紧流形上平方和算子的全局亚椭圆性

Some remarks on polar sets to sums of squares

We argue that a necessary condition for hypoellipticity is that the polar is not a spiral domain.

Analytic Hypoellipticity for a Class of Sums of Squares of Vector Fields with Non-Symplectic Characteristic Variety

The recent example of Hanges: $P = \partial_t^2 + t^2Δ_x + \partial^2_{θ(x)}$ in $R^3$ is analytic hypoelliptic in the sense of germs but not in the strong sense in any neighborhood of the origin. And its characteristic variety is non-symplectic. We give a purely $L^2,$ and hence quite flexible, proof of this result and generalizations, and link it to, and contrast it with, the celebrated Baouendi-Goulaouic operator. We point out that the results are consistent with the conjecture of Treves.

Analytic hypoellipticity for $\square_b + c$ on the Heisenberg group: an $L^2$ approach

In an interesting note, E.M. Stein observed some 20 years ago that while the Kohn Laplacian $\square_b$ on functions is neither locally solvable nor (analytic) hypoelliptic, the addition of a non-zero complex constant reversed these conclusions at least on the Heisenberg group, and Kwon reproved and generalized this result using the method of concatenations. Recently Hanges and Cordaro have studied this situation on the Heisenberg group in detail. Here we give a purely $L^2$ proof of Stein's result using the author's now classical construction of $(T^p)_ϕ= ϕT^p +...,$ where $T$ is the 'missing…

Singular Sums of Squares of Degenerate Vector Fields

We simplify and give an alternative proof of hypoellipticity for generalizations of the singular sum of squares of complex vector fields studied by Kohn, with an appendix by Derridj and Tartakoff, in the Annals of Mathematics, vol. 162 no. 2, 2005, pp. 943-986. The main generalization is that the complex vector fields now come from domains of finite type. We also prove real analytic hypoellipticity and the optimality of our estimates.

Local Real Analyticity of Solutions for sums of squares of non-linear vector fields

We show that all smooth solutions of model non-linear sums of squares of vector fields are locally real analytic. A global result for more general operators is presented in a paper by Makhlouf Derridj and the first author under the title "Global Analytic Hypoellipticity for a Class of Quasilinear Sums of Squares of Vector Fields".

A Class of Sums of Squares with a Given Poisson-Treves Stratification

We study a class of sum of squares exhibiting the same Poisson-Treves stratification as the Oleinik-Radkevič operator. We find three types of operators having distinct microlocal structures. For one of these we prove a Gevrey hypoellipticity theorem analogous to our recent result for the corresponding Oleinik-Radkevič operator.

Hypoellipticity and loss of derivatives with Appendix Analyticity and loss of derivatives

For each value of k, two complex vector fields satisfying the bracket condition are exhibited the sum of whose squares is hypoelliptic but not subelliptic - in fact the operator loses k-1 derivatives in Sobolev norms. In the Appendix it is proven that this operator is analytic hypoelliptic.

Analyticity for Singular Sums of Squares of Degenerate Vector Fields

In this paper we prove local analytic hypoellipticity for a degenerate sum of squares of complex vector fields generalizing those of Kohn in "Hypoellipticity and Loss of Derivatives". Kohn's article is to appear in the Annals of Mathematics with an appendix by Derridj and Tartakoff proving local analyticity in that case. Here we consider vector fields corresponding to a CR manifold of finite type so there is a kind of double degeneracy, and prove local analyticity using our balanced localization of high derivatives in the "missing" direction T. In a forthcoming paper, Bove, Derridj, Kohn and T…

环面上的一类实向量场系统的正则性传播

Propagation of regularity for a class of systems of real vector fields on torus

本文研究环面上一类实向量场系统，通过证明解的正则性传播定理，刻画了系统的全局亚椭圆性、几乎亚椭圆性和可解性及相应平方和算子的性质。

On the Gevrey regularity for Sums of Squares of vector fields, study of some models

The micro-local Gevrey regularity of a class of "sums of squares" with real analytic coefficients is studied in detail. Some partial regularity result is also given.

Some non-analytic-hypoelliptic sums of squares of vector fields

Certain second-order partial differential operators, which are expressed as sums of squares of real-analytic vector fields in $\Bbb R^3$ and which are well known to be $C^\infty$ hypoelliptic, fail to be analytic hypoelliptic.

关于一类具有非负特征形式的全局解析亚椭圆算子

On a class of globally analytic Hypoelliptic operators with non-negative characteristic form

本文证明了一类定义在环面上、具有非负特征形式的二阶偏微分方程的全局解析亚椭圆性，并在一定程度上推广了Bove-Chinni和Cordaro-Himonas的平方和类结果。

On a Class of Globally Analytic Hypoelliptic Sums of Squares

We consider sums of squares operators globally defined on the torus. We show that if some assumptions are satisfied the operators are globally analytic hypoelliptic. The purpose of the assumptions is to rule out the existence of a Hamilton leaf on the characteristic variety lying along the fiber of the cotangent bundle, i.e. the case of the (global) Métivier operator.

紧李群上一类全局解析亚椭圆算子

A class of globally analytic hypoelliptic operators on compact Lie groups

本文研究紧李群上一类微分算子的全局解析亚椭圆性，在Hörmander有限型条件、闭子群不变性及横截方向椭圆性三个条件下证明了该性质，并指出这为研究主纤维丛上平方和的 regularity 奠定了基础。

Global estimates in Sobolev spaces for homogeneous Hörmander sums of squares

Let $\mathcal{L}=\sum_{j=1}^m X_j^2$ be a Hörmander sum of squares of vector fields in space $\mathbb{R}^n$, where any $X_j$ is homogeneous of degree $1$ with respect to a family of non-isotropic dilations in space. In this paper we prove global estimates and regularity properties for $\mathcal{L}$ in the $X$-Sobolev spaces $W^{k,p}_X(\mathbb{R}^n)$, where $X = \{X_1,\ldots,X_m\}$. In our approach, we combine local results for general Hörmander sums of squares, the homogeneity property of the $X_j$'s, plus a global lifting technique for homogeneous vector fields.

Existence of positive definite noncoercive sums of squares

Positive definite forms $f$ which are sums of squares are constructed to have the additional property that the members of any collection of forms whose squares sum to $f$ must share a nontrivial complex root.

Local Hardy and Rellich inequalities for sums of squares of vector fields

We prove local refined versions of Hardy's and Rellich's inequalities as well as of uncertainty principles for sums of squares of vector fields on bounded sets of smooth manifolds under certain assumptions on the vector fields. We also give some explicit examples, in particular, for sums of squares of vector fields on Euclidean spaces and for sub-Laplacians on stratified Lie groups.

A remark on sums of squares of complex vector fields

This note is a comment on a recent paper by J. J. Kohn. We give an example of a second order partial differential operator, expressed as a sum of squares of complex vector fields satisfying the bracket condition, that is not hypoelliptic.