HYPOELLIPTIC ESTIMATE FOR SOME COMPLEX VECTOR FIELDS

LI Wei-xi,LIU Lv-qiao,ZENG Juan

(1.School of Mathematics and Statistics;Statistics and Computational Science Hubei Key Laboratory,Wuhan University,Wuhan 430072,China)

(2.School of Mathematics and Statistics,Wuhan University,Wuhan 430072,China)

1 Introduction and Main Results

Let Ω⊂Rnbe a neighborhood of 0,and denote byithe square root of-1.We consider the following system of complex vector fields

whereφ(x)is a real-valued function defined in Ω.This system was first studied by Treves[4],and considered therein is more general case fortvaries in Rmrather than R.Denote by(ξ,τ)the dual variables of(x,t).Then the principle symbolσfor the system{Pj}1≤j≤nis

with(x,t;ξ,τ)∈T*(Ω×Rt){0},and thus the characteristic set is

Since outside the characteristic set the system{Pj}1≤j≤nis(microlocally)elliptic,we only need to study the microlocal hypoellipticity in the two components{τ＞0}and{τ＜0}under the assumption that

Note we may assumeφ(0)=0 if replacingφbyφ-φ(0).Throughout the paper we will always supposeφsatisfies the following condition of finite type

for some positive integerk.In view of(1.2)it suffices to consider the nontrivial case ofk≥2 for the maximal hypoellipticity.By maximal hypoellipticity(in the sense of Helffer-Nourrigat[2]),it means the existence of a neighborhood⊂Ω of 0 and a constantCsuch that for anywe have

where and throughout paper we use the notationfor vectorvalued functions=(a1,···,an).Note that the maximal hypoellipticity along with the condition(1.3)yields the following subellptic estimate

Thus the subellipticity is in some sense intermediate between the maximal hypoellipticity and the local hypoellipticity.

Observe the system{Pj}1≤j≤nis translation invariant fort.So we may perform partial Fourier transform with respect tot,and study the maximal microhypoellipticity,in the two directionsτ＞0 andτ＜0.Indeed we only need consider without loss of generality the maximal microhypoellipticity in positive directionτ＞0,since the other directionτ＜0 can be treated similarly by replacingφby-φ.Consider the resulting system as follows after taking partial Fourier transform fort∈R.

and we will show the maximal microhypoellipticity at 0 in the positive direction inτ＞0,which means the existence of a positive numberτ0＞0,a constantC＞0 and a neighborhood⊂Ω of 0 such that

where and throughout the paper we denotefor short if no confusion occurs.We remark the operators defined in(1.5)is closely related to the semi-classical Witten Laplacianwithτ-1the semi-classical parameter,by the relationship

where(·,·)L2stands for the inner product inL2(Rn).Helffer-Nier[1]conjecturedis subelliptic near 0 ifφis analytic and has no local minimum near 0,and this still remains open so far.Note(1.6)is a local estimate concerning the sharp regularity near 0∈Rnforτ＞0,and we have also its global counterpart,which is of independent interest for analyzing the spectral property of the resolvent and the semi-classical lower bound of Witten Laplacian.We refer to Helffer-Nier’s work[1]for the detailed presentation on the topic of global maximal hypoellipticity and its application to the spectral analysis on Witten Laplacian.

Theorem 1.1(Maximal microhypoellipticity forτ＞0)Letφbe a polynomial satisfying condition(1.3)withk≥2.Denote byλj,1≤j≤n,the eigenvalues of the Hessian matrix(∂xi∂xjφ)n×n.Suppose there exists a constantC*＞0 such that for anyx∈Ω,we have the following estimates:ifk=2,then

and ifk＞2,then

where∈0＞0 is an arbitrarily small number andμβare given numbers withμβ＞(k-2)/(k-|β|)for 2≤|β|≤k-1.Then the systemPjdefined in(1.1)is maximally microhypoelliptic in positive positionτ＞0,that is,estimate(1.6)holds.

Replacingφby-φwe can get the maximal microhypoellipticity forτ＜0,and thus the maximal hypoellipticity for allτ.

Corollary 1.2(Maximal hypoellipticity)Under the same assumption as Theorem 1.1 with(1.7)and(1.8)replaced by the estimate that for anyx∈Ω,

the systemPjdefined in(1.1)is maximally hypoelliptic,that is,estimate(1.4)holds.

Remark 1.3We need only verify conditions(1.7)and(1.8)for these points where Δφis positive,since it obviously holds for the points where Δφ≤0.

The details of the proof for the main result were given by[3].