Oscillatory and Fourier Integral operators with degenerate canonical relations

We mostly survey results concerning the $L^2$ boundedness of oscillatory and Fourier integral operators. This article does not intend to give a broad overview; it mainly focusses on a few topics directly related to the work of the authors.

In (1.1) it is assumed that the real-valued phase function Φ is smooth in Ω L × Ω R where Ω L , Ω R are open subsets of R d and amplitude σ ∈ C ∞ 0 (Ω L × Ω R ). (The assumption that dim(Ω L ) = dim(Ω R ) is only for convenience; many of the definitions, techniques and results described below have some analogues in the nonequidimensional setting.) The L 2 boundedness properties of T λ are determined by the geometry of the canonical relation C = {(x, Φ x , y, −Φ y ) : (x, y) ∈ supp σ} ⊂ T * Ω L × T * Ω R .
The best possible situation occurs when C is locally the graph of a canonical transformation; i.e., the projections π L , π R to T * Ω L , T * Ω R , resp., are locally diffeomorphisms. In this case Hörmander [37], [38] proved that the norm of T λ as a bounded operator on L 2 (R d ) satisfies The proof consists in applying Schur's test to the kernel of T * λ T λ ; see the argument following (1.6) below.
It is also useful to study a more general class of oscillatory integrals which naturally arises when composing two different operators T λ , T λ and which is also closely related to the concept of Fourier integral operator. We consider the oscillatory integral kernel with frequency variable ϑ ∈ Θ (an open subset of R N ), defined by (1.3) K λ (x, y) = e ıλΨ(x,y,ϑ) a(x, y, ϑ)dϑ where Ψ ∈ C ∞ (Ω L × Ω R × Θ) is real-valued and a ∈ C ∞ 0 (Ω L × Ω R × Θ). Let T λ be the associated integral operator, (1.4) T λ f (x) = K λ (x, y)f (y)dy.
Again the L 2 mapping properties of T λ are determined by the geometric properties of the canonical relation It is always assumed that C is an immersed manifold, which is a consequence of the linear independence of the vectors ∇ (x,y,ϑ) Ψ ϑi , i = 1, . . . , N at {Ψ ϑ = 0}. In other words, Ψ is a nondegenerate phase in the sense of Hörmander [37], although Ψ is not assumed to be homogeneous. As before, the best possible situation for L 2 estimates arises when C is locally the graph of a canonical transformation. Analytically this means that (1.5) det Ψ xy Ψ xϑ Ψ ϑy Ψ ϑϑ = 0 Under this assumption the L 2 result becomes (1.6) T λ L 2 →L 2 λ −(d+N )/2 so that we discover (1.2) when N = 0. The proof of (1.6) could be given by using methods in [37] or alternatively by a straightforward modification of the argument in [38]. Indeed consider the Schwartz kernel H λ of the operator T * λ T λ which is given by H λ (u, y) = e −ıλ[Ψ(x,u,w)−Ψ(x,y,ϑ)] γ(x, u, w, y, ϑ)dwdϑdx where γ is smooth and compactly supported. By using partitions of unity we may assume that σ in (1.1) has small support; thus γ has small support. Change variables w = ϑ + h, and, after interchanging the order of integration, integrate parts with respect to the variables (ϑ, x). Since this yields, in view of the small support of γ, It follows that T * λ T λ L 2 →L 2 λ −N −d and hence (1.6).

Reduction of frequency variables.
Alternatively, as in the theory of Fourier integral operators, one may compose T λ with unitary operators associated to canonical transformations, and together with stationary phase calculations, deduce estimates for operators of the form (1.3-4) from operators of the form (1.1), which involve no frequency variables; in fact this procedure turns out to be very useful when estimating operators with degenerate canonical relations.
We briefly describe the idea based on [37], for details see [25].
Consider the operator T λ with kernel R N e ıλφ(x,y,z) a(x, y, z)dz. Let A i , i = 1, 2, be symmetric d × d matrices and define clearly S λ,i are unitary operators on L 2 (R d ). A computation yields that the operator λ −d S λ,1 T λ S * λ,2 can be written as the sum of an oscillatory integral operator with kernel O λ (x, y) plus an operator with L 2 norm O(λ −M ) for any M . The oscillatory kernel O λ (x, y) is again of the form (1.3) where the phase function is given by Ψ(x, y, ϑ) = y,w − x, w + 1 2 (A 1w ·w − A 2 w · w) + φ(w,w, z) with frequency variables ϑ = (w, z,w) ∈ R d × R N × R d , and the amplitude is compactly supported. One can choose A 1 , A 2 so that for tangent vectors δx, δy ∈ R d at a reference point the vector (δx, A 1 δx, δy, A 2 δy) is tangent to the canonical relation C associated with S λ,1 T λ S * λ,2 . Let π space be the projection C → Ω L × Ω R which with our choice of A 1 , A 2 has invertible differential. Since the number of frequency variables (N + 2d) minus the rank of φ ϑϑ is equal to 2d − rank dπ space , we deduce that det φ ϑϑ = 0.
In the integral defining the kernel of S λ,1 T λ S * λ,2 we can now apply the method of stationary phase to reduce the number of frequency variables to zero, and gain a factor of λ −(2d+N )/2 . Thus we may write S λ,1 T λ S * λ,2 = λ −N/2 T λ + R λ where T λ is an oscillatory integral operator (without frequency variables) and R λ is an operator with L 2 norm O(λ −M ) for any large M . Since S λ,i are unitary the L 2 bounds for λ N/2 T λ and T λ are equivalent.

Fourier integral operators.
The kernel of a Fourier integral operators F : C ∞ 0 (Ω R ) → D ′ (Ω L ) of order µ, F ∈ I µ (Ω L , Ω R ; C) is locally given as a finite sum of oscillatory integrals (1.7) e ıΨ(x,y,θ) a(x, y, θ)dθ, where now Ψ is nondegenerate in the sense of Hörmander [37], satisfies the homogeneity condition Ψ(x, y, tθ) = tΨ(x, y, θ) for |θ| = 1 and t ≫ 1, and a is a symbol of order µ + (d − N )/2. We assume in what follows that a(x, y, θ) vanishes for (x, y) outside a fixed compact set. The canonical relation is locally given by C = {(x, Ψ x , y, −Ψ y ), Ψ θ = 0} and we assume that where 0 L , 0 R denote the zero-sections in T * Ω L and T * Ω R . Staying away from the zero sections implies (1.8) |Ψ x (x, y, θ)| ≈ |θ| ≈ |Ψ y (x, y, θ)| for large θ (when Ψ θ is small). Let β ∈ C ∞ 0 (1/2, 2) and a k (x, y, θ) = β(2 −k |θ|)a(x, y, θ) and let F k be the dyadic localization of F ; i.e. (1.7) but with a replaced by a k . The assumptions Ψ x = 0 and Ψ y = 0 can be used to show that for k, l ≥ 1 the operators F k are almost orthogonal, in the sense that F * k F l and F k F * l have operator norms O(min{2 −kM , 2 −lM }) for any M , provided that |k − l| ≥ C for some large but fixed constant C. This follows from a straightforward integration by parts argument based on (1.8) and the assumption of compact (x, y) support. Using a change of variable θ = λϑ the study of the L 2 boundedness (and L 2 -Sobolev boundedness) properties is reduced to the study of oscillatory integral operators (1.3-4) and, in the nondegenerate case, an application of estimate (1.2) above. The result is that if F is of order µ and if the associated homogeneous canonical transformation is a local canonical graph, then F maps the Sobolev space L 2 α to L 2 α−µ An important subclass is the class of conormal operators associated to phase functions linear in the frequency variables (see [37, §2.4]). The generalized Radon transforms arise as model cases.
Here M x are codimension ℓ submanifolds in R d , and dσ x is a smooth density on M x , varying smoothly in x, and χ ∈ C ∞ 0 (Ω L ×Ω R ). One assumes that the M x are sections of a manifold M ⊂ Ω L × Ω R , so that the projections to Ω L and to Ω R have surjective differential; this assumption insures the L 1 and L ∞ boundedness of the operator R. We refer to M as the associated incidence relation.
Assuming that M is given by an R ℓ valued defining function Φ, then the distribution kernel of R is χ 0 (x, y)δ(Φ(x, y)) where χ 0 ∈ C ∞ 0 (Ω L × Ω R ) and δ is the Dirac measure in R ℓ at the origin. The assumptions on the projections to Ω L , Ω R imply that rank Φ x = rank Φ y = ℓ in a neighborhood of M = {Φ = 0}. The Fourier integral description is then obtained by writing out δ by means of the Fourier inversion formula in R ℓ , this has been used in [35] where R is identified as a Fourier integral operator of order −(d − ℓ)/2, see also [55]. More general conormal operators are obtained by composing Radon transforms with pseudo-differential operators (see [37]). The canonical relation associated to the generalized Radon transform is the twisted conormal bundle of the incidence relation, We can locally (after possibly a change of coordinates) parametrize M as a graph so that Using (1.5) with Ψ(x, y, τ ) = τ · Φ(x, y) one verifies that the condition for N * M ′ being a local canonical graph is equivalent to the nonvanishing of the determinant for all τ ∈ S ℓ−1 . Under this condition R maps L 2 to L 2 (d−ℓ)/2 . We note that the determinant in (1.14) vanishes for some τ if ℓ < d/2. In particular if ℓ = d − 1 then the expression (1.14) is a linear functional of τ and thus, if (x, y) is fixed, it vanishes for all τ in a hyperplane. Therefore degeneracies always occur for averaging over manifolds with high codimension, in particular for curves in three or more dimensions. Here we shall restrict ourselves to maps (or pairs of maps) which have corank ≤ 1.
Let M , N be n-dimensional manifolds, P ∈ M and Q ∈ N , and let f : M → N be a C ∞ map with f (P 0 ) = Q 0 . A vector field V is a kernel field for the map f on a neighborhood U of P 0 if V is smooth on U and if there exists a smooth vector field W on f (U ) so that Df P V = det(Df P )W f (P ) for all P ∈ U. If rank Df P0 ≥ n − 1 then it is easy to see that there is a neighborhood of P and a nonvanishing kernel vector field V for f on U. Moreover if V is another kernel field on U then V = αV − det(Df )W in some neighborhood of P 0 , for some vector field W and smooth ) and a choice for the kernel vector field is Definition. Suppose that M and N are smooth n-dimensional manifolds and that f : M → N is a smooth map with dim ker(Df ) ≤ 1 on M . We say that f is of type k at P if there is a nonvanishing kernel field V near P so that V j (det Df ) P = 0 for j < k but V k (det Df ) P = 0.
This definition was proposed by Comech [13], [15] who assumes in addition that Df drops rank simply on the singular variety {det Df = 0}.
The finite type condition is satisfied for the class of Morin singularities (folds, cusps, swallowtails, ...) which we shall now discuss.
2.2 Morin singularities. We consider as above maps f : M → N of corank ≤ 1.
We say that f drops rank simply at P 0 if rank Df P0 = n − 1 and if d(det Df ) P = 0. Then near P 0 the variety S 1 (f ) = {x : rank Df = n − 1} is a hypersurface and we say that f has an S 1 singularity at P with singularity manifold S 1 (f ).
Next let S be a hypersurface in a manifold U and let V be a vector field defined on S with values in T U (meaning that v P ∈ T P U for P ∈ S). We say that v is transversal to S at P ∈ S if v P / ∈ T P S. We say that v is simply tangent to S at P 0 if there is a one-form ω annihilating vectors tangent to S so that ω, v S vanishes of exactly first order at P 0 . This condition does not depend on the particular choice of ω. Next let P → ℓ(P ) ⊂ T P (V ) be a smooth field of lines defined on S. Let v be a nonvanishing vector field so that ℓ(P ) = Rv P . The definitions of transversality and simple tangency carry over to field of lines (and the notions do not depend on the particular choice of the vector field).
Next consider F : U → N where dim U = k ≥ 2 and dim N = n ≥ k and assume that rank DF ≥ k −1. Suppose that S is a hypersurface in U such that rank DF = k − 1 on S. Suppose that Ker DF is simply tangent to S at P ∈ S. Then there is a neighborhood U of P in S such that the variety {Q ∈ U : rank DF With these notions we can now recall the definition of Morin singularities ( [78], [47]).
Definition. Let 1 ≤ r ≤ n. Let S 1 , . . . , S r be submanifolds of an open set U ⊂ M so that S k is of dimension n−k in V and S 1 ⊃ S 2 ⊃ · · · ⊃ S r ; we also set S 0 := U.
We say that f has an S 1r singularity in U, with a descending flag of singularity manifolds (S 1 , . . . , S r ) if the following conditions hold in U.
(i) For P ∈ U, either Df P is bijective or f drops rank simply at P .
Definition. We say that f has an S 1r,0 singularity at P , if the following conditions hold.
(i) There exists a neighborhood U of P submanifolds S k of dimension n − k in U so that P ∈ S r ⊂ S r−1 ⊂ · · · ⊂ S 1 and so that f : U → N has an S 1r singularity in U, with singularity manifolds (S 1 , . . . , S r ).
The singularity manifolds S k are denoted by S 1 k (f ) in singularity theory (if the neighborhood is understood). An S 1,0 (or S 11,0 ) singularity is a Whitney fold; an S 1,1,0 (or S 12,0 ) singularity is referred to as a Whitney or simple cusp.
If f is given in adapted coordinates vanishing at P , i.e.
then f has an S 1r singularity in a neighborhood of P = 0 if and only if and the gradients are linearly independent at 0. Moreover f has an S 1r ,0 singularity at P if in addition The singularity manifolds are then given by In these coordinates the kernel field for f is ∂/∂t n and the map f is of type r at P . Normal forms of S 1r singularities are due to Morin [47], who showed that there exists adapted coordinate systems so that (2.2) holds with Finally we mention the situation of maximal degeneracy for S 1 singularities which occurs when the kernel of Df is everywhere tangential to the singularity surface S 1 (f ). In this case we say that f is a blowdown; see example 2.3.3 below.

Examples.
We now discuss some model examples. The first set of examples concern translation invariant averages over curves, the second set restricted X-ray transforms for rigid line complexes. The map f above will always be one of the projections π L : C → T * Ω L or π R : C → T * Ω R . Note that S 1 (π L ) = S 1 (π R ).
2.3.1. Consider the operator on functions in R d Then the canonical relation is given by Consider the projection π L then it is not hard to see that S 1 k (π L ) is the submanifold of C where in addition ξ, Γ (j) (α) = 0} for 2 ≤ j ≤ k + 1. Clearly then S 1 d−1 (π L ) = ∅ so that we have an S 1 d−2,0 singularity. The behavior of π R is of course exactly the same; moreover for small perturbations the projections π L and π R still have at most S 1 d−2,0 singularities. Note that in the translation invariant setting we have S 1 k (π L ) = S 1 k (π R ), but for for small variable perturbations the manifolds S 1 k (π L ), S 1 k (π R ) are typically different if k ≥ 2. By Fourier transform arguments and van der Corput's lemma it is easy to see that A maps L 2 (R d ) to the Sobolev-space L 2 1/d (R d ) and it is conjectured that this estimate remains true for variable coefficient perturbations. This is known in dimensions d ≤ 4 (cf. §5 below).
2.3.2. Consider the example (2.7) with d = 3 and where m, n are integers with 1 < m < n.

2.3.3.
For an example for a one-sided behavior we consider the restricted X-ray transform where γ is now the regular parametrization of a curve in R d−1 and χ 0 , χ are smooth and compactly supported. We say that R is associated to a d dimensional line complex which is referred to as rigid because of the translation invariance in the x ′ variables. The canonical relation is now given by and the singular set S 1 (π L ) = S 1 (π R ) is the submanifold on which τ · γ ′ (x d ) = 0. One computes that V L = ∂/∂y d is a kernel vector field for π L and V R = ∂/∂x d is a kernel vector field for π R . Clearly V L is tangential to S 1 (π L ) everywhere so that π L is a blowdown. The behavior of the projection π R depends on assumptions on γ. The best case occurs when γ ′ (x d ), . . . , γ (d−1) (x d ) are linearly independent everywhere. The singularity manifolds S k = S 1 k (π R ) are then given by the equations τ · γ (j) (x d ) = 0, j = 1, . . . , k, and thus S 1 d−1 (π R ) = ∅ and π R has (at most) S 1 d−2 ,0 singularities. For the model case given here it is easy to derive the sharp L 2 -Sobolev estimates. Observe that defines (modulo the cutoff function) a translation invariant operator. By van der Corput's Lemma it is easy to see that and one deduces that R maps L 2 to L 2 1/(2d−2) . It is conjectured that the X-ray transform for general well-curved line complexes satisfies locally the same estimate; here the support of χ is supported in (−ε, ε) for small ε and it is assumed that for each fixed

Strong Morin singularities.
We now discuss the notion of strong Morin singularities, or S + 1r singularities for maps into a fiber bundle W over a base manifold B, with projection π B . Here it is assumed that dim W = n and dim(B) = q ≤ n − r, so that the fibers W b = π −1 B b are n − q dimensional manifolds (see [26]). The relevant W is T * Ω R , the cotangent bundle of the base B = Ω R .
We remark that for the examples in 2.3.1 both π L and π R have strong Morin singularities while for the example in 2.3.3 π R has strong Morin singularities. This remains true for small perturbations of these examples.
In order to verify the occurence of strong Morin singularities for canonical relations which come up in studying averages on curves the following simple lemma is useful.
Lemma. Let I be an open interval, let ψ : I → R n be a smooth parametrization of a regular curve not passing through 0 and let Let π : M → R n be defined by π(t, η) = η.
For the proof assume first the linear independence of ψ (j) (t). We may work near t = 0 and by a linear change of variables, we may assume that Hence, (η ′ , t) and (ξ ′ , ξ 1 ) form adapted coordinates (cf. (2.2)) for the map π, and in these coordinates are linearly independent. Thus π has at most S 1n−2,0 singularities. Conversely, assume that π has at most S 1n−2,0 singularities. Since ψ does not pass through the origin, we may assume that ψ n (t) = 0 locally. Then the map π is given in adapted coordinates by and the linear independence follows easily from (2.3-5).

Mixed finite type conditions. We briefly discuss mixed conditions for pairs of maps
, there is a nonvanishing function α so that det Df L = α det Df R in the domain under consideration. Let V L , V R be nonvanishing kernel fields on M for the maps f L , f R . Let U be a neighborhood of P in M . We define D j,k (U ) to be the linear space of differential operators generated spanned by operators of the form where V i are kernel fields for the maps f L or f R in U , and k of them are kernel fields for f L and j of them are kernel fields for f R . Let h be a real valued function defined in a neighborhood of P ∈ M ; we say that h vanishes of order at P ∈ M and if there is an operator L ∈ D j,k so that Lh P = 0. Because of the assumption of volume equivalence det Df L in this definition can be replaced by det Df R . In the canonical example of interest here we have M = C ⊂ T * Ω L × T * Ω R , a canonical relation, and f L ≡ π L , f R ≡ π R are the projections to T * Ω L and T * Ω R , respectively.

Fourier integral operators in two dimensions
In this section we examine the regularity of Fourier integral operators in two dimensions, in which case one can get the sharp L 2 regularity properties with the possible exception of endpoint estimates. We shall assume that Ω L , is a homogeneous canonical relation and F ∈ I −1/2 (Ω L , Ω R , C), with compactly supported distribution kernels; we assume that the rank of the projection π space : C → Ω L × Ω R is ≥ 2 everywhere. The generalized Radon transform (1.11) (with ℓ = 1, d = 2) is a model case in which rank (dπ space ) = 3.
In order to formulate the L 2 results we shall work with the Newton polygon, as in [58] where oscillatory integral operators in one dimension are considered. We recall that for a set E of pairs (a, b) of nonnegative numbers the Newton polygon associated to E is the closed convex hull of all quadrants Definition. For c ∈ C let N (c) be the Newton polygon associated to the set Using the notion of type (j, k) in §2.5 we can now formulate 3.1. Theorem. Let Ω L , Ω R ⊂ R 2 and C as above and let F ∈ I −1/2 (Ω L , Ω R ; C), with compactly supported distribution kernel. Let α = min c (2t c ) −1 .
Then the operator F maps L 2 boundedly to L 2 α−ε for all ε > 0 . In the present two-dimensional situation one can reduce matters to operators with phase functions that are linear in the frequency variables (i.e., the conormal situation). We briefly describe this reduction.
First, our operator can be written modulo smoothing operators as a finite sum of operators of the form where a is of order −1/2, and has compact x support. We may also assume that a(x, ξ) has ξ-support in an annulus {ξ : |ξ| ≈ λ} for large λ. By scaling we can reduce matters to show that the L 2 operator norm for the oscillatory integral operator T λ defined by ; here χ has compact support and vanishes for ξ near 0. We introduce polar coordinates in the last integral, ξ = σ(cos y 1 , sin y 1 ) and put Then the asserted bound for T λ is equivalent to the same bound for the L 2 norm of T λ defined by for suitable χ; here we have used the homogeneity of ϕ. Now we rescale again and apply a Fourier transform in σ and see that the bound follows from the L 2 → L 2 α bound for the conormal Fourier integral operator with distribution kernel where Φ(x, y) = S(x, y 1 ) − y 2 , and b is a symbol of order 0, supported in {|τ | ≈ λ} and compactly supported in x.
Thus it suffices to discuss conormal operators of this form; in fact for them one can prove almost sharp L p → L p α estimates. Before stating these results we shall first reformulate the mixed finite type assumption from §2.5 in the present situation.
3.2. Mixed finite type conditions in the conormal situation. We now look at operators with distribution kernels of the form (3.3). The singular support of such operators is given by and it is assumed that Φ x = 0, Φ y = 0. The canonical relation is the twisted conormal bundle N * M ′ as in (1.12). In view of the homogeneity the type condition at c 0 = (x 0 , y 0 , ξ 0 , η 0 ) ∈ N * M ′ is equivalent with the type condition at (x 0 , y 0 , rξ 0 , rη 0 ) for any r > 0 and since the fibers in N * M ′ are one-dimensional it seems natural to formulate finite type conditions in terms of vector fields tangent to M, and their commutators. We now describe these conditions but refer for a more detailed discussion to [67]. Related ideas have been used in the study of subelliptic operators ( [36], [63]), in complex analysis ( [41], [2]) and, more recently, in the study of singular Radon transforms ( [11]).
Two types of vector fields play a special role: We say that a vector field V on M is of type ( . The notation is suggested by an analogous situation in several complex variables ( [41], [55]).
Note that at every point P ∈ M the vector fields of type (1, 0) and (0, 1) span a two-dimensional subspace of the three-dimensional tangent space T P M. Thus we can pick a nonvanishing 1−form ω which annihilates vector fields of type (1, 0) and (0, 1); we may choose are (1, 0) and (0, 1) vector fields, respectively. With this choice does not vanish. The quantity (3.4) is often referred to as "rotational curvature" (cf. [55]). Now let µ and ν be two positive integers. For a neighborhood U of P let W µ,ν (U ) be the module generated by vector fields adW 1 adW 2 . . . adW µ+ν−1 (W µ+ν ) where µ of these vector fields are of type (1, 0) and ν are of type (0, 1). The finite type condition in (2.4) can be reformulated as follows. Let P ∈ M and let c ∈ N * M ′ with base point P . Then C is of type (j, k) at c if there is a neighborhood U of P so that for all vector fields W ∈ W j+1,k (U ) ∪ W j,k+1 (U ) we have ω, W P = 0 but there is a vector field W in W j+1,k+1 for which ω, W P = 0. 1 Now coordinates can be chosen so that Φ(x, y) = −y 2 + S(x, y 1 ) and the generalized Radon transform is given by (3.5) Rf (x) = χ(x, y 1 , S(x, y 1 ))f (y 1 , S(x, y 1 ))dy 1 then at P = (x, y 1 , S(x, y 1 )) the mixed finite type condition amounts to For the equivalence of these conditions see [67].
We now relate the last condition to the finite type condition above. Notice that and using coordinates (x 1 , x 2 , y 1 , τ ) a kernel vector field for the projection π R is given by V R = S x2 ∂ x1 − S x1 ∂ x2 ; this can be identified with the vector field X on M. Moreover a kernel vector field for the projection π L is given by at the point c (with coordinates (x, y 1 , τ )) if conditions (3.6), (3.7) are satisfied, and this is just a condition at the base-point P .
We shall now return to the proof of Theorem 3.1 and formulate an L p version for the conormal situation.
, the convex hull of the points (1, 1), (0, 0) and The L 2 estimate of Theorem 3.1 for conormal operators follows as a special case, and for the general situation we use the above reduction. Theorem 3.3 is sharp up to the open endpoint cases (cf. also §3.5.1-3 below).
We now sketch the main ingredients of the proof of Theorem 3.3. We may assume that S x2 is near 1 and |S x1 | ≪ 1. Suppose that Q = (x 0 , y 0 ) ∈ M and suppose that the type (j ′ , k ′ ) condition holds for some choice of (j ′ , k ′ ) with j ′ ≤ j and k ′ ≤ k at Q, and suppose that this type assumption is still valid in a neighborhood on the support of the cutoff function χ in (3.5) (otherwise we work with partitions of unity).
Since we do not attempt to obtain an endpoint result, it is sufficient to prove the required estimate for operators with the frequency variable localized to |τ | ≈ λ for large λ. We then make an additional dyadic decomposition in terms of the size of |∆| (i.e., the rotational curvature). Define a Fourier integral operator F λ,l0 by then by interpolation arguments our goal will be achieved by proving the following crucial estimates: A variant of this interpolation argument goes back to investigations on maximal operators in [18] and [71], [72], and (3.9) can be thought of a version of an estimate for damped oscillatory integrals. The type assumption is only used for the estimate (3.8). We note that by integration by parts with respect to the frequency variable the kernel of F λ,l0 is bounded by (3.10) λ(1 + λ|y 2 − S(x, y 1 )|) −N χ(2 l0 ∆(x, y 1 )).
We can use a well known sublevel set estimate related to van der Corput's lemma (see [8]) to see that for each fixed x the set of all y 1 such that |∆(x, The two sublevel set estimates together with (3.10) and straightforward applications of Hölder's inequality yield (3.8), see [67].
We now turn to the harder L 2 estimate (3.9). We sketch the ideas of the proof (see [66] and also [67] for some corrections).
If we tried to use the standard T T * argument we would have to have good lower bounds for S y1 (w, y 1 ) − S y1 (x, y 1 ) in the situation where S(w, y 1 ) − S(x, y 1 ) is small, but the appropriate lower bounds fail to hold if the rotational curvature is too small. Thus it is necessary to work with finer decompositions. Solve the equation S(w, y 1 ) − S(x, y 1 ) = 0 by w 2 = u(w 1 , x, y 1 ) and expand and (3.14) where M ≫ 100/ε. In particular γ 0 (x, y 1 ) = S y1x1 (x, y 1 ) + u w1 (x 1 , x, y 1 )S y1,x2 (x, y 1 ) = ∆(x, y 1 ) S x2 (x, y 1 ) ; where the α k are smooth and V is the (1, 0) vector field ∂ x1 −S x1 /S x2 ∂ x2 . We introduce an additional localization in terms the size of γ j (x, y 1 ). For l = (l 0 , . . . , l M ), with l j < l 0 for j = 1, . . . , M define which describes a localization to the sets where |γ j (x, y 1 )| ≈ 2 −lj . A modification of the definition is required if |γ j | ≤ 2 −l0 for some j ∈ {1, ..., M }.
Since we consider at most O((1 + l 0 ) M ) = O(2 εl0 ) such operators it suffices to bound any individual T λ, l , and the main estimate is In what follows we fix λ and l and set Note that while µ i and ν i are close there may be 'large' gaps between µ i and ν i+1 for which the favorable lower bound (iii) holds. The elementary but somewhat lengthy proof of the Lemma based on induction is in [66]. A shorter and more elegant proof (of a closely related inequality) based on a compactness argument is due to Rychkov [64].
In order to descibe the orthogonality argument we need some terminology. Let I be a subinterval of [0, 1]. We say that β is a normalized cutoff function associated to I if β is supported in I and |β (j) (t)| ≤ |I| −j , for j = 1, . . . , 5 and denote by A(I) the set of all normalized cutoff functions associated to I.
Fix I and β in A(I); then we define another localization of T = T λ, l by It follows quickly from the definition and the property ν s ≤ µ s ≤ Cν s that This is because for any interval I of length µ s a function β ∈ A( I) can be written as a sum of a bounded number of functions associated to subintervals of length ν s .
One uses the Cotlar-Stein Lemma in the form , for a (finite) sum of operators j A j on a Hilbert space. (See [73, ch. VII.2]; as pointed out in [7] and elsewhere, the version (3.19) follows by a slight modification of the standard proof). Now if J is an interval of length ν s /8 and β ∈ A(J) then we split β = n β n where for a fixed absolute constant C the function C −1 β n belongs to A(I n ) and the I n are intervals of length µ s−1 ; I n and I n ′ are disjoint if |n − n ′ | > 3 and the sum extends over no more than O(ν s /µ s−1 ) terms and thus over no more than O(2 l0 ) terms. Now let |I n | = |I n ′ | ≈ µ s−1 and dist(I n , I n ′ ) ≈ |n−n ′ ||I| and assume |n−n ′ ||I| ≤ ν s /8. Let β n , β n ′ be normalized cutoff functions associated to I n , I n ′ . Then (3.20) T by the disjointness of the intervals I n , I ′ n . The crucial estimate is (3.21) T (3.20/21) allows us to apply (3.19) with θ = 0 (the standard version does not apply, as is erroneously quoted in [66]). This yields the bound To see (3.21) one examines the kernel K of T [β n ]T [β n ′ ] * which is given by and by definition of µ s−1 , ν s , I n , I n ′ and the above Lemma we have To analyze the kernel K and prove (3.21) by Schur's test one integrates by parts once in y 1 and then many times in τ , for the somewhat lengthy details see [66], [67]. Analogous arguments also apply to the estimation of T [β]T [β] * when β is associated to an interval of length ≪ ν 1 , this gives (3.18). Namely, if Rf (x) = f (y 1 , x 2 + s(x 1 , y 1 ))χ(x, y 1 )dy 1 then the endpoint L 2 → L 2 α estimate in Theorem 3.1 holds true. An only slightly weaker result for the case s ∈ C ∞ has been obtained by Rychkov [64]. For related work see also some recent papers by Greenblatt [23], [24].

3.5.2.
It is not known exactly which endpoint bounds hold in the general case of Theorem 3.3. As an easy case the L p → L p 1/p estimate holds if p > n and a type (0, n − 2) condition is satisfied (in the terminology of Theorem 3.3). A similar statement for 1 < p < n/(n − 1) is obtained for type (n − 2, 0) conditions by passing to the adjoint operator.

Some endpoint inequalities in Theorem 3.3 fail: M. Christ [9]
showed that the convolution with a compactly supported density on (t, t n ) fails to map L n → L n 1/n . The best possible substitute is an L n,2 → L n 1/n estimate in [69]; here L n,2 is the Lorentz space.
3.5.4. Interpolation of the bounds in Theorem 3.3 with trivial L 1 → L ∞ bounds (with loss of one derivative) yields almost sharp L p → L q bounds ( [66], [67]). Endpoint estimates for the case of two-sided finite type conditions are in [1]. For endpoint L p → L q estimates in the case (3.24), with real-analytic s, see [57], [79], [42].
3.5.5. It would be desirable to obtain almost sharp L 2 versions such as Theorem 3.1 for more general oscillatory integral operators with a corank one assumption. Sharp endpoint L 2 results where one projection is a Whitney fold (type 1) and the other projection satisfies a finite type condition are due to Comech [15].
3.5.6. Interesting bounds for the semi-translation invariant case (3.24) where only lower bounds on s xy (or higher derivatives) are assumed were obtained by Carbery, Christ and Wright [6]. Related is the work by Phong, Stein and Sturm ( [60], [62], [61]), with important contributions concerning the stability of estimates.

Operators with one-sided finite type conditions
We now discuss operators of the form (1.1) and assume that one of the projections, π L , is of type ≤ r but make no assumption on the other projection, π R . The role of the projections can be interchanged by passing to the adjoint operator.

4.1.Theorem [25],[26],[28]
. Suppose π L is of corank ≤ 1 and type ≤ r, and suppose that det dπ L vanishes simply. If r ∈ {1, 2, 3} then It is conjectured that this bound also holds for r > 3. The estimate (4.1) is sharp in cases where the other projection exhibits maximal degeneracy. In fact if π L is a fold and π R is a blowdown then more information is available such as a rather precise description of the kernel of T λ T * λ , cf. Greenleaf and Uhlmann [32], [33]. Applications include the restricted X-ray transform in three dimensions for the case where the line complexes are admissible in the sense of Gelfand ([21], [30], [34]); for an early construction and application of a Fourier integral operator with this structure see also [43].
In the discussion that follows we shall replace the assumption that det dπ L vanishes simply (i.e., ∇ x,z det π L = 0) by the more restrictive assumption In the case r = 1 this is automatically satisfied, and it is shown in [26], [28] that in the cases r = 2 and r = 3 one can apply canonical transformations to reduce matters to this situation. For the oscillatory integral operators coming from the restricted X-ray transform for well-curved line complexes, the condition (4.2) is certainly satisfied. We shall show that for general r the estimate (4.1) is a consequence of sharp estimates for oscillatory integral operators satisfying two-sided finite type conditions of order r − 1, in d − 1 dimensions. The argument is closely related to Strichartz estimates and can also be used to derive L 2 → L q estimates (an early version can be found in Oberlin [48]).
We shall now outline this argument. After initial changes of variables in x and z separately we may assume that moreover by our assumption on the type we may assume that We may assume that the amplitudes are supported where |x| + |z| ≤ ε 0 ≪ 1.
We form the operator T λ T * λ and write and by an integration by parts argument we get Thus the corresponding operator R x d y d is bounded on L 2 (R d−1 ) and satisfies for any N . For the main contribution H x d y d we are aiming for the estimate From (4.4), (4.5), (4.6) and the L 2 (R) boundedness of the operator with kernel the bound (4.1) follows in a straightforward way. Now observe that the operator H x d y d is local on cubes of diameter ≈ |x d − y d | and we can use a trivial orthogonality argument to put the localizations to cubes together. For a single cube we may then apply a rescaling argument. Specifically, let c ∈ R d and define Then for the corresponding operators we have Note that H x d y d c does not vanish only for small c. A calculation shows that the kernel of H x d y d c is given by an oscillatory integral with small parameters c, α = |x d − y d |, y d , and the phase function is given by Here the choice of Ψ + is taken if x d > y d and Ψ − is taken if x d < y d ; for the error we have ρ ± = O(α(|y d | + c)) in the C ∞ topology. Observe in particular that for α = 0 we get essentially the localization of a translation invariant operator. We now examine the canonical relation associated to the oscillatory integral, when α = 0. In view of (4.3.1) the critical set {∇ z Ψ ± = 0} for the phase function at α = 0 is given by ; in view of (4.2) this defines a smooth manifold. Consequently the canonical relation is a smooth manifold. By (4.2) we may assume (after performing a rotation) that Φ x d z d z1 = 0 and then solve the equation Φ x d z d (0, z) = 0 near the origin in terms of a function z 1 = z 1 (z ′′ , z d ). The projection π L is given by and ∂/∂z d is a kernel field for π L . Implicit differentiation reveals that ∂ k belongs to the ideal generated by Φ x d z j d , j ≤ k and thus, by our assumption (4.3.2) we see that π L is of type ≤ r − 1. The same holds true for π R , by symmetry considerations. Although we have verified these conditions for α = 0 they remain true for small α since Morin singularities are stable under small perturbations.
We now discuss estimates for the oscillatory integral operator S ± µ whose kernel is given by (4.8) (we suppress the dependence on c, α, y d .) The number of frequency variables is N = d and thus we can expect the uniform bound for small α. Indeed, the case α = 0 of (4.9) is easy to verify; because of the translation invariance we may apply Fourier transform arguments together with the method of stationary phase and van der Corput's lemma. Given (4.9) we obtain from (4.7) and from (4.9) with µ = λ|x d − y d | that Of course the Fourier transform argument does not extend to the case where α is merely small. However if r = 1 the estimate follows from (1.6) (with d replaced by d − 1 and N = d) since then C Ψ ± is a local canonical graph. Similarly, if r = 2 then the canonical relation C Ψ ± projects with two-sided fold singularities so that the desired estimate follows from known estimates for this situation (see the pioneering paper by Melrose and Taylor [44], and also [53], [19], [27]). For the case r = 3, inequality (4.6) follows from a recent result by the authors [28] discussed in the next section, plus the reduction outlined in §1.2. The case r ≥ 4 is currently open.

Remarks.
4.2.1. The argument above can also be used to prove L 2 → L q estimates (see [48], [25], [26]). Assume r = 1 and thus assume that π L : C → T * Ω L projects with Whitney folds. Then a stationary phase argument gives that and interpolation with (4.5-6) yields L q ′ → L q estimates for K x d y d and then L 2 → L q bounds for T λ . The result [25] is The estimate (4.10) may be improved under the presence of some curvature assumption. Assume that the projection of the fold surface S 1 (π L ) to Ω L is a submersion, then for each x ∈ Ω L the projection of S 1 (π L ) to the fibers is a hypersurface Σ x in T * x Ω L . Suppose that for every x this hypersurface has l nonvanishing principal curvatures (this assumption is reminiscent of the so-called cinematic curvature hypothesis in [46]). Then (4.10) can be replaced by and (4.11) holds true for a larger range of exponents, namely (4.12) The version of this estimate for Fourier integral operators [25], with l = 1, yields Oberlin's sharp L p → L q estimates [48] for the averaging operator (2.7) in three dimension (assuming that Γ is nondegenerate), as well as variable coefficient perturbations. It also yields sharp results for certain convolution operators associated to curves on the Heisenberg group ( [65], see §7.3 below) and for estimates for restricted X-ray transforms associated to well curved line complexes in R 3 ([25]).
In dimensions d > 3 the method yields L 2 → L q bounds ([26]) which should be considered as partial results, since in most interesting cases the endpoint L p → L q estimates do not involve the exponent 2.
4.2.2. The analogy with the cinematic curvature hypothesis has been exploited by Oberlin, Smith and Sogge [52] to prove nontrivial L 4 → L 4 α estimates for translation invariant operators associated to nondegenerate curves in R 3 . Here it is crucial to apply a square function estimate due to Bourgain [3] that he used in proving bounds for cone multipliers. The article [51] contains an interesting counterexample for the failure of L p → L p 1/p−ε estimates when p < 4. Oberlin [49] to obtain essentially sharp L p → L q estimates for the operator (2.7) in four dimension, see also [29] for a related argument for the restricted X-ray transform in four dimensions, in the rigid case (2.8).

Techniques of oscillatory integrals have been used by
4.2.4. More recently, a powerful combinatorial method was developed by Christ [10] who proved essentially sharp L p → L q estimates for the translation invariant model operator (2.7) in all dimensions (for nondegenerate Γ). L p → L q bounds for the X-ray transform in higher dimensions, in the model case (2.8), have been obtained by Burak-Erdogan and Christ ([4], [5]); these papers contain even stronger mixed norm estimates. Christ's combinatorial method has been further developed by Tao and Wright [75] who obtained almost sharp L p → L q estimates for variable coefficient analogues.

Two-sided type two singularities
We consider again the operator (1.1) and discuss the proof of the following result mentioned in the last section. [28]. Suppose that both π L and π R are of type ≤ 2. Then for λ ≥ 1

A slightly weaker version of this result is due to Comech and Cuccagna [17] who obtained the bound
The proof of the endpoint estimate is based on various localizations and almost orthogonality arguments. As in §2 we start with localizing the determinant of dπ L/R and its derivatives with respect to a kernel vector field. The form (5.2) below of this first decomposition can already be found in [15], [17].
We assume that the amplitude is supported near the origin and assume that (4.3.1) holds.
then kernel vector fields for the projections π L are given by respectively. Also let h(x, z) = det Φ xz and by the type two assumption we can assume that |V 2 L h|, |V 2 R h| are bounded below. Emphasizing the amplitude in (1.1) we write T λ [σ] for the operator T λ and will introduce various decompositions of the amplitude.
Let β 0 ∈ C ∞ (R) be an even function supported in (−1, 1), and equal to one in (−1/2, 1/2) and for j ≥ 1 let β j (s) = β 0 (2 −j s) − β 0 (2 −j+1 s). Denote by ℓ 0 that is the largest integer ℓ so that 2 ℓ ≤ λ 1/2 (we assume that λ is large). Define thus if j, k > 0 then |h| ≈ 2 −l , |V L h| ≈ 2 k−l/2 , |V R h| ≈ 2 j−l/2 on the support of σ j,k,l . It is not hard to see that the estimate of Theorem 5.1 follows from 5.2. Proposition. We have the following bounds: We shall only discuss (5.3) as (5.4) is proved similarly. In what follows j, k, l will be fixed and we shall discuss the main case where 0 < k ≤ j ≤ l/2, 2 l ≤ λ 1/2 . As in the argument in §2 standard T * T arguments do not work and further localizations and almost orthogonality arguments are needed. These are less straightforward in the higher dimensional situation considered here, and the amplitudes will be localized to nonisotropic boxes of various sides depending on the geometry of the kernel vector fields. For aP , π ⊥ bP be the orthogonal projections to the orthogonal complement of Ra P in T x 0 Ω L and Rb P in T z 0 R d , respectively. Suppose 0 < γ 1 ≤ γ 2 ≪ 1 and 0 < δ 1 ≤ δ 2 ≪ 1 and let We always assume We say that χ ∈ C ∞ 0 is a normalized cutoff function associated to B P (γ 1 , γ 2 , δ 1 , δ 2 ) if it is supported in B P (γ 1 , γ 2 , δ 1 , δ 2 ) and satisfies the (natural) estimates We denote by A P (γ 1 , γ 2 , δ 1 , δ 2 ) the class of all normalized cutoff functions associated to B P (γ 1 , γ 2 , δ 1 , δ 2 ).
In the argument it is crucial that we assume (5.11) min{ γ 1,small γ 2,small , δ 1,small δ 2,small } max{γ 2,large , δ 2,large } since from (5.11) one can see that the orientation of small boxes B Q (γ small , δ small ) does not significantly change if Q varies in the large box B P (γ large , δ large ).
A combination of these estimates (with 5.8) yields the desired bound (5.3); here the outline of the argument is similar to the one given in §3. For each instance we are given a cutoff function ζ ∈ A P (γ large , δ large ) and we decompose where the ζ XZ is, up to a constant, a normalized cutoff function associated to a box of dimensions (γ small , δ small ); the various boxes have bounded overlap, and comparable orientation. More precisely if (P, Q) is a reference point in the big box B P (γ large , δ large ) then each of the small boxes is comparable to a box defined by the conditions |π ⊥ If T XZ denotes the operator T λ [ζ XZ σ j,k,l ] then in each case we have to show that for large N For the estimation in the case (5.12) it is crucial that in any fixed large box V L h does not change by more than O(ε2 k−l/2 ) and thus is comparable to 2 k−l/2 in the entire box; similarly V R h is comparable to 2 j−l/2 in the entire box. For the orthogonality we use that Φ x ′ z ′ is close to the identity. In the other extreme case (5.14) V R h and V L h change significantly in the direction of kernel fields and this can be exploited in the orthogonality argument. (5.13) is an intermediate case. This description is an oversimplification and we refer the reader to [28] for the detailed discussion of each case.

Geometrical conditions on families of curves
We illustrate some of the results mentioned before by relating conditions involving strong Morin singularities to various conditions on vector fields and their commutators. vector field X and a nonvanishing (0, 1) vector field Y are given by Let π L,x0 the restriction of π L to N L,x0 as a map to T * x0 Ω L , then π L has strong Morin singularities if for fixed x 0 the map π L,x0 has Morin singularities.
Similarly, if y 0 ∈ Ω L , let M y 0 = {x ∈ Ω L : (x, y 0 ) ∈ M} then the adjoint operator R * is an integral operator along the curves M y0 ; now we define N R,y 0 as the set of all (x, λ) where x ∈ M y 0 , λ ∈ T * ,⊥ (x,y 0 ) M, and π R,y 0 : N R,y 0 → T * y 0 Ω R is the restriction of the map π R .

Proposition.
(a) Let x 0 ∈ Ω L and y 0 ∈ M x 0 and let P = (x 0 , y 0 ). The following statements are equivalent.
(i) Near P , the only singularities of π L,x 0 are S 1 k ,0 singularities, for k ≤ d − 2.
(b) Let y 0 ∈ Ω R and x 0 ∈ M y 0 and let P = (x 0 , y 0 ). The following statements are equivalent.
(i) Near P , the only singularities of π R,y 0 are S 1 k ,0 singularities, for k ≤ d − 2.
It suffices to verify statement (a). There are coordinate systems x = (x ′ , x d ) near x 0 , vanishing at x 0 and y = (y ′ , y d ) near y 0 , vanishing at y 0 so that near P the manifold M is given by y ′ = S(x, y d ) with where g(0) = 0.
In these coordinates we compute the vector fields X and Y in (6.1) and find By induction one verifies that for m = 1, 2, . . .
Consequently we see that the linear independence of the vector fields (adY ) m X at P is equivalent with the linear independence of ∂ m g j /(∂y m d ) at y d = 0. Next, the map π L,x 0 : N L,x 0 → T * x 0 Ω L is in the above coordinates given by and from (2.3-5) we see that the statement (i) is also equivalent with the linear independence of the vectors ∂ m g j /(∂y m d ) at y d = 0. This proves the proposition.
6.2. Families of curves defined by exponentials of vector fields. Let now {γ t (·)} t∈I be a one-parameter family of diffeomorphisms of R n which we can also consider as a family of parametrized curves, t → γ t (x) := γ(x, t).
We shall assume that x varies in an open set Ω, the open parameter interval I is a small neighborhood of 0 and that γ 0 = Id andγ = 0, whereγ denotes d dt (γ t ). Thus for each x, t → γ(x, t) defines a regular curve passing through x. As in the article by Christ, Nagel, Stein and Wainger [11], we may write such a family as for some vector fields X 1 , X 2 , ..., and N ∈ N. The generalized Radon transform is now defined by Rf (x) = f (γ(x, t))χ(t)dt and incidence relation M is given by Besides using the projections π L and π R , there are other ways of describing what it means for the family {γ t (·)} to be maximally nondegenerate, in either a one-or two-sided fashion. One is given in terms the structure of the pullback map with respect to the diffeomorphisms γ t (·), and another is given by the linear independence of certain linear combinations of the vector fields X j and their iterated commutators. We formulate the conditions on the right, with the analogous conditions on the left being easily obtained by symmetry.
6.3. Strong Morin singularities and pull-back conditions. We are working with (6.2) and formulate the pullback condition (P ) R . Form the curve Proposition. Let c 0 = (x 0 , ξ 0 , x 0 , η 0 ) ∈ N * M ′ . Then condition (P ) R is satisfied at x 0 if and only if π R has only S + 1 k ,0 singularities at c, with k ≤ d − 2. To see this, note that M ⊂ R n × R n is the image of the immersion (x, t) → (x, γ(x, t)). Thus (x, ξ; y, η) belongs to N * M ′ if and only if y = γ(x, t) for some t ∈ R and (DΦ (x,t) ) * (ξ, −η) = (0, 0) ∈ T * (x,t) R n+1 . This yields For each fixed t, let y = γ t (x), so that x = γ −1 t (y) andγ t (x) =γ t (γ −1 t (y)) = d ds (γ t+s • γ −1 t (y)) = Γ R (y, t). We thus have a parametrization of the canonical relation, which is favorable for analyzing the projection π R . Indeed the equivalence of (P ) R with the strong cusp condition follows immediately from the Lemma in §2.4.
6.4. Pullback and commutator conditions. The bracket condition (B) R for families of curves (6.2) states the linear independence of vector fields X i , i = 1, . . . , n where X 1 = X 1 , X 2 = X 2 and for k = 2, . . . , n (6.6) with universal coefficients a I,k which can be computed from the coefficients of the Campbell-Hausdorff formula ([40,Ch.V.5], see also the exposition in [11]). In particular (6.7) See [56], [26] for the computation of the vector fields X 3 , X 4 and their relevance for folds and cusps. Assuming (P ) R we shall now show that (B) R holds and how one can determine the coefficients in (6.6). By Taylor's theorem in the s variable From this it follows that Γ R (x, t) = ψ(t, X 1 , . . . , X n , . . . ) and thus condition (P ) R becomes the linear independence of ψ, ψ ′ , ..., ψ (n−1) . We will work modulo O(s 2 ) + O(st n+1 ) and so can assume that there are only n vector fields, X 1 , . . . , X n . Compute The first few terms are given by For notational convenience, we let the sum start at m = 2 instead of m = 3 and setc ∅ = 1/2, and for the higher coefficients we getc (1) = −c (2) = 1/12 and c (1,2) =c (2,1) = −1/48. These are enough to calculate the coefficients in (B) R in dimensions less than or equal to five which is the situation corresponding to at most S + 1,1,1,0 (strong swallowtail) singularities. Returning to (P ) R , since we have C 1 = A, C 2 = B, we can use the Kronecker delta notation to write C j = (−1) j n i=1 (t + δ j2 is)t i−1 X i . Now From this we obtain Since thec J 's are known (cf. [40,Ch.V.5], [77]) this allows one to compute the X i 's and this shows that the condition (P ) R is equivalent with a bracket condition (B) R for some coefficients a I,k .
To illustrate this, we restrict to n ≤ 5 and to get a manageable expression we work mod O(t 5 ) and use (6.10); the expression for Γ R (x, t) becomes then which becomes = X 1 + 2t X 2 + 3t 2 X 3 + 4t 3 X 4 + 5t 4 X 5 where the X i are given in (6.7). Thus condition (B) R in dimension n ≤ 5 is the linear independence of the X i for 0 ≤ i ≤ n − 1.

Curves on some nilpotent groups.
Let G be an n dimensional nilpotent Lie group with Lie algebra g. Let γ : R → G be a smooth curve and define G R (t) = (DR γ(t) ) −1 (γ ′ (t)), where DR g denotes the differential of right-translation by g ∈ G. Note that G R : R → T 0 G = g defines a curve in the Lie algebra g.
Lemma. The pullback condition (P ) R for the family of curves t → x·γ(t) −1 is satisfied if and only if the vectors G R (t), G ′ R (t), . . . , G (n−1) R (t) are linearly independent everywhere.