Introduction and statement of the result

Rend. Sem. Mat. Univ. Pol. Torino Vol. 58, 4 2000

G. De Donno - L. Rodino GEVREY HYPOELLIPTICITY FOR PARTIAL DIFFERENTIAL

EQUATIONS WITH CHARACTERISTICS OF HIGHER MULTIPLICITY Abstract. We consider a class of partial differential equations with characteristics of constant multiplicity m ≥ 4. We prove for these equations a result of hypoellip- ticity and Gevrey hypoellipticity, by using classical Fourier integral operators and S m ρ,δ arguments.

1. Introduction and statement of the result

This paper concerns the Gevrey hypoellipticity of linear partial differential operators: 1 P = X |α|≤M c α x D α . We use in 1 standard notations, and we assume that the coefficients c α x are analytic, defined in a neighborhood  of a point x ∈ R n . More generally, P could be assumed in the following to be a classical analytic pseudo-differential operator, defined as, for example, in Rodino [15], Tr`eves [16]. We recall that P is said to be hypoelliptic at a neighborhood  of the point x when 2 si ng supp Pu = si ng supp u f or all u ∈ D ′  and Gevrey d-hypoelliptic, 1 d +∞, when 3 d − si ng supp Pu = d − si ng supp u f or all u ∈ D ′ . In 3 the d-singular support of a distribution u is defined as the smallest closed set in the com- plement of which u is a G d function, 1 d +∞, i.e.: it satisfies locally estimates of the type |D α f x| ≤ C |α|+1 α d . We want to study the multiple characteristics case. Namely, consider the principal symbol: p M x, ξ = X |α|=M c α x ξ α . Arguing microlocally, we fix ξ 6= 0 and set: 435 436 G. De Donno - L. Rodino D EFINITION 2. We say that P is an operator with characteristics of constant multiplicity m ≥ 2 at x , ξ if in a conic neighborhood Ŵ ⊂  × R n \0 of x , ξ we may write p M x, ξ = e M−m x, ξ a 1 x, ξ m , where e M−m x, ξ is an analytic elliptic symbol, homogeneous of order M − m, and the first order analytic symbol a 1 x, ξ is real- valued and of microlocal principal type, i.e. d x ,ξ a 1 x, ξ never vanishes and it is not parallel to P n j =1 ξ j d x j on 6 = { x, ξ ∈ Ŵ , a 1 x, ξ = 0}. Observe that 6 is also characteristic manifold of p M x, ξ ; we understand x , ξ ∈ 6 . For P satisfying such definition, we want to study hypoellipticity or, more precisely, micro- hypoellipticity at x , ξ , defined by 4 Ŵ \ W F Pu = Ŵ \ W F u f or all u ∈ D ′  and d-micro-hypoellipticity, defined by 5 Ŵ \ W F d Pu = Ŵ \ W F d u f or all u ∈ D ′ , for a sufficiently small neighborhood Ŵ of x , ξ . See for example H¨ormander [4], Rodino [15] for the definition of the wave front set W F u and Gevrey wave front set W F d u of a distribution u. We observe that 4, 5 imply respectively 2, 3, when satisfied in a conic neighborhood Ŵ of x , ξ for all ξ 6= 0. To express our result we need the so-called subprincipal symbol of P: p ′ M−1 x, ξ = X |α|=M−1 c α x ξ α − 1 2i n X j =1 ∂ 2 ∂ x j ∂ξ j p M x, ξ . We recall that p ′ M−1 has a geometric invariant meaning at 6, see for example H¨ormander [4]; we shall write in the following J x, ξ = p ′ M−1 x, ξ | 6 . Let us assume for simplicity in Ŵ : 6 p M x, ξ i s r eal − valued and, when m i s even, non − negati ve this is not restrictive, if we are allowed to multiply by an elliptic factor passing to the pseudo- differential frame. It is then known from Liess-Rodino [6] that in the case ℑ J x , ξ 6= 0 we have micro- hypoellipticity and d-micro-hypoellipticity at x , ξ for d ≥ m m−1 . In this paper we shall allow ℑ J x , ξ = 0, but assume ℜ J x , ξ 6= 0. To be definite, let us set 7 ℜ J x, ξ 0 f or x, ξ ∈ 6. Fixing attention here on the higher multiplicity case m ≥ 3, we need to consider some other invariants associated to p ′ M−1 , cf. Liess-Rodino [7], Mascarello-Rodino [8]: J r x, ξ, X = 1 r χ r p ′ M−1 x, ξ , Gevrey hypoellipticity 437 for x, ξ, X ∈ N 6, 1 ≤ r ≤ m − 2, where N 6 is the normal bundle to the characteristic manifold 6 and χ is a vector field in Ŵ such that χ x, ξ at x, ξ ∈ 6 is in the equivalence class of X ∈ N x ,ξ 6 . Obviously we have: 8 J r x, ξ, −X = −1 r J r x, ξ, X . For uniformity of notation we shall also regard J as a function on N 6, independent of X at x, ξ . T HEOREM 1. Let P be an operator with characteristics of constant multiplicity m, satisfy- ing 6, 7. Assume moreover there exists r ∗ , 0 r ∗ m−1 2 , such that i ℑ J r ∗ x, ξ, X 6= 0, for all x, ξ, X ∈ N 6, X 6= 0, ii ℑ J r ∗ x, ξ, X ℑ J r x, ξ, X ≥ 0, for all x, ξ, X ∈ N 6, 0 ≤ r r ∗ . Then P is micro-hypoelliptic and d-micro-hypoelliptic for d ≥ m m−1−r ∗ . Let us compare our result with the existing literature. For the sake of brevity, we limit attention to some models in R 2 , satisfying 6, 7 at x 1 = 0, x 2 = 0, ξ 1 = 0, ξ 2 0. We list first the following examples, representative of general classes already considered by other authors: 9 D m x 1 − D m−1 x 2 m ≥ 2 , 10 D m x 1 − D m−1 x 2 + i x 1 D x 1 D m−2 x 2 m ≥ 3 , 11 D m x 1 − D m−1 x 2 + i x 2h 1 D m−1 x 2 m ≥ 2 , 12 D m x 1 − D m−1 x 2 + i x 2h 1 D m−1 x 2 + i x l 1 D x 1 D m−2 x 2 m ≥ 3 . The operators 9, 10 are not hypoelliptic; observe also that 10 is not locally solvable, cf. Corli [1]. The operator 11 is hypoelliptic for any h ≥ 1, despite the fact that ℑ J x , ξ = 0, cf. Menikoff [9], Popivanov [10], Roberts [14]; the operator 12, having the same J as 11, is not hypoelliptic if h is sufficiently large with respect to l ≥ 1, cf. Popivanov-Popov [12], Popivanov [11]. Theorem 1 gives new conditions on J r , i.e. on the coefficient of the terms D r x 1 D m−r−1 x 2 for models of the preceding type, to guarantee hypoellipticity and Gevrey hypoel- lipticity. We have to assumenow m ≥ 4. Let us observe that, if r ∗ is odd, then i, ii in Theorem 1 and 8 actually imply ℑ J r ≡ 0 for even r r ∗ ; as examples of hypoelliptic operators characterized by Theorem 1 consider in this case 13 D m x 1 − D m−1 x 2 + i D x 1 D m−2 x 2 r ∗ = 1, 14 D m x 1 − D m−1 x 2 + i x 2h 1 D x 1 D m−2 x 2 + i D 3 x 1 D m−4 x 2 r ∗ = 3, 438 G. De Donno - L. Rodino having the same J as the non-hypoelliptic operators 9, 10. If r ∗ is even, then i, ii and 8 imply ℑ J r ≡ 0 for odd r r ∗ ; as corresponding example of hypoelliptic operator consider 15 D m x 1 − D m−1 x 2 + i x 2h 1 D m−1 x 2 + i D 2 x 1 D m−3 x 2 r ∗ = 2, having the same J as 11, 12. In 13, 14, 15, the order m has to be chosen sufficiently large, to satisfy the assumption m−1 2 r ∗ . Returning to general operators, we may regard Theorem 1 as extension of a result of Liess-Rodino[7],Theorem 6.3, which prove the same order of Gevrey hypoellipticity, requiring i and ℑ J r = 0 for all r r ∗ , which is stronger than ii; see also Tulovsky [17] for hypoellipticity in the C ∞ -sense. Observe however that Liess- Rodino [7] allow 0 r ∗ ≤ m − 2, whereas we do not know whether our result is valid for m−1 2 ≤ r ∗ ≤ m − 2. The proof of Theorem 1 will be reduced, after conjugation by classical Fourier integral op- erators, to a simple S m ρ,δ argument let us refer in particular to the result of Kajitani-Wakabayashi [5] in the Gevrey frame.

2. Gevrey hypoellipticity for a class of differential polynomials.