ANALYTIC HYPOELLIPTICITY AND THE TREVES CONJECTURE IPOELLITTICIT `A ANALITICA E CONGETTURA DI TREVES

. We are concerned with the problem of the analytic hypoellipticity; precisely, we focus on the real analytic regularity of the solutions of sums of squares with real analytic coeﬃcients. Treves conjecture states that an operator of this type is analytic hypoelliptic if and only if all the strata in the Poisson-Treves stratiﬁcation are symplectic. We discuss a model operator, P , (ﬁrstly appeared and studied in [3]) having a single symplectic stratum and prove that it is not analytic hypoelliptic. This yields a counterexample to the suﬃcient part of Treves conjecture; the necessary part is still an open problem.


Introduction
This paper is concerned with the problem of the analytic hypoellipticity of a sum of squares operator (1) where X j (x, D) is a vector field with real analytic coefficients defined in an open set Ω ⊂ R n .Precisely, we are interested in studying the analytic regularity of the distribution solutions to the equation where u is a distribution in Ω and f ∈ C ω (Ω), the space of all real analytic functions in Ω.
We say that P is analytic hypoelliptic in Ω if P preserves the analytic singular support; namely, if for every u ∈ D (Ω) and every open set V ⊂ Ω, The problem of the C ∞ (Ω) hypoellipticity of (2) has been solved completely by L. Hörmander in 1967, [20], by proving that P is hypoelliptic if the vector fields defining it verify the Hörmander condition (H) The Lie algebra generated by the vector fields and their commutators has dimension n, equal to the dimension of the ambient space.
We point out that, if the X j in (1) are C ∞ vector fields, the condition (H) is only sufficient but not necessary in order for P to be C ∞ hypoelliptic (see Fedii [14], Morimoto [27]).
However, if the coefficients of the X j in (1) are analytic, as in the present case, M. Derridj, [13], showed that then the condition (H) is also necessary for the C ∞ hypoellipticity of P .
Therefore, the analytic setting seems to be a better choice if we are interested in studying the geometric properties of a sum of squares operator.
As a further step in the analysis of the hypoellipticity of P one may ask if, when condition In 1972 M. S. Baouendi and C. Goulaouic [4] produced an example of a sum of squares satisfying condition (H)-and hence C ∞ hypoelliptic-which is not analytic hypoelliptic.
Precisely, consider in R 3 x,y,t the operator and, for a positive , the function Note that u is a C ∞ function near the origin and an easy computation shows that is not analytic at the origin, being In 1996, [40], F. Treves stated a conjecture for the sums of squares of vector fields that takes into account all the cases known to this date (see [3] for a brief survey on this).The conjecture requires some work to be stated precisely; see to this end the papers [40], [10], [41].In what follows we give a brief, sketchy account of how to formulate it.
Let P be as in (2).Then the characteristic variety of P is Char(P ) = {(x, ξ) | X j (x, ξ) = 0, j = 1, . . ., N }.This is a real analytic variety and, as such, it can be stratified in real analytic manifolds, Σ i , for i in a family of indices, having the property that for i = i , Next one examines the rank of the restriction of the symplectic form to the analytic strata Σ i .If there is a change of rank of the symplectic form on a stratum, we may add to the equations of the stratum the equations of the subvariety where there is a change in rank and stratify the so obtained variety into strata which are real analytic manifolds where the symplectic form has constant rank.
In the final step one considers the multiple Poisson brackets of the symbols of the vector fields.Denote by X j (x, ξ) the symbol of the j-th vector field.Let I = (i 1 , i 2 , . . ., i r ), where i j ∈ {1, . . ., N }.Write |I| = r and define r is called the length of the multiple Poisson bracket X I (x, ξ).For each stratum previously obtained, say Σ ik , we want that all brackets of length lesser than a certain integer, say ik vanish, but that there is at least one bracket of length ik which is non zero on Σ ik .One can show that this makes sense and defines a stratification.
Thus the strata obtained are real analytic manifolds where the symplectic form has constant rank and where all brackets of the vector fields vanish if their length is < ik , and there is at least one microlocally elliptic bracket of length ik , ik depending on the stratum.ik is also called the depth of the stratum.
We now state Treves' conjecture: Conjecture 1 (Treves, [40]).The operator P in (2) is analytic hypoelliptic if and only if every stratum in the above described stratification is symplectic.
We remark that the above statement is in agreement with a number of known results.
We note that Baouendi-Goulaouic operator does not have a symplectic characteristic manifold and so one might expect it not to be analytic hypoelliptic.We just would like to mention that a number of results have been published over the last fifteen years in agreement with the conjecture.As a non exhaustive and certainly incomplete list we mention the papers [11], [12], [15], [16], [17], [38], [39] as well as [2], [9], [7], [6], [31].
In [3] Albano-Bove-M. prove that the sufficient part of the Treves conjecture is actually false by showing a counterexample based on an operator whose stratification has just a single symplectic stratum.The study of that operator requires a precise semiclassical analysis of the spectral properties of suitable anharmonic Schrödinger operators.
The purpose of this paper is to discuss a simplified version P of the counterexample in [3]; this choice lead us to consider harmonic oscillators whose spectral properties are explicitly known.Taking advantage of this, the proof of the non analytic hypoellipticity of P will be a little bit shorter.However we refer the reader to [3] for a more general and detailed discussion of the problem.
Although the operator P we shall consider here is less general than the one presented in [3], it is enough to get that a symplectic stratification does not imply analytic hypoellipticity, at least if the dimension of the stratum is ≥ 4. The necessary part of the conjecture, as far as we know, is still an open problem: If there is a non symplectic stratum, so that Hamilton leaves appear, then the operator P is not analytic hypoelliptic.
Here is the structure of the paper.In Section 2 we state the result by considering an operator having a single simplectic stratum.
Section 3 is devoted to the proof of the optimality of the s 0 Gevrey regularity.We construct a solution to P 1 u = 0 which is not better than Gevrey s 0 > 1; hence P is not analytic hypoelliptic.To obtain u we have to discuss a semiclassical spectral problem for a stationary Schrödinger equation with a symmetric double well potential depending on two parameters.
It is known that, since the bottom of the well is quadratic, for very small values of the Planck constant h the eigenvalues, which are simple and positive, behave like the eigenvalues of a harmonic oscillator.

Statement of the Result
The object of this section is to state the optimal Gevrey regularity result for the operator (1) First of all we remark that both P is a sum of squares of vector fields with real analytic coefficients satisfying Hörmander bracket condition, i.e. the whole ambient space is generated when we take iterated commutators of the vector fields in the definition of P .
In particular P is C ∞ hypoelliptic at the origin.This means that there exists an open neighborhood of the origin, Ω, such that for every open set V Ω, 0 ∈ V , we have, for every distribution u ∈ D (Ω).
The characteristic manifold of P is the real analytic manifold (2) According to Treves conjecture one has to look at the strata associated with P .
According to the conjecture we would expect local real analyticity near the origin for the distribution solutions, u, of P 1 u = f , with a real analytic right hand side.
We are ready to state the theorem that is proved in the next section.
Theorem 2.1.P is not analytic hypoelliptic near the origin.
As a consequence of Theorem 2.1 we have Corollary 2.1.The sufficient part of Treves conjecture does not hold.
We note however that for a single symplectic stratum of codimension 2 the conjecture is true (see [12]).

Proof of Theorem 2.1
In this section we construct a solution to the equation P u = 0 which is not Gevrey s for any s < s 0 = 4 3 and is defined in a neighborhood of the origin.This proves Theorem 2.1.
In fact we look for a function u(x, y, t) defined on R x × R y × (R + t ∪ {0}) and such that µ > 0, z(ρ) and M u > 0 are to be determined.Here we assume that x ∈ U , a suitable neighborhood of the origin whose size will ultimately depend on the upper estimate for z(ρ).
We have Rewriting the r.h.s. of the above relation in terms of the variables we obtain Choose now µ = 1 3 .Then the above relation becomes We make the Ansatz that u .
We shall see that condition (2) will be satisfied. Set . We note that, due to condition (2), the quantity in in such a way we have We then obtain Our next step is to find u 2 as a solution of the differential equation where we wrote u instead of u 2 for the sake of simplicity.( 5) becomes , so that the above equation becomes where z 1 (t) = z(ρ).The latter equation can be turned into a stationary semiclassical Schrödinger equation if we perform the canonical dilation Note that t, h are small and positive for large ρ.Thus we may rewrite the above equation as where z 2 (h) = z 1 (t).
We want to show that there are countably many choices for the function z 2 (h) in such a way that equation ( 8) has a non zero solution in L 2 (R), which actually turns out to be a smooth rapidly decreasing function.
Since the term (1 1 2 is a scalar quantity commuting with the other terms, we consider first the operator This is a Schrödinger operator with a symmetric double well potential.The latter is not positive everywhere; in order to work with a positive double well potential we subtract (and add) its minimum.This is Equation ( 8) becomes Let us make the Ansatz that z 2 is a positive valued function.We make the canonical dilation y = xz 2 .
By [5] (Chapter 2, Theorem 3.1) the above Schrödinger operator has a discrete simple spectrum depending in a real analytic way on the parameter hz 2 (h) −3 , for h > 0. Let us denote by E h z 2 (h) 3   an eigenvalue.Let u = u(x, h) be the corresponding eigenfunction.Then (11) becomes (13) 1 Next we are going to solve the above equation w.r.t.z 2 as a function of h, for small positive values of h.Proposition 3.1.There is h 0 > 0 such that equation (13) implicitly defines a function Therefore we may always assume that Proof.The operator in (12) has a symmetric non negative double well potential with two non degenerate minima and unbounded at infinity.From Theorem 1.1 in [34] we deduce that (15) lim µ→0+ E(µ) µ = e * > 0.
We may then continue the function E, by setting Note that f (0, z) = 0. We want to show that the equation f (h, z) = 0 can be uniquely solved w.r.t.z for h ∈ [0, h 0 ], for a suitable h 0 .
Proof of Lemma 3.1.
From Q µ v = E(µ)v we get Due to the self adjointness of Q µ the first terms on both sides of the above identity are equal, so that for every µ > 0, provided v is normalized.Again from Q µ v = E(µ)v we deduce that (18) for µ → 0+.The existence of the right derivative in µ = 0 is a consequence of (15).
For positive h trivially z(h) is real analytic.Let us show that z(h) ∈ C([0, h 0 [).Arguing by contradiction assume that z(h) → z for h → 0+.Then there is a sequence h k → 0+ such that z(h k ) → ẑ = z.Then 0 = f (h k , z(h k )) → f (0, ẑ) which is false since z is the only zero of f (0, z) = 0.