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Hypoellipticity and loss of derivatives. (English) Zbl 1107.35044
Let $\{X_1,X_2,\dots,X_p \}$ be complex-valued vector fields in $\Bbb R^n$ and assume that they satisfy the bracket condition (i.e. that their Lie algebra spans all vector fields), and $E:=\sum X_i^* X_i$, where $X_i^*$ is $L_2$ adjoint of $X_i$. The operator $E$ is subelliptic at point $P \in \Bbb R^n$ if there exists a neighborhood $U$ of $P$, a real number $\epsilon >0$, and a constant $C=(U<\varepsilon)$, such that $$ \|u \|_{\epsilon} \leq C(|Eu,u)| + \|u \|^2),\tag1 $$ for all $u \in C_0^{\infty}(U)$. 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 $$ \|u \|_{\varepsilon} \leq C(\sum \|X_j^2 \|^2 + \|u \|^2), $$ is satisfied, with $\varepsilon=\frac{1}{2}$. For $k \geq 0$ there exist complex vector fields $X_{1k}$ and $X_2$ on a neighborhood of the origin in $\mathbb R^3$ 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 $u \in H^{-s_0}(\mathbb R^3)$ and if $U \subset \mathbb R^3$ is an open set such that $f \in H^{s_2}(U)$, then $u \in H^{s_2-k+1}_{loc}(U)$. Theorem 2 shows that the loss of derivatives is $k-1$. The author introduces subelliptic multipliers to establish subelliptic estimates for the $\bar{\partial}$-Neumann problem. To prove these theorems he uses subelliptic multipliers.

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