Some Properties of Solutions to Weakly Hypoelliptic Equations

Table 1

Examples of hypoelliptic differential operators.

𝐿	𝑀	𝐸	𝐹	ℋ(𝑀,𝐿)	Type	Reference
𝜕/𝜕𝑧, Cauchy-Riemann	Open subset of ℂ	Trivial ℂ-line bundle	Trivial ℂ-line bundle	Holomorphic functions	Elliptic	[15, Sec. I.5]
𝜕𝐸	Complex manifold	Holomorphic vector bundle	𝐸⊗𝑇^∗𝑀^0,1	Holomorphic sections	Overdet. elliptic	[16, Sec. 0.5]
∇^∗∇+𝑙.𝑜.𝑡., Laplace-type	Riemannian manifold	Riem. vector bundle with connection	𝐸	Harmonic sections	Elliptic	[17, Sec. 2.1]
Δ^2, bi-Laplacian	Riemannian manifold	Trivial ℝ-line bundle	Trivial ℝ-line bundle	Biharmonic functions	Elliptic
𝐷, Dirac	Riemannian spin manifold	Spinor bundle	Spinor bundle	Harmonic spinors	Elliptic	[12, Ch. II, §5]
𝜕/𝜕𝑡−Δ, heat operator	Interval × Riemannian manifold	Trivial ℝ-line bundle	Trivial ℝ-line bundle		Parabolic	[14, Sec. 6.4]
Δ𝑋, sub-Laplacian	Sub-Riemannian manifold (bracket generating)	Trivial ℝ-line bundle	Trivial ℝ-line bundle		Subelliptic	[18, Thm. 1.1]
∑𝑗𝑋^∗𝑗𝑋𝑗, where 𝑋𝑗= complex vector fields (bracket cond.)	ℝ^𝑛	Trivial ℂ-line bundle	Trivial ℂ-line bundle		Sobolev-subell.	[19, Thm. C]
Bismut’s hypoelliptic Laplacian	𝑇^∗𝑋	⊕^𝑛𝑗=0∧^𝑗𝑇^∗𝑇^∗𝑋	𝐸		Hypoelliptic	[20, Ch. 3]
𝜕^2/𝜕𝑥^2+𝑥𝜕/𝜕𝑦−𝜕/𝜕𝑡, Kolmogorov	(0,∞)×ℝ^2	Trivial ℝ-line bundle	Trivial ℝ-line bundle		Hypoelliptic	[21, Sec. 22.2]