# Hypoelliptic second order differential equations

@article{Hrmander1967HypoellipticSO, title={Hypoelliptic second order differential equations}, author={Lars H{\"o}rmander}, journal={Acta Mathematica}, year={1967}, volume={119}, pages={147-171} }

that is, if u must be a C ~ function in every open set where Pu is a C ~ function. Necessary and sufficient conditions for P to be hypoelliptic have been known for quite some time when the coefficients are constant (see [3, Chap. IV]). I t has also been shown tha t such equations remain hypoelliptic after a perturbation by a "weaker" operator with variable coefficients (see [3, Chap. VIII) . Using pseudo-differential operators one can extend the class of admissible perturbations further; in…

## 2,014 Citations

Hypoellipticity in the infinitely degenerate regime

- Mathematics
- 2001

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…

Hypoellipticity for a class of the second order partial differential equations

- Mathematics
- 1977

In this paper, we shall investigate the hypoellipticity for a class of degenerate equations of the second order with complex coefficients as a direct extension of the results obtained in [8]. As is…

Hypoellipticity for a class of degenerate elliptic operators of second order

- Mathematics
- 1992

for Xi^O. Hormander's results in [2] can not be applicable to L when (xi) has a zero of infinite order. Compared with higher dimensional cases, the problem in R2 becomes much simpler. So one can…

Failure of analytic hypoellipticity in a class of differential operators

- Mathematics
- 2003

For the hypoelliptic differential operators P = ∂2 x + ( x∂y − x∂t )2 introduced by T. Hoshiro, generalizing a class of M. Christ, in the cases of k and l left open in the analysis, the operators P…

Hypoellipticity for infinitely degenerate elliptic operators

- Mathematics
- 1987

Introduction. In the recent paper [5] Kusuoka-Strook gave a sufficient condition of hypoellipticity for degenerate elliptic operators of second order, as an application of the Malliavin calculus (see…

A Necessary Condition For Analytic Hypoellipticity

- Mathematics
- 1994

A partial differential operator, L, is said to be analytic hypoelliptic in some open set Ω, if for every open U ⊂ Ω and every distribution u ∈ D′(U), if Lu ∈ C(U), then u ∈ C(U). Let X1, X2 be real…

SOME ASPECTS OF DIFFERENTIAL GEOMETRY ASSOCIATED WITH HYPOELLIPTIC SECOND ORDER OPERATORS

- Mathematics
- 1989

I investigate some aspects of the geometry of the ^characteristics of a class of hypoelliptic second order partial differential operators. The resulting geometry looks quite a bit like Riemannian…

Applications of analysis on nilpotent groups to partial differential equations

- Mathematics
- 1977

The last few years have witnessed the birth of a body of techniques for obtaining refined regularity theorems for certain hypoelliptic differential operators through analysis of homogeneous…

Subelliptic estimates for some systems of complex vector fields: Quasihomogeneous case

- Mathematics
- 2006

For about twenty five years it was a kind of folk theorem that complex vector-fields defined on Ω x R t (with Ω open set in R n ) by L j = ∂ ∂t j + i∂φ ∂t j (t)∂ ∂x, j=1,...,n, t ∈ Ω, x ∈R, with φ…

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