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Hypoellipticity and loss of derivatives. (English) Zbl 1107.35044

Let {X 1 ,X 2 ,,X p } be complex-valued vector fields in n and assume that they satisfy the bracket condition (i.e. that their Lie algebra spans all vector fields), and E:=X i * X i , where X i * is L 2 adjoint of X i . The operator E is subelliptic at point P n if there exists a neighborhood U of P, a real number ϵ>0, and a constant C=(U<ε), such that

u ϵ C(|Eu,u)|+u 2 ),(1)

for all uC 0 (U). The author studies whether E is hypoelliptic and whether it satisfies the subelliptic estimate (1).

If {X i ,{X i ,X j }} spans the complex tangent space at the origin, then a subelliptic estimate

u ε C(X j 2 2 +u 2 ),

is satisfied, with ε=1 2.

For k0 there exist complex vector fields X 1k and X 2 on a neighborhood of the origin in 3 such that the two vectorfields {X 1k ,X 2 } and their commutators of order k+1 span the complexified tangent space at the origin, and when k>0 the subelliptic estimate (1) does not hold. Moreover, when k>1, the operator E k =X 1k * X 1k +X 2 * X 2 loses k derivatives in the sup norms and k-1 derivatives in the Sobolev norms.

If X 1k and X 2 are the vector fields given in Theorem 2 then the operator E k =X 1k * X 1k +X 2 * X 2 is hypoelliptic. More precisely, if u is a distribution solution of Eu=f with uH -s 0 ( 3 ) and if U 3 is an open set such that fH s 2 (U), then uH loc s 2 -k+1 (U). Theorem 2 shows that the loss of derivatives is k-1.

The author introduces subelliptic multipliers to establish subelliptic estimates for the ¯-Neumann problem. To prove these theorems he uses subelliptic multipliers.


MSC:
35H10Hypoelliptic PDE
35H20Subelliptic PDE
35B65Smoothness and regularity of solutions of PDE
58A10Differential forms (global analysis)