Gevrey hypoellipticity for a class of differential polynomials.

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.

In this section we begin to study a pseudo-differential model in suitable simplectic co-ordinates. The conclusion of the proof of Theorem 1 will be given in the subsequent Section 3. As before we denote by x = x 1 , ..., x n the real variables in , open subset of R n ; ξ = ξ 1 , ξ 2 , ..., ξ n , ξ 2 0, the dual variables of x. We consider the conic neighborhood 3 = {0 ξ 2 1 + |ζ | 2 Cξ 2 2 } of the axis ξ 2 0, where ζ = ξ 3 , ..., ξ n ∈ R n−2 , for a suitable constant C. Moreover we take q, m, r, s ∈ N such that m ≥ 2, 1 ≤ q m, and r, s belong to the set I = { r, s ∈ N 2 : 0 qr + ms qm }. Let the function in  × 3 = Ŵ 16 px, ξ = ξ m 1 − h 0,q x, ξ ξ q 2 + X r,s∈I h r,s x, ξ ξ r 1 ξ s 2 , be a differential polynomial, symbol of a micro pseudo-differential operator Px, D, where h ·,· : Ŵ → C , h ·,· = ℜ h ·,· + i ℑ h ·,· , ℜ h ·,· , ℑ h ·,· : Ŵ → R, ℜ h ·,· , ℑ h ·,· ∈ G 1 Ŵ , see below. We define the sets, for k ∈ N, 0 k qm: I k = { r, s ∈ N 2 : qr + ms = k } and fix k = k ∗ such that qm − 1 2 k ∗ qm. We use the notation k − for all k k ∗ and k + for all k k ∗ . We may split I = I − S I k ∗ S I + , with I − = S I k − , I + = S I k + . L EMMA 1. Let px, ξ be the function 16,where h ·,· is assumed to be homogeneous of order zero with respect to ξ and analytic, which implies for some constant L 0 17 |D α x D β ξ h ·,· | ≤ L |α| + |β| + 1 α β 1 + |ξ | −|β| . Assume moreover I k ∗ consists of one couple r ∗ , s ∗ , k ∗ = qr ∗ + ms ∗ , such that: i ℑ h r ∗ , s ∗ x, ξ 6= 0, for all x, ξ ∈ Ŵ, Gevrey hypoellipticity 439 ii ℑ h r ∗ , s ∗ x, ξ ℑ h r,s x, ξ ξ r ∗ +r 1 ξ s+s ∗ 2 ≥ 0, for all x, ξ ∈ Ŵ, k ∗ k + = qr +ms qm , iii ℑ h 0,q x, ξ ℑ h r ∗ , s ∗ x, ξ ξ r ∗ 1 ξ q+s ∗ 2 ≤ 0, for all x, ξ ∈ Ŵ, iv ℜ h 0,q x, ξ 6= 0, for all x, ξ ∈ Ŵ. Then for all α, β ∈ Z n + , for all K ⊂⊂ , we have for new positive constants L and B independent of α , β: 18 |D α x D β ξ px, ξ | |ξ | ρ|β| − δ|α| | px, ξ | ≤ L |α| + |β| + 1 α β , |ξ | B , where ρ = k ∗ − q m−1 m , δ = qm − k ∗ m . Observe that we have δ ρ, since we have assumed k ∗ qm − 1 2 R EMARK 1. Hypothesis ii implies that ℑ h r ∗ , s ∗ x, ξ and ℑ h r,s x, ξ are both positive or both negative ℑ h r,s x, ξ may vanish, too, and that r is according both even or both odd to r ∗ for all r such that k ∗ k + . Otherwise r is not according to r ∗ , ℑ h r,s x, ξ has to vanish in Ŵ. Hypothesis iii induces ℑ h 0,q x, ξ ≡ 0 if r ∗ is odd. R EMARK 2. By formula 18 and by Kajitani-Wakabayashi[5], Theorem 1.9, we have that the operator Px, D, associated to the symbol px, ξ in 16, is G d -microlocally hypoelliptic in Ŵ for d ≥ max n 1 ρ , 1 1−δ o = 1 ρ . R EMARK 3. When ρ 1, and δ 0, one can prove by means of interpolation theory as in Wakabayashi[18], Theorem 2.6 that 18 is valid for any α, β ∈ Z n + , if 18 holds for |α + β| = 1. Hence it is sufficient to verify 18 for |α + β| = 1 because ρ = k ∗ −qm−1 m q m 1, and δ = qm−k ∗ m 0. R EMARK 4. For the proof of Theorem 1 it will be sufficient to apply Lemma 1 for q = m − 1. The general case 1 ≤ q m leads to a more involved geometric invariant statement, which we shall detail in a future paper. Proof of Lemma 1. We first estimate the numerator of 18, then we give some lemmas to esti- mate the denominator of 18. If |α| = 1, |β| = 0, we get |D x j px, ξ | |ξ | −δ = P r,s ∈ I D x j h r,s x, ξ ξ r 1 ξ s 2 − D x j h 0,q x, ξ ξ q 2 |ξ| −δ ≤ L 1 P r,s ∈ I |ξ 1 | r ξ s 2 + ξ q 2 |ξ | −δ , j = 1, ..., n; for a suitable constant L 1 in view of the assuption 17. If |α| = 0, |β| = 1, then |D ξ j px, ξ | |ξ | ρ ≤ L 2   X r,s ∈ I |ξ 1 | r ξ s 2 + ξ q 2   |ξ| ρ 1 + |ξ | −1 , 19 j = 3, ..., n; 440 G. De Donno - L. Rodino for a suitable constant L 2 , in view of 17. Moreover: 20 |D ξ 1 px, ξ | |ξ | ρ ≤ m|ξ 1 | m−1 + L 3 P r,s ∈ I |ξ 1 | r−1 ξ s 2 |ξ | ρ +L 4 P r,s ∈ I |ξ 1 | r ξ s 2 + ξ q 2 |ξ | ρ 1 + |ξ | −1 and 21 |D ξ 2 px, ξ | |ξ | ρ ≤ q L 0,q ξ 2 q−1 + L 5 P r,s ∈ I |ξ 1 | r ξ s−1 2 |ξ | ρ +L 6 P r,s ∈ I |ξ 1 | r ξ s 2 + ξ q 2 |ξ | ρ 1 + |ξ | −1 , for suitable constants L 3 , L 4 , L 5 , L 6 , L 0,q in view of 17. On the other hand, we have: 22 |ξ | ρ 1 + |ξ | −1 ≤ |ξ | −δ , f or all ξ ∈ 3, in fact, by multiplying by |ξ | δ 1 + |ξ | on both sides of 22, we obtain |ξ | − |ξ | p + 1 ≥ 0 , f or all ξ ∈ 3, where p = ρ + δ 1. Then in the right-hand side of 19, 20, 21 we may further estimate |ξ | ρ 1 + |ξ | −1 by |ξ | −δ . Therefore, to prove 18, it will be sufficient to show the boundedness in Ŵ, for |ξ | B, of the functions Q 1 ξ = P r,s ∈ I |ξ 1 | r ξ s 2 + ξ q 2 |ξ | −δ | px, ξ | , Q 2 ξ = m|ξ 1 | m−1 + L 3 P r,s ∈ I |ξ 1 | r−1 ξ s 2 |ξ | ρ | px, ξ | , Q 3 ξ = q L 0,q |ξ 2 | q−1 + L 5 P r,s ∈ I |ξ 1 | r ξ s−1 2 |ξ | ρ | px, ξ | we observe that terms of the type Q 2 , Q 3 were already considered in De Donno [2]. First introduce in the cone 3, three regions: 23 R 1 : c ξ q 2 ≤ |ξ 1 | m ≤ C ξ q 2 , R 2 : |ξ 1 | m ≥ C ξ q 2 , R 3 : |ξ 1 | m ≤ c ξ q 2 ; where the constants c, C satisfy c min n 1 2 mi n x ,ξ ∈Ŵ |ℜ h 0,q x, ξ |, 1 o , and C max 2 max x ,ξ ∈Ŵ |ℜ h 0,q x, ξ |, 1 . The following inequalities then hold: 24 |ξ | −δ ≤      C δ q |ξ 1 | −δ m q , ξ ∈ 3 T R 1 I |ξ 1 | −δ , ξ ∈ 3 T R 2 I I ξ −δ 2 , ξ ∈ 3 T R 3 ; I I I note that II and III hold for all ξ ∈ 3, but for our aim we may limit ourselves to consider them respectively in 3 T R 2 and in 3 T R 3 . By abuse of notation, in the following we shall also denote by R 1 , R 2 , R 3 the sets  × R 1 ,  × R 2 ,  × R 3 ; recall that Ŵ =  × 3. Gevrey hypoellipticity 441 We will show in Lemma 2, Lemma 3 and Lemma 4, that there are positive constants K 1 1, K 2 1, K 3 1, B, such that: 25 | px, ξ | ≥ K 1 ℑh r ∗ , s ∗ x, ξ |ξ 1 | r ∗ ξ s ∗ 2 , i n Ŵ \ R 1 , |ξ | B, 26 | px, ξ | ≥ K 2 |ξ 1 | m , i n Ŵ \ R 2 , |ξ | B, 27 | px, ξ | ≥ K 3 ξ 2 q , i n Ŵ \ R 3 , |ξ | B. In 25 we may further estimate ℑh r ∗ , s ∗ x, ξ λ for λ 0, in view of i in Lemma 1. We first consider Q 1 ξ separately in the regions R 1 , R 2 , R 3 , to prove boundedness. In R 1 by 24,25, we get easily, writing as before k = qr + ms: Q 1 ξ ≤ const   X k 1 |ξ 1 | m− k q + 1   , |ξ| B , where m − k q 0 by definition of I and I k − set s. In the regions R 2 , R 3 by using respectively 24,26 and 24,27, we have for a constant ǫ 0 which we may take as small as we want by fixing B sufficiently large: Q 1 ξ ≤ const   X k 1 |ξ 1 | m+q− k q − k∗ m + 1 |ξ 1 | δ   ǫ , |ξ| B , and Q 1 ξ ≤ const   X k 1 |ξ 2 | 2q− k m − k∗ m + 1 |ξ 2 | δ   ǫ , |ξ| B . We have therefore proved that Q 1 ξ is bounded. Let us estimate Q 2 ξ , Q 3 ξ . As above in the regions R 1 , R 2 , R 3 , we obtain Q 2 ξ ≤ const  1 + X k 1 |ξ 1 | m− k q   , Q 3 ξ ≤ const   1 |ξ 1 | m q −1 + X k 1 |ξ 1 | m+ m q −1− k q   ǫ , in R 1 for |ξ | B, Q 2 ξ ≤ const   1 |ξ 1 | m− k∗ q + X k 1 |ξ 1 | 2m− k q − k∗ q   ǫ , Q 3 ξ ≤ const   1 |ξ 1 | m+ m q −1− k∗ q + X k 1 |ξ 1 | 2m+ m q −1− k q + k∗ q   ǫ , 442 G. De Donno - L. Rodino in R 2 for |ξ | B, Q 2 ξ ≤ const   1 |ξ 2 | q− k∗ m + X k 1 |ξ 2 | 2q− k m − k∗ m   ǫ , Q 3 ξ ≤ const   1 |ξ 2 | 1+q− q m − k∗ m + X k 1 |ξ 2 | 2q+1− q m − k m − k∗ m   ǫ , in R 3 for |ξ | B. Now Lemma 2, Lemma 3 and Lemma 4 complete the proof. L EMMA 2. Let px, ξ be the function 16, such that 17 and i, ii, iii in Lemma 1 hold. Then there are positive constants K 1 1, B, such that: | px, ξ | ≥ K 1 ℑh r ∗ , s ∗ x, ξ |ξ 1 | r ∗ ξ s ∗ 2 , x, ξ ∈ Ŵ \ R 1 , |ξ | B. Proof. We have that 28 | px, ξ | 2 = ξ m 1 − ℜ h 0,q x, ξ ξ q 2 + P r,s∈I ℜ h r,s x, ξ ξ r 1 ξ s 2 2 + + ℑ h r ∗ , s ∗ x, ξ ξ r ∗ 1 ξ s ∗ 2 + P r,s∈I − ℑ h r,s x, ξ ξ r 1 ξ s 2 + + P r,s∈I + ℑ h r,s x, ξ ξ r 1 ξ s 2 − ℑ h 0,q x, ξ ξ q 2 2 ; by removing the terms rising from the real part of px, ξ , we can write | px, ξ | 2 ≥ ℑ h r ∗ , s ∗ x, ξ 2 ξ 2r ∗ 1 ξ 2s ∗ 2 + 4 X j =1 J j x, ξ where J 1 x, ξ =   X r,s∈I − ℑ h r,s x, ξ ξ r 1 ξ s 2 + 29 X r,s∈I + ℑh r,s x, ξ ξ r 1 ξ s 2 − ℑh 0,q x, ξ ξ q 2   2 , 30 J 2 x, ξ = 2ℑ h r ∗ , s ∗ x, ξ X r,s∈I − ℑ h r,s x, ξ ξ r ∗ +r 1 ξ s ∗ +s 2 , 31 J 3 x, ξ = 2ℑ h r ∗ , s ∗ x, ξ X r,s∈I + ℑ h r,s x, ξ ξ r ∗ +r 1 ξ s ∗ +s 2 , 32 J 4 x, ξ = −2ℑ h r ∗ , s ∗ x, ξ ℑ h 0,q x, ξ ξ r ∗ 1 ξ s ∗ +q 2 . Gevrey hypoellipticity 443 29 is non-negative for all x, ξ ∈ Ŵ, 31 and 32 are also non negative by hypotheses ii , iii for all x, ξ ∈ Ŵ. Let us fix attention on J 2 x, ξ defined by 30. We have for all ǫ 0 ℑh r ∗ , s ∗ x, ξ 2 ξ 1 2r ∗ ξ 2s ∗ 2 + J 2 x, ξ ≥ 1 − ǫ ℑh r ∗ , s ∗ x, ξ 2 ξ 1 2r ∗ ξ 2s ∗ 2 , in Ŵ T R 1 , |ξ | B. In fact, assuming for simplicity ξ 1 ≥ 0,by 17, 23 in Ŵ T R 1 and hypothesis i, for all ǫ 0 we get for B sufficiently large |J 2 x, ξ | ℑh r ∗ , s ∗ x, ξ 2 ξ 1 2r ∗ ξ 2s ∗ 2 ≤ const X r,s∈I − ξ r ∗ +r 1 ξ s ∗ +s 2 ξ 1 2r ∗ ξ 2s ∗ 2 ≤ ≤ const X r,s∈I − ξ r ∗ +r+s ∗ +s m q 1 ξ 2r ∗ +2s ∗ m q 1 ǫ , |ξ | B; we remark that k ∗ = qr ∗ + ms ∗ k − = qr + ms. Then, | px, ξ | ≥ K 1 ℑh r ∗ , s ∗ x, ξ |ξ 1 | r ∗ ξ s ∗ 2 , x, ξ ∈ Ŵ \ R 1 , |ξ | B, for a suitable constant K 1 . L EMMA 3. Let px, ξ be the function 16,such that 17 holds. Then there are positive constants K 2 1, B, such that: | px, ξ | ≥ K 2 |ξ 1 | m , x, ξ ∈ Ŵ \ R 2 , |ξ | B. Proof. We write | px, ξ | 2 as in 28; by removing the terms arising from the imaginary part of px, ξ , we get 33 | px, ξ | 2 ≥ ξ m 1 − ℜ h 0,q x, ξ ξ q 2 2 + W 1 x, ξ + W 2 x, ξ where 34 W 1 x, ξ =   X r,s∈I ℜ h r,s x, ξ ξ r 1 ξ s 2   2 , 35 W 2 x, ξ = 2 X r,s∈I ℜ h r,s x, ξ ξ r+m 1 ξ s 2 − 2ℜ h 0,q x, ξ X r,s∈I ℜ h r,s x, ξ ξ r 1 ξ s+q 2 . Observe first that for λ 0 sufficiently small ξ m 1 − ℜ h 0,q x, ξ ξ q 2 2 λ ξ 2m 1 ; in fact ξ m 1 − ℜ h 0,q x, ξ ξ q 2 2 ≥ ξ 2m 1 − 2ℜ h 0,q x, ξ ξ m 1 ξ q 2 , 444 G. De Donno - L. Rodino and using 23 in Ŵ T R 2 , we have for ℜ h 0,q ξ 1 ≥ 0 ξ 2m 1 − 2ℜ h 0,q x, ξ ξ m 1 ξ q 2 ≥ 1 − 2 C ℜ h 0,q x, ξ ξ 2m 1 λ ξ 2m 1 , since C 2 max x ,ξ ∈Ŵ |ℜ h 0,q x, ξ |. 34 is non negative for all x, ξ ∈ Ŵ. We denote 35 by ϒ 1 x, ξ − ϒ 2 x, ξ , then | px, ξ | 2 ≥ λξ 2m 1 + ϒ 1 x, ξ − ϒ 2 x, ξ . Arguing on ϒ 1 , ϒ 2 in the same way as we have done in Lemma 2, it is possible to show that for all ǫ 0 λξ 2m 1 + ϒ 1 x, ξ − ϒ 2 x, ξ ≥ λ − ǫξ 2m 1 , x, ξ ∈ Ŵ \ R 2 , |ξ | B , then | px, ξ | ≥ K 2 |ξ 1 | m , x, ξ ∈ Ŵ \ R 2 , |ξ | B , where K 2 = λ − ǫ 1 2 . L EMMA 4. Let px, ξ be the function 16, such that 17 and iv in Lemma 1 hold. Then there are positive constants K 3 1, B, such that: | px, ξ | ≥ K 3 ξ 2 q , x, ξ ∈ Ŵ \ R 3 , |ξ | B. Proof. We apply again 33, 34, 35 to | px, ξ | 2 . Observe that in Ŵ T R 3 , arguing as above, since c 1 2 min x ,ξ ∈Ŵ |ℜ h 0,q x, ξ |, we obtain for a suitable constant µ 0 ξ m 1 − ℜ h 0,q x, ξ ξ q 2 2 µ ξ 2q 2 . About the terms in 34 and 35, the remarks we have done in Lemma 3 hold by replacing λ ξ 2m 1 with µ ξ 2q 2 , then we have | px, ξ | ≥ K 3 ξ 2 q , x, ξ ∈ Ŵ \ R 3 , |ξ | B , where K 3 = µ − ǫ 1 2 .

3. Fourier integral operators and proof of Theorem 1