Hypoellipticity of the Kohn Laplacian for Three-dimensional Tubular Cr Structures

(1) ∂̄b = ∂x + i(∂y − φ(x)∂t) where φ ∈ C∞(R) is real-valued. Such CR structures may be realized as the boundaries of tube domains {z : Im z2 > φ(Re z1)} in C. The Levi form may be identified with the function φ′′(x). We always assume that φ is convex, so that the structure is pseudoconvex. By ∂̄∗ b we mean the adjoint of ∂̄b with respect to L (R, dx dy dt); thus ∂̄∗ b = −∂x + i(∂y − φ(x)∂t). The purpose of this note is to characterize hypoellipticity of the Kohn Laplacian ∂̄b ∂̄ ∗ b for this class of CR structures. Main Theorem. For any C∞ pseudoconvex tubular three-dimensional CR structure, the following four conditions are equivalent in any open set. (α) ∂̄b is C ∞ hypoelliptic, modulo its nullspace. (β) The CR structure is not exponentially degenerate. (γ) The pair {∂̄b , ∂̄∗ b} satisfies a superlogarithmic estimate. (δ) There exists s > 0 such that ∂̄b is H s hypoelliptic, modulo its nullspace. The main new result here is the implication [exponential degeneracy] ⇒ [nonhypoellipticity]. The implication [not exponentially degenerate] ⇒ [hypoelliptic] is a sharpening of the known sufficient condition x log φ′′(x)→ 0 as |x| → 0. This work is part of a broader investigation of related problems, concerning both hypoellipticity and global regularity in C∞, C, and Gevrey classes. See [3] and [15] for speculation on some of these matters in a wider context. The essential novelty in this paper is a characterization for a natural, though restricted, class of structures, as opposed to isolated examples; tube domains have long served as prototypical examples. The author had not anticipated obtaining such a characterization, because the behavior of smooth functions vanishing to infinite order can be so wild. For arbitrary smooth, pseudoconvex three-dimensional CR structures, a superlogarithmic estimate for {∂̄b , ∂̄∗ b} implies hypoellipticity [4], but the the converse is false in general [5]. The notions appearing in this characterization are defined as follows.