Hypoellipticity in the Infinitely Degenerate Regime
نویسنده
چکیده
Let {Xj} be a collection of real vector fields with C∞ coefficients, defined in a neighborhood of a point x0 ∈ R. Consider a second order differential operator L = − ∑ j X 2 j + ∑ j αjXj + β where αj, β are C ∞ real coefficients. A well known sufficient condition [17] for L to be C∞ hypoelliptic is that the Lie algebra generated by {Xj} should span the tangent space to R at x0. This bracket condition is by no means necessary; no satisfactory characterization of hypoellipticity exists, and it appears unlikely that one could be found. The purposes of this note are: (1) To establish sufficient conditions for hypoellipticity for operators such as − ∑ j X 2 j , for the Kohn Laplacian on pseudoconvex three-dimensional CR manifolds, and for the ∂̄–Neumann problem in C, in the case of infinite type. (2) To point out an inequality weaker than the subelliptic estimates which implies hypoellipticity, and which is the weakest possible such inequality. (3) To popularize within the ∂̄–Neumann community certain ideas developed in another context. (4) To emphasize the parallel between the theories of C∞ and analytic/Gevrey class hypoellipticity. This paper has undergone several revisions since its first version, written in the Spring of 1996. Although the results and methods employed here are new within the context of the ∂̄–Neumann problem, I have subsequently learned that when viewed in the wider context of sums of squares of vector fields and related operators, they substantially overlap works of other authors, some earlier and some contemporaneous, including but not limited to
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On Local Properties of Some Classes of Infinitely Degenerate Elliptic Differential Operators
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