# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Hypoellipticity and loss of derivatives. (English) Zbl 1107.35044

Let $\left\{{X}_{1},{X}_{2},\cdots ,{X}_{p}\right\}$ be complex-valued vector fields in ${ℝ}^{n}$ 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 $P\in {ℝ}^{n}$ if there exists a neighborhood $U$ of $P$, a real number $ϵ>0$, and a constant $C=\left(U<\epsilon \right)$, such that

${\parallel u\parallel }_{ϵ}\le {C\left(|Eu,u\right)|+\parallel u\parallel }^{2}\right),\phantom{\rule{2.em}{0ex}}\left(1\right)$

for all $u\in {C}_{0}^{\infty }\left(U\right)$. The author studies whether $E$ is hypoelliptic and whether it satisfies the subelliptic estimate (1).

If $\left\{{X}_{i},\left\{{X}_{i},{X}_{j}\right\}\right\}$ spans the complex tangent space at the origin, then a subelliptic estimate

${\parallel u\parallel }_{\epsilon }\le C\left(\sum \parallel {X}_{j}^{2}{\parallel }^{2}+{\parallel u\parallel }^{2}\right),$

is satisfied, with $\epsilon =\frac{1}{2}$.

For $k\ge 0$ there exist complex vector fields ${X}_{1k}$ and ${X}_{2}$ on a neighborhood of the origin in ${ℝ}^{3}$ such that the two vectorfields $\left\{{X}_{1k},{X}_{2}\right\}$ 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 $u\in {H}^{-{s}_{0}}\left({ℝ}^{3}\right)$ and if $U\subset {ℝ}^{3}$ is an open set such that $f\in {H}^{{s}_{2}}\left(U\right)$, then $u\in {H}_{loc}^{{s}_{2}-k+1}\left(U\right)$. Theorem 2 shows that the loss of derivatives is $k-1$.

The author introduces subelliptic multipliers to establish subelliptic estimates for the $\overline{\partial }$-Neumann problem. To prove these theorems he uses subelliptic multipliers.

##### MSC:
 35H10 Hypoelliptic PDE 35H20 Subelliptic PDE 35B65 Smoothness and regularity of solutions of PDE 58A10 Differential forms (global analysis)