Sums of squares III: hypoellipticity in the infinitely degenerate regime

This is the third in a series of papers dealing with sums of squares and hypoellipticity in the infinitely degenerate regime. We establish a C^2,delta generalization of M. Christ's sum of squares theorem, and use a bootstrap argument with the sum of squares theorem for matrix functions in the second paper of this series, in order to prove a hypoellipticity theorem generalizing work in the infinitely degenerate regime to include nondiagonal operators and more general degeneracies.


Introduction
The regularity theory of second order subelliptic linear equations with smooth coefficients is well established, see e.g.Hörmander [Ho] and Fefferman and Phong [FePh].In [Ho], Hörmander obtained hypoellipticity of sums of squares of smooth vector fields plus a lower order term, whose Lie algebra spans at every point.In [FePh], Fefferman and Phong considered general nonnegative semidefinite smooth self-adjoint linear operators, and characterized subellipticity in terms of a containment condition involving Euclidean balls and "subunit" balls related to the geometry of the nonnegative semidefinite form associated to the operator.Of course subelliptic operators L with smooth coefficients are hypoelliptic, namely every distribution solution u of Lu = φ is smooth when φ is smooth.In the converse direction, Hörmander also showed in [Ho] that a sum of squares of smooth vector fields in R n , with constant rank Lie algebras, is hypoelliptic if and only if the rank is n.See Trèves [Tre] for a treatment of further results on characterizing hypoellipticity in certain special cases.
However, the question of hypoellipticity in general remains largely a mystery.A possible form for a characterization involving the effective symbol σ (x, ξ) (when it exists) is given by Christ in [Chr2], motivated by his main hypoellipticity theorem for sums of squares in the infinitely degenerate regime in [Chr,see Main Theorem 2.3].We will generalize this latter theorem of Christ to hold for C 2,δ symbols, which will play a major role in Theorem 9 below on hypoellipticity in the infinitely degenerate regime.
Thus a basic obstacle to understanding hypoellipticity in general arises when ellipticity degenerates to infinite order in some directions, and we briefly review what is known in this infinite regime here.The theory has only had its surface scratched so far, as evidenced by the results of Fedii [Fe], Kusuoka and Strook [KuStr], Kohn [Koh], Koike [Koi], Korobenko and Rios [KoRi], Morimoto [Mor], Akhunov, Korobenko and Rios [AkKoRi], and the aforementioned paper of Christ [Chr], to name just a few.In the rough infinitely differentiable regime, Rios, Sawyer and Wheeden [RiSaWh] had earlier obtained results analogous to those in [KoRi], where L is 'rough' hypoelliptic if every weak solution u of Lu = φ is continuous when φ is bounded.
In [Fe], Fedii proved that the two-dimensional operator ∂ ∂x 2 + f (x) 2 ∂ ∂y 2 is hypoelliptic merely under the assumption that f is smooth and positive away from x = 0.In [KuStr], Kusuoka and Strook showed using probabilistic methods that under the same conditions on f (x), the three-dimensional analogue ∂z 2 of Fedii's operator is hypoelliptic if and only if lim x ln f (x) = 0.
Morimoto [Mor] and Koike [Koi] introduced the use of nonprobabilistic methods, and further refinements of this approach were obtained in Christ [Chr], using a general theorem on hypoellipticity of sums of squares of smooth vector fields in the infinite regime, i.e.where the Lie algebra does not span at all points.In particular, for the operator then L 3 is hypoelliptic.Moreover, he showed that if some partial derivative of b is nonzero at x = 0, then L 3 is hypoelliptic if and only if the above condition holds.
On the other hand, the novelty in Kohn [Koh], which was generalized in [KoRi], was the absence of any assumption regarding sums of squares of vector fields.This is relevant since it is an open problem whether or not there are smooth nonnegative functions λ on the real line vanishing only at the origin, and to infinite order there, such that they cannot be written as a finite sum λ = N n=1 f 2 n of squares of smooth functions f n .The existence of such examples are attributed to Paul Cohen in both [Bru] and [BoCoRo], but apparently no example has ever appeared in the literature, and the existence of such an example is an open problem, see [Pie,Remark 5.1] 1 .This extends moreover to matrices since if a matrix is a sum of squares (equivalently a sum of positive rank one matrices), then each of its diagonal elements is as well.On the other hand, Kohn makes the additional assumption that λ (x) vanishes only at the origin in R m , something not necessarily assumed in the other aforementioned works.More importantly, Kohn's theorem applies only to operators of Grushin type L (x, D) + λ (x) L (y, D), where the degeneracy λ (x) factors out of the operator λ (x) L (y, D), a restriction that this paper will in part work to remove.
Missing then is a treatment of more general smooth operators L = ∇A (x) ∇ + lower order terms, whose matrix A (x) is comparable to an operator in diagonal form of the types considered above -see Definition 1 below.Our purpose in this paper is to address this more general case in the following setting of realvalued differential operators.Suppose 1 ≤ m < p ≤ n.Let L = ∇A (x) ∇ where A (x) ∼ D λ (x) with x = (x 1 , ..., x m ), x = (x 1 , ..., x n ) and where D λ (x) has C 2 nonnegative diagonal entries λ 1 (x) , ..., λ n (x) depending only on x and positive away from the origin in R m : We will refer to a diagonal matrix having this form for any m < p ≤ n as a Grushin matrix function of type m.Note that the comparability A (x) ∼ D λ (x) impies that a k,k (x) ≈ λ k (x) for all the diagonal entries, so that λ k (x) ≈ a k,k (x, 0) may be assumed smooth without loss of generality.Moreover A (x) ∼ A diag (x, 0) (see [KoSa2,after Definition 10]).All of our theorems will apply to operators L having a Grushin matrix function A (x) of type m that is also elliptical in the sense that A (x) is positive definite for x = 0.Moreover, we will require in addition that the intermediate diagonal entries {a k,k (x)} p−1 k=m+1 (there won't be any such entries in the case p = m + 1) are smooth and strongly C 4,2δ (see [KoSa1]) for some δ > 0 (we show in [KoSa2] that such functions can be written as a sum of squares of C 2,δ functions, and moreover give a sharp ω-monotonicity criterion for strongly C 4,2δ ), and that the off diagonal entries of A (x) satisfy certain strongly subordinate inequalities (which are shown to be sharp in a certain case, see [KoSa2,Theorem 42]).We emphasize that no additional assumptions are made on the last n − p + 1 entries of D (x), which are all equal to λ p (x).
Our approach is broadly divided into four separate steps, the first and second of which are the subject of the first two papers in this series: (1) First, a proof that a C 3,1 function can be written as a finite sum of squares of C 1,1 functions first appeared in Guan [Gua], who attributed the result to Fefferman.In [KoSa1] we adapted treatments of this result from Tataru [Tat] and Bony [Bon] to establish conditions under which a C 4,2δ nonnegative function can be written as a finite sum of squares of C 2,δ functions for some δ > 0. The methods of Tataru and Bony were in turn modelled on a localized splitting of a nonnegative symbol a, due to Fefferman and Phong [FePh], who used it to establish a strong form of Gårding's inequality, and is the main idea behind the result of Fefferman appearing in [Gua].That splitting used the implicit function theorem to write a nonnegative symbol a as a sum of squares plus a symbol depending on fewer variables, so that induction could be applied.This same scheme was used in [KoSa1] to obtain a sum of squares of C 2,δ functions, but taking care to arrange assumptions so that the implicit function theorem applied.(2) Second, in [KoSa2], we showed that under analogous conditions on the diagonal entries of a matrixvalued function M , and strong subordinate-type inequalities on the off diagonal entries, M can then be written as a finite sum of squares of C 2,δ vector fields for some δ > 0.
(3) Third, we here extend a theorem of M. Christ on hypoellipticity of sums of smooth squares of vector fields to the setting of C 2,δ vector fields, with the appropriate notion of gain in a range of Sobolev spaces.(4) Fourth, we here adapt arguments of M. Christ together with the above steps to obtain hypoellipticity of linear operators L of the form where the matrix A and scalar D are smooth functions of x ∈ R n , and with x = (x 1 , ..., x m ), we have , where I m is the m × m identity matrix, and D λ (x) is the (n − m) × (n − m) diagonal matrix with the components of λ (x) = (λ m+1 (x) , ..., λ n (x)) along the diagonal.The component functions λ ℓ (x) satisfy certain natural conditions described explicitly below.We will end this section by stating our main results on hypoellipticity.Then in the next section, we use a result on calculus of rough symbols from the 1980's [Saw] to derive a rough version of M. Christ's hypoellipticity theorem for sums of smooth vector fields in the infinitely degenerate regime, where symbol splitting is inadequate.Finally in the last sections, we use a bootstrap argument that exploits the C 2,δ regularity of the vector fields, to bring all of these results to bear on proving hypoellipticity for linear partial differential operators L of the form (1.1).
But first we recall the main results from the second paper in this series [KoSa2] on sums of squares of matrix functions that we will use here.
Definition 1.Let A and B be real symmetric positive semidefinite n×n matrices.We define A B if B −A is positive semidefinite.Let β < α be positive constants.A real symmetric positive semidefinite n × n matrix A is said to be (β, α)-comparable to a symmetric n × n matrix B, written A ∼ β,α B, if βB A α B, i.e. (1.3) Note that if A is comparable to B, then both A and B are positive semidefinite.Indeed, both 0 ≤ (α − β) ξ tr Bξ and 0 tr ∂A ∂x k (x) CA (x).Finally recall the following seminorm from [Bon], (1.4) [h] α,δ (x) ≡ lim sup y,z→x Here is the sum of squares decomposition with a quasiformal block of order (n − p + 1) × (n − p + 1), where 1 < p ≤ n.We say that a symmetric matrix function Q p (x) is quasiconformal if the eigenvalues λ i (x) of Q p (x) are nonnegative and comparable. and Suppose that A (x) is a C 4,2δ symmetric n × n matrix function of a variable x ∈ R M , which is comparable to a diagonal matrix function D (x), hence comparable to its associated diagonal matrix function A diag (x).
(1) Moreover, assume a p,p (x) ≈ a p+1,p+1 (x) ≈ ... ≈ a n,n (x) and that the diagonal entries a 1,1 (x) , ..., a p−1,p−1 (x) satisfy the following differential estimates up to fourth order, (2) Furthermore, assume the off diagonal entries a k,j (x) satisfy the following differential estimates up to fourth order, (3) Then there is a positive integer I ∈ N such that the matrix function A can be written as a finite sum of squares of C 2,δ vectors X k,j , plus a matrix function A p , where the vectors , and 2 A more general assumption is that of semisubordinaticity, namely ∂A CA for j = 1, 2 and k = 1, 2, ..., n, whose importance arises from the fact that semisubordinaticity of Qp can be used in place of subordinaticity Remark 4. If in addition a k,k (x) ≈ 1 for 1 ≤ k ≤ m < p, then the conditions (1.5) and (1.6) in (1) and (2) are vacuous for 1 ≤ k ≤ m, and moreover the proof shows that the vectors X k,i are actually in These remarks yield the following corollary in which conditions (1.5) and (1.6) in ( 1) and (2) play no role.
Corollary 5. Suppose A (x) is a C 4,δ R M symmetric n×n matrix function that is comparable to a diagonal matrix function.In addition suppose that a k,k (x) ≈ 1 for 1 ≤ k ≤ p−1 and a k,k (x) ≈ a p,p (x) for p ≤ k ≤ n.Then Remark 6.If the diagonal entry a k,k (x) is smooth and ω s -montone on R n for some s > 1 − ε, then the diagonal differential estimates (1.5) above hold for KoSa2,Theorem 18]).
Remark 7. If in Theorem 3, we drop the hypothesis (1.5) that the diagonal entries satisfy the differential estimates, and even slightly weaken the off diagonal hypotheses (1.6), then using the Fefferman-Phong theorem for sums of squares of scalar functions, the proof of Theorem 3 shows that the operator L = ∇ tr A∇ can be written as L = N j=1 X tr j X j where the vector fields X j are C 1,1 for j = 1, 2, ..., N .However, unlike the situation for scalar functions, the example in Theorem 38 of [KoSa2] shows that we cannot dispense entirely with the off diagonal hypotheses (1.6) in (2).Moreover, the space C 1,1 seems not to be sufficient for gaining a positive degree δ of smoothness for solutions to a second order operator, and so this result will neither be used nor proved here.
In this paper we will apply the sums of squares representations for matrix functions obtained in [KoSa2] to a rough generalization of a theorem of M. Christ, that then leads to our main hypoellipticity theorem via a bootstrap argument.

Statement of main hypoellipticity theorems
We begin with the following general hypoellipticity theorem in the infinitely degenerate regime as in Step (4) of the introduction.We emphasize that we make no assumptions regarding the order of vanishing of the matrix function A (x) at the origin.Since we only consider degeneracies at the origin, it is useful to make the following definition.Definition 8. We say that a q × q matrix function f : R n → R q 2 on R n is elliptical if f (x) is positive definite for x = 0.A scalar function f corresponds to the case q = 1.
Theorem 9. Suppose 1 ≤ m < p ≤ n.Let L be a second order real self-adjoint divergence form partial differential operator in R n given by where the matrix A and scalar D are smooth real functions of x ∈ R n , and A (x) is subordinate, i.e. ∂A ∂x k ∇ is subunit with respect to ∇ tr A (x) ∇.
(1) Suppose further that with x = (x 1 , ..., x m ) we have the following Grushin assumption, of Qp in the proof of Theorem 9 below.However, the semisubordinate condition is much harder to pass through the 1-SD in [KoSa2] than is the subordinate condition, and it is ultimately as difficult to deal with as the sum of squares decomposition itself.
(2) Then L is hypoelliptic if Remark 10.Note that when m = 1, it suffices to assume only smoothness of the diagonal entries λ ℓ (x) in place of (1.5), in view of Bony's sum of squares theorem [Bon,Théorème 1].
Here is a variation, without any special hypotheses on the diagonal entries, that will be used to prove Theorem 9 in conjunction with the sum of squares decomposition in Theorem 3.However, the proof of this next result will require a generalization of M. Christ's sum of squares theorem to include C 2,δ vector fields.
(2) Suppose further that there are elliptical scalar functions λ m+1 (x) , ...λ p (x) ∈ C 2 (R n ) with 0 ≤ λ j ≤ 1 for all j, such that Q p (x) ∼ λ p (x) I n−p+1 and such that the following inequalities hold for all Lipschitz functions v: The diagonal inequalities become more demanding the smaller ε is, while the off diagonal inequalities become less demanding.
(3) Finally set and define the Koike functional µ (t, g) for any function g (x) by (4) Then the operator L is hypoelliptic if This is sharp in the sense that (2.7) holds if L is both hypoelliptic and diagonal with monotone entries.
Here is our rough version, in the setting of sums of squares of real vector fields, of M. Christ's hypoellipticity theorem as needed in Step (3) of the introduction.Note in particular that the vector fields X j appearing below are only assumed to be C 2,δ , while the sum of their squares j X tr j X j is assumed to be smooth.
be any ray, and assume that the operator L has the form where the vector fields ) matrix that is subordinate and quasiconformal, and (1) Assume further that (c) Finally, suppose that for each small conic neighborhood Γ of R there exist scalar valued symbols ψ, p ∈ S 0 1,0 such that ψ is everywhere nonnegative, ψ does not depend on ξ in Γ, ψ ≡ 0 in some smaller conic neighborhood of R, ψ ≥ 1 on T * V \Γ, p ≡ 0 in a conic neighborhood of the closure of Γ, and such that for each δ > 0 there exists C δ < ∞ such that for any relatively compact open subset U ⋐ V and for all u ∈ C 2 0 (U ) and each index i, where ξ = ξ p , . . ., ξ n .(2) Then there exists γ > 0 such that for any u ∈ L 2 loc we have , needed in the bootstrap procedure.Indeed, we have denoting Using rough pseudodifferential calculus we have Denoting . We end this section on statements of the main hypoellipticity theorems, by outlining the four steps taken in order to get to the point where we can apply Theorems 3, 11 and 12 to obtain our hypoellipticity Theorem 9.
2.1.Summary of the steps.Consider the operator L = ∇A (x) ∇ + D (x) with smooth coefficients.
(1) We first apply Theorem 3 to write ∇A (x) ∇ = X tr X plus a quasiconformal subordinate term ∇ • Q p (x) ∇, where the vector fields X belong to C 2,δ S 1 1,0 for some δ > 0, and (2) We then use the smooth pseudodifferential calculus to write 1,0 is subunit with respect to the quasiconformal term, and where (4) Finally, we apply Theorem 12 and Theorem 11 to obtain hypoellipticity of L.
Remark 14.Note that if we apply symbol splitting as in [Tay] to the vector fields X to obtain X , then the subunit property of the vector field X is not inherited by the smooth vector field X ♮ .Indeed, the definition of X ♮ shows that it is obtained by applying a mollification of size 2 −jη to a Littlewood-Paley projection onto frequencies of size 2 j , and such mollifications are not comparable when applied to infinitely degenerate fields, even suitably away from the degeneracies.

A rough variant of M. Christ's theorem
We now prove our extension of M. Christ's hypoellipticity theorem, namely Theorem 12, to the case of a sum of squares of rough vector fields, whose sum of squares is nevertheless smooth.We will assume the rough symbols are in the classes C 2,δ S α 1,0 , but we could just as well formulate and prove a variant for the symbol classes C 2,δ S α ρ,η , which we leave for the interested reader, as we will not use such a variant in our applications.The proof of this rough theorem is accomplished by adapting the sum of squares argument of Christ [Chr] in the smooth case.For this we begin with some preliminaries.
3.1.1.Symbols.We begin by recalling in R n , the definition of symbols S m ρ,η from Stein [Ste, Chapter VI], the definition of symbols S m,k ρ,η and S m+ ρ,η from Christ [Chr], and then some results on rough versions of the symbol classes S m ρ,η from [Saw] and [Tay].See also Treves [Tre] for symbols defined in open sets Ω ⊂ R n .Definition 15.Let a (x, ξ) be a smooth function on (1) Define a ∈ S m ρ,η , referred to as a symbol of type (ρ, η) and order m, if For a symbol a ∈ S m ρ,η , the associated pseudodifferential operator A : S (R n ) → S (R n ), also denoted by A = Op a, is defined on the space of rapidly decreasing functions It follows with some work (see e.g.[Ste]) that Op a : S (R n ) → S (R n ) is continuous, and moreover, if a k converges pointwise to a on R n , and (3.1) holds for a = a k uniformly in k, then a ∈ S m ρ,η as well.By duality Op a : S ′ (R n ) → S ′ (R n ) is a continuous map from the space of tempered distributions S ′ (R n ) to itself, and the asymptotic formulas for adjoints and compositions holds without restriction, e.g. if a ∈ S m1 ρ,η and b ∈ S m2 ρ,η , then Op a .
It follows immediately from the definitions that the asymptotic formulas for adjoints and compositions extend to the symbol classes S m+ ρ,η .For example, by uniqueness of the expansions, we have for each ε > 0, and so . Now S m+ ρ,η ⊂ S m,k ρ,η , and it turns out that for our purposes, we apply the pseudodifferential calculus to the symbol classes S m+ ρ,η , as well as to the classes S m,k ρ,η that arise naturally from the hypotheses of the theorems.We will not necessarily make explicit mention of this distinction in the sequel however.
3.1.2.Parametrices.Let a (x, ξ) ∈ S m 1,η be elliptic of order m, i.e. there are strictly positive continuous functions ρ (x) and c (x) in Ω such that the symbol a (x, ξ) satisfies which make sense only for |ξ| ≥ ρ (x).The first three such symbols are given by To deal with the requirement that |ξ| ≥ ρ (x), we select a monotone increasing sequence of continuous functions ρ j+1 (x) > ρ j (x) > ρ (x) and a sequence of smooth cutoff functions .
One can easily prove by induction on j that χ j b j ∈ S −m−j (Ω), and moreover that for carefully chosen such j=1 is a standard exhausting sequence of compact sets for Ω, and if the constants then we need only require in addition that The converse is an easy exercise using only the consequence where 3.1.3.Rough symbols.The following definitions are taken from [Tay] and [Saw].
Definition 18.A symbol σ : R n × R n → R belongs to the rough symbol class C M S m ρ,δ (where M ∈ Z + and 0 ≤ ρ, δ ≤ 1) if for all multiindices α, β with |α| ≤ M , there are constants C α,β such that Here the subscript comp means compactly supported distributions in the space, while the subscript loc means distributions locally in the space.
The following result of Bourdaud is well known, see also [Tay,Section 2.1] and [Saw,Theorem 3].
Theorem 20 ( [Bou, Bou]).For all real m, and all ν > 0 and 0 ≤ δ < 1 we have 3.1.4.Rough pseudodifferential calculus.While symbol smoothing is a very effective and relatively simple tool for use in elliptic and finite type situations, it fails to sufficiently preserve the subunit property of vector fields in the infinitely degenerate regime.For this reason we will instead use the pseudodifferential calculus from [Saw], to which we now turn.
and the double integral on the right hand side is absolutely convergent under the compact support assumption, thus justifying the claim.Given such symbols without the assumption of compact support, we may then consider instead the symbols σ ε and τ ε where ) is 1 on the unit ball, the symbols a ε are uniformly in the same symbol class as a, and hence the above formula persists in the limit when the operators are restricted to acting on the space S of rapidly decreasing functions.Of course it may happen that the resulting symbol σ • τ fails to belong to any reasonable rough symbol class C M+µ S m ρ,δ -see [Saw,Subsection 5.3].Nevertheless, we have the following useful symbol expansion of σ • τ valid up to an error operator in an appropriate class , for every ε > 0.
There is an analogous expansion for the symbol of the adjoint operator (Op σ) tr .
3.1.5.Smooth distributions and wave front sets.The following definitions are taken from Treves [Tre].
, and an open cone Γ 0 ⊂ R n containing ξ 0 such that for every M > 0 there is a positive constant C M satisfying For γ ∈ R, the H γ wave front set of u is defined analogously, where H γ is the Sobolev space of order γ.
3.2.Proof of Theorem 12, the limited smoothness variant of Christ's theorem.Now we can begin our proof of the limited smoothness variant Theorem 12, in the setting of real vector fields, of M. Christ's theorem.Let u ∈ D ′ (V ) and 0 < γ < δ be given.Suppose that the H γ wave front set of Lu is disjoint from some open conic neighbourhood Γ 0 of a point (x 0 , ξ 0 ) ∈ T * V .Without loss of generality we may assume that u ∈ E ′ (V ).Fix an integer K ∈ Z (possibly quite large) such that u ∈ H −K .We will show that (x 0 , ξ 0 ) / ∈ W F H γ (u) by first constructing a pseudodifferential operator Λ, that is elliptic of order γ in a conic neighbourhood of (x 0 , ξ 0 ), and then showing that Λu ∈ H 0 R d .
To do this, let ψ be as in part (1) (c) of Theorem 12. Recall the definitions of the symbol classes S m ρ,η , S m,k ρ,η and S m+ ρ,η : Then following Christ we define a symbol of nonconstant order, depending on parameters γ and N 0 by The nonnegativity of ψ implies that λ ∈ S γ+ 1,0 .Moreover, λ ∈ S γ,0 1,0 .With γ fixed, there exists θ > 0 such that for each N 0 , we have λ ∈ S −θN0+ 1,0 on the closure of the complement of Γ 1 .Now choose N 0 so that −θN 0 < −K.Then with Λ = Op (λ) , we have Λu ∈ H −K+θN0 ⊂ H 0 microlocally on the complement of Γ 1 .Define cutoff functions η 1 , η 2 ∈ C c R d such that η 2 ≡ 1 in a neighbourhood of the support of u, η 1 ≡ 1 in a neighbourhood of the support of η 2 , and Supp η 1 ⊂ V .
Recall that if a ∈ S m ρ,η and b ∈ S n ρ,η , and ρ > η, then Op (a) • Op (b) has a symbol a ⊙ b with an asymptotic expansion The notation ∼ means that for every N , the operator is smoothing of order m + n − N (ρ − η) in the scale of Sobolev spaces.The next lemma is taken verbatim from [Chr], as it involves only symbols of type (1, 0).
Lemma 24 (Lemma 4.1 in [Chr]).There exists an operator Λ −1 ∈ S m+ 1,0 for some m = m (γ) depending on γ, such that Λ • Λ −1 − is smoothing of infinite order.Moreover, such an operator may be constructed with a symbol of the form using the asymptotic expansion (3.5) and the usual iterative procedure as given in (3.3).One obtains f 1 ∈ S −1,2 , and by induction, each f k ∈ S −k+ 1,0 .Choose Λ to be an operator whose full symbol has expansion ∞ k=1 f k , so that the error is smoothing of all orders in the scale of Sobolev spaces.
To prove an analogue of Lemma 4.2 in [Chr] we will need an auxiliary lemma.
Lemma 25.Let P ∈ Op(C ν S m,l 1,0 ) where m, l ∈ N, and let Λ be the operator in (3.4), where where we recall that ψ is everywhere nonnegative, vanishes identically in a small conic neighbourhood of (x 0 , ξ 0 ), and is strictly positive on the complement of Γ 1 .Then for every m ≤ M < ν and some 0 < ε < 1.Moreover, the operator R 1 has the form R 1 = Op ({log λ, σ (P )}) .
Proof.Using Theorem 21 we see that the symbol of ΛP − P Λ divided by λ equals for every M < ν and some 0 < ε < 1; and {log λ, σ (P )} is the Poisson bracket of log λ and σ (P ), and is a symbol in This next lemma is our first analogue of Lemma 4.5 in [Chr].
Lemma 27 (Lemma 4.6 in [Chr]).Suppose that L, ψ, p satisfy the hypotheses of Theorem 12. Then for any N ≥ 0, and for any fixed relatively compact subset U ⊂ V , any δ > 0 and any f ∈ C γ+3 supported in U , the operator G constructed in Lemma 26 satisfies Proof.We first note σ(b j ) = {log λ, σ(X tr j )} = −N 0 log |ξ|{ψ, σ(X tr j )} + symbol in C 1,δ S 0 1,0 , and similarly for b j .Using this together with (3.8 ) and hypothesis (2.10) with δ = δ 0 we therefore obtain The rest of the terms in (3.7) are handled in the same way, giving ( 3.9).
To handle the Grushin type term ∇ • Q p (x) ∇ in (2.8) we will need the following two lemmas Lemma 28.There holds where R ∈ O −ε (−δ,δ) , and with ξ = ξ p , . . ., ξ n , the matrix operator E takes the form Proof.In constructing the symbol of E we will work formally, ignoring the cutoff functions η 1 and η 2 .This is permissible by pseudolocality since Next using Lemma 25 we have where H = {log λ, ξ} ∈ Op S 0,1 1,0 , and H 3 ∈ Op S −1,1 1,0 . To estimate ΛQ p ∇Λ −1 we will need a refinement of Lemma 25, namely, the estimate obtained in the proof where S ∈ O −1−ε (−ν,ν) for some 0 < ε < 1 and 0 < ν < δ.Now σ(P where the last equality holds since ψ does not depend on ξ in Γ, and therefore no logarithmic terms arise from differentiation of log λ with respect to ξ. Altogether we thus have where we note that ξD α ξ log λ ∈ S 0 1,0 for each α with |α| = 1 since ψ does not depend on ξ, and therefore no logarithmic terms arise from differentiation of log λ with respect to ξ.This gives Lemma 29.Let E be a pseudodifferential operator of the form (3.10 ).Then for any fixed relatively compact subset U ⊂ V , any δ > 0 and any f ∈ C ∞ c supported in U , we have Proof.
Here is where we will need to use that the matrix Q p is subordinate -in the case p = n, then Q n is simply a scalar and the subordinate inequality is that of Malgrange.We will use (3.10) and the notation Finally, using the definition of λ we obtain Combining with estimate (2.10) as in the proof of Lemma 27 we conclude (3.12).
Finally, we obtain an estimate on the subunit term R 1 .
We are now ready to prove a generalization of Lemma 4.4 in [Chr], which is the main estimate we need.
Lemma 31 (Lemma 4.4 in [Chr]).Let L take the form (2.8) and satisfy (2.9) and (2.10).Let 0 < γ < δ be fixed.If N 0 is chosen sufficiently large in the definition of Λ, then for any fixed relatively compact U ⋐ V and any u ∈ C 2,δ (U ), (3.15) Proof.Recall that where we used the notation we have since the operators A j and A j have order 0. Similarly We also have from Lemmas 27, 28, and 30 that since η 2 u = u, and hence adding together Thus from (3.9), (3.12), (3.14), and the above we conclude that Combining this with the inequality and the condition Q p ≈ aI n−p+1 we obtain, choosing δ smaller if necessary, Absorbing the terms δ j X j v 2 L 2 and δ|| √ a ∇v|| 2 L 2 into the left hand side, and then using that the order of the error term R is −ε, we obtain where the term involving the H 1 norm of Op (p) Λu may be absorbed into u 2 H −ε since Λ may be made to be regularizing of arbitrary high order in a conic neighborhood of the symbol p, by choosing N 0 to be sufficiently large.Next we write where for the last inequality we used (2.9).Let δ = C/w 2 (N ) and note that δ can be made arbitrarily small by choosing N sufficiently large, we combine the above equality with (3.16) to obtain Choosing δ sufficiently small to absorb the norm v 2 L 2 to the left hand side we conclude for a constant C γ depending on γ.

3.2.1.
Removal of the smoothness assumption.It remains to remove the smoothness assumption u ∈ C 2,δ (U ) in Lemma 31, and to convert the above a priori estimate (3.15) to the desired conclusion Λu ∈ H 0 of Theorem 12.For this we fix a strictly positive smooth function r and we fix a large exponent q.For ε > 0 small define a mollified symbol where r ε (ξ) ≡ r (εξ) q .
where λ (x, ξ) = |ξ| γ e −N0(log|ξ|)φ(x,ξ) for |ξ| ≥ e as in (3.4).Let Λ ε = Op λ ε .The symbols r ε (ξ) satisfy (3.17) If q is chosen sufficiently large relative to the order of the distribution u, then Λ ε u ∈ C 2 for all ε > 0, and since Λ ε is elliptic of order γ in a conic neighbourhood of (x 0 , ξ 0 ), it suffices to show that the L 2 norm of η 1 Λ ε u remains uniformly bounded as ε ց 0. However, Lemma 31 fails to apply since we do not know that the distribution u is a function in C 2,δ (U ), and we now work to circumvent this difficulty.
The parameter N 0 in (3.4) can be chosen sufficiently large that η 1 ΛLu ∈ L 2 because φ is strictly positive in a conic neighbourhood of the H γ wave front set of u, and hence Λ is regularizing there of order at least γ − σN 0 for some constant σ > 0. The L 2 norm of η 1 Λ ε Lu is bounded uniformly in ε > 0 and tends to the L 2 norm of η 1 ΛLu.
As in the proof of Lemma 31, we have for each ε > 0, an operator G ε and an identity with both sides of the equation in C 2 for each ε > 0.Moreover, the differential inequalities (3.17) ensure that the proof of Lemma 31 carries through for each ε > 0 with Λ replaced by Λ ε , so that G ε takes the form (3.7), i.e.
, where the pseudodifferential operator coefficients B 0,ε , B j,ε and B j,ε lie uniformly in the indicated operator classes.A similar argument holds for E ε and J ε .All functions have sufficient differentiability for the proof of Lemma 31 to apply, and this proof, together with the above identity, yield uniformly in ε > 0. We conclude as desired that the L 2 norm of η 1 Λ ε u remains bounded as ε ց 0. Thus we have proved that for any distribution u ∈ D ′ (V ), and any 0 < γ < δ, there is a symbol Λ as in (3.4) that is elliptic of order γ on the conical set Γ, and satisfies The proof of Theorem 12 is now complete.
Combined with the bootstrapping argument above, this shows that u ∈ H s loc (R) for all s ∈ R. Indeed, η 2 u ∈ H −M (R) for some M sufficiently large, and thus we can begin the bootstrapping argument at s = −M .

Proof of Theorem 11
We now prove Theorem 11.The first step is to use a bootstrapping argument to reduce matters to the level of L 2 (R n ).Consider the general second order divergence form operator where A and D are real and smooth, and where A (x) satisfies appropriate form comparability conditions.In order to conclude hypoellipticity of L it is enough to show that there is γ > 0 such that for every s ∈ R, we have the bootstrapping inequality , and γ > 0 fixed, it suffices to show For s ≥ 0 we use and for s ≤ 0 we use to conclude that it suffices to prove The second step is to use the sum of squares assumption in part (1) of Theorem 11 to show that it is sufficient to establish the conditions of Theorem 12.So define and suppose for the moment that the operator L has the simple form where L ∈ S 2 1,0 is smooth and X j ∈ C 2,δ .We first establish the properties of G we need using the rough version of asymptotic expansion from [Saw] given in Theorem 21 above, which we repeat here for the reader's convenience. Suppose , for every ε > 0.
Lemma 32.Let L and G be as in (4.3) and (4.2).Then (−δ/2,δ/2) for every ε > 0, and and so we investigate operators [Λ s , X tr j ] and [Λ s , X j ].The analysis is similar, so we only give details for . Composing with Λ −s and using Op Now we start with an operator L ∈ S 2 1,0 of the more general form where X j ∈ C 2,δ and A 0 ∈ S 1 1,0 .Using Lemma 32 for any operator L in the form (4.5) and Remark 13 we can show that the operator Λ s LΛ −s has the form where X j , B j , Bj , and B 0 are as in Lemma 32 and R 1 is as in Theorem 12. Thus to show hypoellipticity of the operator (4.5), it is sufficient to show that it satisfies the hypotheses of Theorem 12, which completes the second step of the proof.We prepare for the final step of the proof with an auxiliary Lemma (see [Chr,Lemma 5.1]), and its corollary to be used later for showing condition (2.9).
simply positive, and s > 0. Then for any l ∈ {1, . . ., n} there exists a constant C l independent of s such that where the minimum is taken over all x ∈ suppϕ s.t.|x| ≥ s.
Proof.Fix s > 0, for any x ∈ R n we have Switching the order of integration in the last term on the right and making a change of variables y = (x 1 , . . ., x l−1 , tx l , x l+1 , . . ., x n ) we obtain |∂ l ϕ(y)y l | 2 t −1/2 dy t dt s 2 |∂ l ϕ(y)| 2 dy, which combining with the above gives Finally, and thus altogether which implies (4.6).
Lemma 34.Let ϕ and f as in Lemma 33.There exists a strictly positive continuous function w satisfying w(τ ) → ∞ as τ → ∞ such that for every l ∈ {1, . . ., n} and some constant C l > 0 (4.7) Proof.For all s ≥ 0 define f 0 (s) ≡ min and note that f 0 (0) = 0, f 0 (s) > 0 for s = 0, and f 0 is nondecreasing on [0, ∞).Let r = r(τ ) > 0 be the unique point satisfying Define the function w by since 1/s is nonincreasing and f 0 (s) nondecreasing in s we have w(τ ) ≈ 1/r where r is given by (4.8).Therefore, w(τ ) → ∞ as τ → ∞ and using (4.6) with s = r we obtain 4.1.Sufficiency.We can now proceed to complete the sufficiency part of Theorem 11.We note that without loss of generality we may assume that the diagonal entries λ k (x) are smooth.Indeed, from A (x) ∼ D λ (x) we obtain A (x) ∼ A diag (x) and hence (4.9) where the functions a k,k (x, 0, 0) are smooth for 1 ≤ k ≤ n by assumption.
Thus ψ is 1 outside a large ball of radius 3ρ, vanishes inside a small ball of radius ρ, and makes the transition from 0 to 1 in the strip while depending only on the variables x in the strip x, ξ |η| ≤ 2ρ.In the strip x, ξ |η| < ρ, ψ is a function of variables x m+1 , . . ., x n only, and the main step of Christ's application of his theorem occurs now: for each j = 1, . . ., k there exist a j ℓ (x), ℓ = m + 1, . . ., n such that λ p (x), ℓ = p, . . ., n using conditions (2.5), and {ψ, η} = i ∇ψ.
Using the condition |ξ| ≤ ρ |η| this gives for each j = 1, . . ., N upon using the definition of Λ sum (x).To show (2.10) it is therefore sufficient to establish the first inequality in the following display (since the second follows directly from (2.5)) Using the definitions of Λ sum (x) and Λ product (x) we conclude that it is sufficient to show This implies (4.10) by splitting the region of integration into |η| sufficiently large so that δ( η ) ≤ δ, and the region where |η| is bounded, and thus the left hand side of (4.10) is bounded by C||u|| 2 .
To establish (4.11), we first recall for convenience the Koike condition (4.12) lim x→0 µ(|x|, Λ sum ) ln Λ product (x) = 0. Now let φ ∈ C 1 0 (B(0, r)).Then we then have with φ ỹ (ρ) ≡ φ (ρỹ), where in the last line we have applied Hardy's inequality.Fix ϕ ∈ C 1 0 (R m ) as in (4.11).Let χ ∈ C 1 0 (R 1 ) satisfy χ(t) = 1 for |t| ≤ 1 and χ(t) = 0 for |t| ≥ 2, and define the function and the set We can write To estimate the second integral we notice that it vanishes outside the set I(τ ) and thus To estimate the first integral on the right hand side of (4.15) we define Since Supp ϕ is compact, the supremum above is attained at some point z ∈ Supp ϕ, and moreover we have both |z| = r and τ = 2 Λ product (z) .
Combining the above estimate with (4.17) and (4.13) we conclude that shows that p > m + 1 (since lim sup x→0 µ |x| , λ p (x) ln 1 λp(x) = 0) and that there is a pair of distinct indices k, j ∈ {m + 1, ..., p} such that lim sup Our sharpness assertion in Theorem 11 now follows immediately from Proposition 36 and Theorem 37 below.
To prove the Proposition and Theorem, we will need the following lemma (see Hoshiro [Hos,(2.7)]),whose short proof we include here for the reader's convenience.
Lemma 35 (T.Hoshiro [Hos]).Let L be a hypoelliptic operator on R n .For any multiindex β and any subsets Ω, Ω ′ of R n such that Ω ′ ⋐ Ω, there exists N ∈ N and C > 0 such that Proof.Fix Ω ′ ⋐ Ω and consider the set The family of seminorms ||u|| L 2 (Ω ′ ) , ||D α Lu|| L 2 (Ω ′ ) , |α| ∈ N, makes it a Fréchet space.Since L is hypoelliptic we have S ⊂ C ∞ (Ω ′ ), and in particular S ⊂ C M (Ω ′ ) for any M > 0. Now consider the inclusion map This implies u = v, i.e.T is closed.By the closed graph theorem T is continuous, and therefore there exists N ∈ N and C > 0 such that Since the choice of M was arbitrary, this implies (4.18).
The least eigenvalue is given by the Rayleigh quotient formula Let By strong monotonicity of f and h we have This implies using (4.21) where we used (4.22) and the definition of η n for the last two inequalities.It also follows from (4.21) and the fact that Now let v 0 (x, η n ) be an eigenfunction on the ball B = B(0, 1) associated with λ 0 (1, η n ) i.e.

Proof of Theorem 9
Finally, we prove Theorem 9 by showing that the requirements of Theorem 11 are satisified.Let L be as in (2.1).We apply Theorem 3 to obtain A = N j=1 Y j Y tr j + A p , and write the second order term in L as where X j = Y tr j ∇, and then note that condition (2.7) is satisfied by the assumption (2.4) of Theorem 9.Moreover, condition (2.5) follows from (1.7).
6. Open problems 6.1.First problem.In Theorem 9 we have shown that the Koike condition is sufficient for the hypoellipticity of an operator L with n × n matrix A (x) satisfying certain conditions on both its diagonal and nondiagonal entries.However, in the converse direction we only showed that failure of the Koike condition implies failure of hypoellipticity if in addition L is diagonal with strongly monotone entries.In fact the proof shows that we need only assume in addition that A (x) has the block form .
where just a m+1,m+1 (x) and a n,n (x) are assumed to be strongly monotone and satisfy (4.20).
Problem 38.Is the Koike condition actually necessary and sufficient for hypoellipticity under the assumptions of Theorem 9, without assuming the above block form for A (x)?
6.2.Second problem.Recall that the main theorem in [KoRi] extends Kohn's theorem in [Koh] to apply with finitely many blocks instead of the two blocks used in [Koh].These operators are restricted by being of a certain block form, but they are more general in that the elliptic blocks are multiplied by smooth functions that are positive outside the origin, and have more variables than in our theorems, and furthermore that need not be finite sums of squares of regular functions.
Problem 39.Can Theorem 11 be extended to more general operators that include the operators appearing in [KoRi]?6.3.Third problem.What sort of smooth lower order terms of the form B (x) ∇ and ∇ tr C (x) can we add to the operator L in the main Theorem 9? The natural hypothesis to make on the vector fields B (x) ∇ and C (x) ∇ is that they are subunit with respect to ∇ tr A (x) ∇.However, if we use Theorem 11 in the proof, we require more, namely that B (x) ∇ and C (x) ∇ are linear combinations, with C 2,δ coefficients, of the C 2,δ vector fields X j (x) arising in the sum of squares Theorem 3, something which seems difficult to arrange more generally. x→0 even, elliptic, nondecreasing on [0, ∞), and a(x) ≥ b(x) for all x, and if in addition lim sup x→0 |x ln a(x)| = 0, and the coefficient b satisfies lim x→0 b(x)x| ln a(x)| = 0, m−1)×m D {λ m+1 (x),...,λp−1(x)} 0 (p−m−1)×(n−p+1) 0 (n−p+1)×m 0 (n−p+1)×(p−m−1) λ p (x) I n−p+1   .
Now we use the crucial fact that Q p is subordinate, i.e.Q ′