Systems Of Complex Vector Fields Research Articles

Hypoellipticity for certain systems of complex vector fields

In this work we study the smooth and Gevrey hypoellipticity for systems of n real-analytic, complex vector fields in n+1 variables. Our results are stated in terms of the properties of a first integral of the system. In particular, we give a partial answer to a conjecture stated by F. Treves in 1981.

Global Denjoy–Carleman hypoellipticity for a class of systems of complex vector fields and perturbations

We characterize global Denjoy–Carleman hypoellipticity for the family of systems acting on the torus $${\mathbb {T}}_{t}^{N} \times {\mathbb {T}}_{x}$$ given by $$L_{j} = \frac{\partial }{\partial t_{j}} + \left( a_{j}(t_{j}) + ib_{j}(t_{j}) \right) \frac{\partial }{\partial x} + \lambda _j$$ , where $$a_j, b_j$$ are real-valued $$2\pi$$ -periodic elements of the classes and $$j = 1, 2,\ldots , N$$ .

Local Hardy-Littlewood-Sobolev inequalities for canceling elliptic differential operators

Let A(x,D) be an elliptic linear differential operator of order ν with smooth complex coefficients in Ω⊂RN from a complex vector space E to a complex vector space F. In this paper we show that if ℓ∈R satisfies 0<ℓ<N and ℓ≤ν, then the estimate(∫RN|Pν−ℓ(x,D)u(x)|q|x|−N+(N−ℓ)qdx)1/q≤C‖A(x,D)u‖L1 holds locally for every u∈Cc∞(U;E) and 1≤q<N/(N−ℓ) assuming A(x,D) is canceling, i.e. ⋂ξ∈RN∖{0}A(x0,ξ)[E]={0} for each x0∈Ω. Here Pν−ℓ(x,D) is a properly supported pseudo-differential operator in Hörmander's class OpS1,δν−ℓ(Ω), 0≤δ<1. This statement is inspired in a new characterization of Hardy-Littlewood-Sobolev inequalities for elliptic and canceling homogeneous operators A(D) with constant coefficients that extends and unifies several results stemming from the classical Hardy-Sobolev estimates. Variants and applications are presented with focus on operators associated to elliptic systems of complex vector fields.

Global Gevrey hypoellipticity on the torus for a class of systems of complex vector fields

Let $L_j = \partial_{t_j} + (a_j+ib_j)(t_j) \partial_x, \, j = 1, \dots, n,$ be a system of vector fields defined on the torus $\mathbb{T}_t^{n}\times\mathbb{T}_x^1$, where the coefficients $a_j$ and $b_j$ are real-valued functions belonging to the Gevrey class $G^s(\mathbb{T}^1)$, with $s>1$. In this paper we were able to characterize the global $s-$hypoellipticity of this system in terms of Diophantine approximations and the Nirenberg-Treves condition (P).

Continuous solutions for divergence-type equations associated to elliptic systems of complex vector fields

In this paper, we characterize all the distributions F∈D′(U) such that there exists a continuous weak solution v∈C(U,Cn) (with U⊂Ω) to the divergence-type equationL1⁎v1+...+Ln⁎vn=F, where {L1,…,Ln} is an elliptic system of linearly independent vector fields with smooth complex coefficients defined on Ω⊂RN. In case where (L1,…,Ln) is the usual gradient field on RN, we recover the classical result for the divergence equation proved by T. De Pauw and W. Pfeffer. Its proof is based on the closed range theorem and inspired by [3] and [6] in the classical case. Our method slightly differs from theirs by relying on the Banach–Grothendieck theorem and introducing tools from pseudodifferential operators, useful in our local setting of a system of complex vector fields with variable coefficients.

Local solvability of a class of degenerate second order operators

ABSTRACTWe study the local solvability of a class of operators with multiple characteristics. The class introduced in this paper exhibits a degeneracy due to the interplay between the singularity associated with the characteristic set of a system of vector fields and the vanishing of a smooth function. It generalizes in part the class introduced by Colombini, Cordaro and Pernazza, for which they proved L2 local solvability. It is shown here, among other results, that local solvability with right-hand side in or and solution in is also possible. In addition, we extend the solvability result to certain systems of complex vector fields, and show that also in this case one has local solvability with right-hand side in , with s = 0 or s = −1/r, r ≥ 1 an integer, with solution in .

Div–curl type estimates for elliptic systems of complex vector fields

In this work, we extend the classical div–curl lemma proved by Coifman, Lions, Meyer and Semmes to the setup of elliptic complexes of complex vector fields with smooth coefficients. The tools include local versions of the Gagliardo–Nirenberg estimate and Rellich–Kondrachov compactness theorem for compactly supported distributions in the local Hardy space hp(RN) that we prove to hold for 0<p≤1.

Local $\mathbf {L}^{1}$ estimates for elliptic systems of complex vector fields

Local $\mathbf {L}^{1}$ estimates for elliptic systems of complex vector fields

Denjoy–Carleman Classes: Boundary Values, Approximate Solutions and Applications

This paper investigates and extends results proved in Petzsche and Vogt (1984) to solutions of systems of complex vector fields. A key ingredient is the existence of approximate solutions for such systems. The edge-of-the-wedge theory in such structures and microlocal analysis of solutions of nonlinear system of vector fields are presented as applications.

On the Borel property for solutions to systems of complex vector fields

In this work we study the Borel property for smooth solutions to systems of complex vector fields associated to locally integrable structures. Inspired by the recent article [6], in which the Borel property was studied for generic submanifolds of the complex space, we prove similar results in this more general set up. In particular we obtain, for the case of corank one structures, a necessary and sufficient condition for the validity of the Borel property.

Strong unique continuation for systems of complex vector fields

In this work we study the property of strong unique continuation, at a given point, for Gevrey solutions to homogeneous systems of PDE defined by complex, real-analytic vector fields in involution. We show that when the system is minimal at the point then the strong unique continuation property holds for Gevrey solutions of order σ∈[1,2] and, furthermore, when the minimality property fails to hold then there are non-trivial Gevrey flat solutions of any given order σ>1. The case of Gevrey order σ>2 is also studied for some particular classes of involutive systems.

Globally solvable systems of complex vector fields

We consider a class of involutive systems of n smooth vector fields on the n+1 dimensional torus. We obtain a complete characterization for the global solvability of this class in terms of Liouville forms and of the connectedness of all sublevel and superlevel sets of the primitive of a certain 1-form in the minimal covering space.

Loss of derivatives for systems of complex vector fields and sums of squares

We discuss, both for systems of complex vector fields and for sums of squares, the phenomenon discovered by Kohn of hypoellipticity with loss of derivatives.

Complex vector fields and hypoelliptic partial differential operators

We prove a subelliptic estimate for systems of complex vector fields under some assumptions that generalize the essential pseudoconcavity for CR manifolds, that was first introduced by two of the authors; and the Hormander's bracket condition for real vector fields. Applications are given to prove the hypoellipticity of first order systems and second order partial differential operators. Finally we describe a class of compact homogeneous CR manifolds for which the distribution of (0,1) vector fields satisfies a subelliptic estimate.

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

For about twenty five years it was a kind of folk theorem that complex vector-fields defined on Ω × R t \Omega \times \mathbb R_t (with Ω \Omega open set in R n \mathbb R^n ) by \[ L j = ∂ ∂ t j + i ∂ φ ∂ t j ( t ) ∂ ∂ x , j = 1 , … , n , t ∈ Ω , x ∈ R , L_j = \frac {\partial }{\partial t_j} + i \frac {\partial \varphi }{\partial t_j}(\mathbf {t})\, \frac {\partial }{\partial x}\;,\; j=1,\dots , n\;,\; \mathbf {t}\in \Omega , x\in \mathbb R, \] with φ \varphi analytic, were subelliptic as soon as they were hypoelliptic. This was the case when n = 1 n=1 , but in the case n &gt; 1 n&gt;1 , an inaccurate reading of the proof given by Maire (see also Trèves) of the hypoellipticity of such systems, under the condition that φ \varphi does not admit any local maximum or minimum (through a nonstandard subelliptic estimate), was supporting the belief for this folk theorem. Quite recently, J.L. Journé and J.M. Trépreau show by examples that there are very simple systems (with polynomial φ \varphi ’s) which are hypoelliptic but not subelliptic in the standard L 2 L^2 -sense. So it is natural to analyze this problem of subellipticity which is in some sense intermediate (at least when φ \varphi is C ∞ C^\infty ) between the maximal hypoellipticity (which was analyzed by Helffer-Nourrigat and Nourrigat) and the simple local hypoellipticity (or local microhypoellipticity) and to start first with the easiest nontrivial examples. The analysis presented here is a continuation of a previous work by the first author and is devoted to the case of quasihomogeneous functions.

Necessary and sufficient conditions for the local solvability in hyperfunctions of a class of systems of complex vector fields

A necessary and sufficient condition is proved for the validity of the Poincare Lemma in degreeq≧1, in the differential complex attached to a locally integrable structure of codimension one, in spaces of hyperfunctions. The base manifold ℳ is only assumed to be smooth. The hyperfunctions are defined in the hypo-analytic structure associated to a smooth “first integral” Z. The condition is that the singular homology of the fibres of the map Z be trivial in dimensionq-1. By the approximation formula of [BT]|the germ of this fibration at a point is independent of the choice of the first integral Z.

Liouville's theorem and the maximum modulus principle for a system of complex vector fields

This work is concerned with Liouville's theorem and the maximum principle for the homogeneous solutions of systems of complex vector fields . Necessary and sufficient conditions are provedfor tube structures to have the Liouville property. Maximum principles are proved for a general system of complex vector fields which are integrable. As an application, in the case of vector fields, we get new characterizations of the local solvability property (P) of Nirenberg andTreves. Another application concerns a solvability condition (Pn-1) introduced by P.Cordaro and J. Hounie in differential complexes associated to locally integrable structures.

Hypoelliptic systems of complex vector fields

0. Introduction. Let X1, . . . , X2n be C∞ real vector fields on Ω open in R and put Lj = 12 (Xj + iXj+n) , 1 ≤ j ≤ n , L = (L1, . . . , Ln) . The following hypotheses are assumed throughout: (H1) (Hormander condition) the brackets of length at most r of the Xj generate TxΩ, ∀x ∈ Ω; (H2) dxξX(x, ξ) are linearly independent; (H3) [Lj , Lk] = 0, ∀j, k ∈ [1, n]. Under (H1), it is well known that the system {Xj}1≤j≤2n is (1 − 1/r)subelliptic, i.e.