Table 1: Examples of hypoelliptic differential operators.

Type Reference

, Cauchy-Riemann Open subset of Trivial -line bundle Trivial -line bundle Holomorphic functions Elliptic [15, Sec. I.5]
Complex manifold Holomorphic vector bundle Holomorphic sectionsOverdet. elliptic [16, Sec. 0.5]
., Laplace-type Riemannian manifold Riem. vector bundle with connection Harmonic sections Elliptic [17, Sec. 2.1]
, bi-Laplacian Riemannian manifold Trivial -line bundle Trivial -line bundle Biharmonic functions Elliptic
, Dirac Riemannian spin manifold Spinor bundle Spinor bundle Harmonic spinors Elliptic [12, Ch. II, 5]
, heat operator Interval Riemannian manifold Trivial -line bundle Trivial -line bundle Parabolic [14, Sec. 6.4]
, sub-Laplacian Sub-Riemannian manifold (bracket generating) Trivial -line bundle Trivial -line bundle Subelliptic [18, Thm. 1.1]
, where complex vector fields (bracket cond.) Trivial -line bundle Trivial -line bundle Sobolev-subell. [19, Thm. C]
Bismut’s hypoelliptic Laplacian Hypoelliptic [20, Ch. 3]
, Kolmogorov Trivial -line bundle Trivial -line bundle Hypoelliptic [21, Sec. 22.2]