ 1/2 implies | G h,0 (u θ ), N j | h δ 1/2 for 1 ≤ j ≤ n − d and, consequently, (G Q (uθ ),Ṽ ) h (δ/ρ) 1/2 . Thus, the claim follows by applying the decoupling inequality from the previous step to the function Q at scale ∼ δ/ρ. Following [28], the general case of Proposition 7.3 may be deduced from the quadratic case via an induction-on-scale procedure, using Lemma 7.5.

Proof of Proposition 7.3: general case
Fix h : B n−1 → R of signature σ , a vector subspace V ⊆ R n of dimension d satisfying (7.2) and a Lebesgue exponent 2 ≤ p ≤ ∞. For 0 < δ < 1 define the decoupling constant D h,V , p (δ) to be the infimum over all constants C ≥ 1 for which the inequality holds for all δ 1/2 −slab decompositions (V , δ) on [h] along V and all tuples of functions (F θ ) θ ∈ (V ,δ) satisfying suppF θ ⊆ θ for all θ ∈ (V , δ). With this notation, given ε > 0 the problem is to show that Fixing ε > 0, the argument proceeds by induction on the scale δ, using the prototypical cases proved above to facilitate the induction step. In particular, let δ • = δ • (h, ε) > 0 be a fixed small parameter, depending only on h and ε and chosen sufficiently small for the purpose of the forthcoming argument. If 1 > δ ≥ δ • , then the desired bound (7.18) follows immediately from Hölder's inequality. This serves as the base case for the induction.
Here C h,ε is a fixed constant, which depends only on the admissible objects h and ε, chosen sufficiently large for the purpose of the forthcoming argument. In particular, it suffices to take C h,ε so that (7.19) holds in the base case 1 > δ ≥ δ • for the choice of δ • determined below.
Fix (V , δ) a δ 1/2 -slab decomposition on [h] along V . Let δ ρ < 1 be a second small parameter. Later in the argument ρ is fixed by taking ρ ∼ h δ 2/3 , but for now it is helpful to keep it a free parameter. Fix a ρ 1/2 -slab decomposition (V , ρ) with the property that every θ ∈ (V , δ) lies in at least one α ∈ (V , ρ).
Clearly, suppF α ⊆ α and so, applying the induction hypothesis (7.19) with δ = ρ ≥ 2δ, one deduces that Fixing α ∈ (V , ρ), the problem is now to decouple the norm To achieve this, h is locally approximated by a quadratic which facilitates application of the decoupling for quadratic surfaces derived in the previous steps. Let u α ∈ B n−1 denote the centre of α and consider the second order approximation h α : R n−1 → R to h around u α , given by h α (u) := 1 2 ∂ 2 uu h(u α )(u − u α ), u − u α + ∂ u h(u α ), u − u α + h(u α ).
By Taylor's theorem and the hypothesis θ ∈ (α), one deduces that Thus, taking ρ := c h δ 2/3 for a suitably small constant c h > 0, depending only on the magnitude of the third order derivatives of h, one concludes thatη n ∈ [−2, 2].
The previous observations show that each F θ for θ ∈ (α) has Fourier support in a (2δ) 1/2 -slab defined with respect to the quadratic surface [h α ]. One may therefore apply the (rescaled version of the) decoupling inequality (7.17) to conclude that Combining (7.20) and (7.21) with the definition of the decoupling constant, where C h,ε is an amalgamation of the implicit constants arising in the above argument. Since, δ/ρ ≤ c −1 h δ 1/3 • , by choosing δ • from the outset to be sufficiently small, depending only on h and ε, one may ensure that C h,ε (δ/ρ) ε/2 ≤ 1 and so the induction closes.

Variable coefficient decoupling
Proposition 7.3 can be used to study Hörmander-type operators, provided that the operator is sufficiently localised. In particular, given a reduced phase function φ λ of signature σ , recall from Sect. 4.2 that for a fixed vectorx ∈ B(0, λ) in the spatial domain, h λ x (u) := ∂ x n φ λ x; λ x (u) is a smooth function of signature σ on its domain. Moreover, if the corresponding operator T λ f is localised to a small ball aroundx, then the Fourier transform of this localised function is supported in a neighbourhood of the surface [h λ x ]. This facilitates application of the decoupling inequality from Proposition 7.3 in this setting.
To make the above discussion precise, fix a Hörmander-type operator T λ and a function f ∈ L 1 (B n−1 ). Let T be a decomposition of the domain B n−1 into finitely-overlapping balls τ ⊆ R n−1 of radius K −1 , each with some centre ω τ ∈ B n−1 , and fix a smooth partition of unity {ψ τ } τ ∈T subordinate to T. Correspondingly, decompose f = τ ∈T f τ where each f τ = f · ψ τ ; in particular, each f τ satisfies supp f τ ⊆ τ .
Lemma 7.6 Given ε > 0 and R ε ε K 2 ≤ λ, with the above definitions, Once this lemma is established, one may immediately apply Proposition 7.3 to deduce the following (pseudo) variable coefficient decoupling inequality. For a strengthening of the result see [29]. whenever R ε ε K 2 ≤ λ. Here the sums are over all caps τ for which (G λ (x, τ ), V ) ≤ K −1 wherex is the centre of B K 2 .
Proof Defining the slabs θ(τ ) with ε replaced with ε := ε/100n in (7.22), the functions By transferring the B K 2 -localisation between the L p -norm and the operator and applying the approximation from Lemma 7.6, the desired bound readily follows from the above display.

Proof of Theorem 1.2 For each k satisfying the constraint 2 ·
n − e(n, σ, k − 1) n − 1 − e(n, σ, k − 1) ≤p(n, σ, k) one may apply Proposition 8.1 withp(n, σ, k) ≤ p ≤ p dec (n, σ, k−1) to obtain a (potentially empty) range of estimates for the linear problem. It is not difficult to check that the optimal choice is given by for n even n+1 2 for n odd and one may readily verify thatp(n, σ, k * ) ≤ p dec (n, σ, k * − 1). Thus, the linear estimate holds for all p ≥p(n, σ, k * ). This corresponds to the range of estimates stated in Theorem 1.2.

Proof of Proposition 8.1
The proof of Proposition 8.1 relies on the induction-on-scales argument originating in [12]. The details are identical to those of the proof of [21, Proposition 11.2] except that Corollary 7.7 is now used in place of [21,Theorem 11.5], and there are corresponding changes to the numerology. The reader is therefore referred to [21] (see also [20]) for the details.
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