Hypoellipticity and loss of derivatives

• 成果类型: Article
• 署名作者: Kohn, JJ
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2005.162.943
• 发表日期: 2005
• 页码: 943-982
• 关键词: pseudo-convex domains; Neumann problem; Operators

摘要

Let {X-1,..., X-p} 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). Our object is to study the operator E = Sigma X-i*X-i, where X-i* is the L-2 adjoint of X-i. A result of Hormander is that when the X-i are real then E is hypoelliptic and furthemore it is subelliptic (the restriction of a destribution u to an open set U is smoother then the restriction of Eu to U). When the X-i are complex-valued if the bracket condition of order one is satisfied (i.e. if the {X-i, [(X)i, X-j]} span), then we prove that the operator E is still subelliptic. This is no longer true if brackets of higher order are needed to span. For each k >= 1 we give an example of two complex-valued vector fields, X-1, and X-2, such that the bracket condition of order k + 1 is satisfied and we prove that the operator E = X-1*X-1 + X-2*X-2 is hypoelliptic but that it is not subelliptic. In fact it loses k derivatives in the sense that, for each m, there exists a distribution u whose restriction to an open set U has the property that the (DEu)-Eu-alpha are bounded on U whenever vertical bar alpha vertical bar <= Tn and for some beta, with vertical bar beta vertical bar = m - k + 1, the restriction of D(beta)u to U is not locally bounded.