Squares Of Vector Fields Research Articles

On a class of sums of squares related to Hamiltonians with a non periodic magnetic field

We consider the operator in (1.1) and prove that it is analytic hypoelliptic. This operator is linked to a stationary Schrödinger equation with a magnetic field and an anharmonic type potential. It is also a sum of squares of vector fields exhibiting a symplectic characteristic variety. This aspect is discussed in the introduction.

On the microlocal regularity of the analytic vectors for “sums of squares” of vector fields

We prove via FBI-transform a result concerning the microlocal Gevrey regularity of analytic vectors for operators sums of squares of vector fields with real-valued real analytic coefficients of Hörmander type, thus providing a microlocal version, in the analytic category, of a result due to Derridj (Pac J Math 302(2):511–543, 2019) concerning the problem of the local regularity for the Gevrey vectors for sums of squares of vector fields with real-valued real analytic/Gevrey coefficients.

Global solvability and propagation of regularity of sums of squares on compact manifolds

We investigate global solvability, in the framework of smooth functions and Schwartz distributions, of certain sums of squares of vector fields defined on a product of compact Riemannian manifolds T × G, where G is further assumed to be a Lie group. As in a recent article due to the authors, our analysis is carried out in terms of a system of left-invariant vector fields on G naturally associated with the operator under study, a simpler object which nevertheless conveys enough information about the original operator so as to fully encode its solvability. As a welcome side effect of the tools developed for our main purpose, we easily prove a general result on propagation of regularity for such operators.

Global estimates for the fundamental solution of homogeneous Hörmander operators

Let $${\mathcal {L}}=\sum _{j=1}^{m}X_{j}^{2}$$ be a Hörmander sum of squares of vector fields in $${\mathbb {R}}^{n}$$ , where any $$X_{j}$$ is homogeneous of degree 1 with respect to a family of non-isotropic dilations in $${\mathbb {R}}^{n}$$ . Then, $${\mathcal {L}}$$ is known to admit a global fundamental solution $$\Gamma (x;y)$$ that can be represented as the integral of a fundamental solution of a sublaplacian operator on a lifting space $${\mathbb {R}}^{n}\times {\mathbb {R}}^{p}$$ , equipped with a Carnot group structure. The aim of this paper is to prove global pointwise (upper and lower) estimates of $$\Gamma$$ , in terms of the Carnot–Carathéodory distance induced by $$X=\{X_{1},\ldots ,X_{m}\}$$ on $${\mathbb {R}}^{n}$$ , as well as global pointwise (upper) estimates for the X-derivatives of any order of $$\Gamma$$ , together with suitable integral representations of these derivatives. The least dimensional case $$n=2$$ presents several peculiarities which are also investigated. Applications to the potential theory for $${\mathcal {L}}$$ and to singular-integral estimates for the kernel $$X_{i}X_{j}\Gamma$$ are also provided. Finally, most of the results about $$\Gamma$$ are extended to the case of Hörmander operators with drift $$\sum _{j=1}^{m}X_{j}^{2}+X_{0}$$ , where $$X_{0}$$ is 2-homogeneous and $$X_{1},\ldots ,X_{m}$$ are 1-homogeneous.

Global estimates in Sobolev spaces for homogeneous Hörmander sums of squares

Let L=∑j=1mXj2 be a Hörmander sum of squares of vector fields in space Rn, where any Xj 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 L in the X-Sobolev spaces WXk,p(Rn), where X={X1,…,Xm}. In our approach, we combine local results for general Hörmander sums of squares, the homogeneity property of the Xj's, plus a global lifting technique for homogeneous vector fields.

Gevrey regularity for a class of sums of squares of monomial vector fields

Analytic or Gevrey hypoellipticity is proved for a class of sums of squares of vector fields having a symplectic characteristic manifold of dimension 2 and arbitrary (even) codimension. We note that this class contains examples for which the Treves stratification seems to work as well as examples for which the Treves stratification does not identify properly the non symplectic stratum.

Analytic regularity for solutions to sums of squares: an assessment

We present a brief survey on the state of the theory of the real analytic regularity (real analytic hypoellipticity) for the solutions to sums of squares of vector fields satisfying the Hormander condition.

On Li–Yau gradient estimate for sum of squares of vector fields up to higher step

In this paper, we generalize the Cao-Yau's gradient estimate for the sum of squares of vector fields up to higher step under assumption of the generalized curvature-dimension inequality. With its applications, by deriving a curvature-dimension inequality, we are able to obtain the Li-Yau gradient estimate for the CR heat equation in a closed pseudohermitian manifold of nonvanishing torsion tensors. As consequences, we obtain the Harnack inequality and upper bound estimate for the CR heat kernel.

On extension problem, trace Hardy and Hardy's inequalities for some fractional Laplacians

We obtain generalised trace Hardy inequalities for fractional powers of general operators given by sums of squares of vector fields. Such inequalities are derived by means of particular solutions of an extended equation associated to the above-mentioned operators. As a consequence, Hardy inequalities are also deduced. Particular cases include Laplacians on stratified groups, Euclidean motion groups and special Hermite operators. Fairly explicit expressions for the constants are provided. Moreover, we show several characterisations of the solutions of the extension problems associated to operators with discrete spectrum, namely Laplacians on compact Lie groups, Hermite and special Hermite operators.

On the Gevrey regularity for sums of squares of vector fields, study of some models

The Gevrey hypoellipticity of a class of “sums of squares” with real analytic coefficients is studied in detail.

Analytic and Gevrey hypoellipticity for perturbed sums of squares operators

We prove a couple of results concerning pseudodifferential perturbations of differential operators being sums of squares of vector fields and satisfying Hormander’s condition. The first is on the minimal Gevrey regularity: if a sum of squares with analytic coefficients is perturbed with a pseudodifferential operator of order strictly less than its subelliptic index it still has the Gevrey minimal regularity. We also prove a statement concerning real analytic hypoellipticity for the same type of pseudodifferential perturbations, provided the operator satisfies to some extra conditions (see Theorem 1.2 below) that ensure the analytic hypoellipticity.

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.

Wave equations on graded groups and hypoelliptic Gevrey spaces

The overall goal of this dissertation is to investigate certain classical results from harmonic analysis, replacing the Euclidean setting, the abelian structure and the elliptic Laplace operator with a non-commutative environment and hypoelliptic operators. More specifically, we consider wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups with time-dependent non-negative propagation speeds that are H\older-regular or even more so. The corresponding Euclidean problem has been extensively studied in the `80s and some additional results have been recently obtained by Garetto and Ruzhansky in the case of a compact Lie group. We establish sharp well-posedness results in the spirit of the classical result by Colombini, De Giorgi and Spagnolo. In this investigation, a structure reminiscent of Gevrey regularity appears, inspiring deeper investigation of certain classes of functions and a comparison with Gevrey classes. In the latter part of this thesis we discuss such Gevrey spaces associated to the sums of squares of vector fields satisfying the H\ormander condition on manifolds. This provides a deeper understanding of the Gevrey hypoellipticity of sub-Laplacians. It is known that if $\mathcal L$ is a Laplacian on a closed manifold $M$ then the standard Gevrey space $\gamma^s$ on $M$ defined in local coordinates can be characterised by the condition that $\|e^{D\mathcal L^{\frac{1}{2s}}}\phi\|_{L^2(M)} 0$. The aim in this part is to discuss the conjecture that a similar characterisation holds true if $\mathcal L$ is H\ormander's sum of squares of vector fields, with a sub-Laplacian version of the Gevrey spaces involving these vector fields only. We prove this conjecture in one direction, while in the other we show it holds for sub-Laplacians on $SU(2)$ and on the Heisenberg group $\mathbb{H}_n$.

Analytic Hypoellipticity for a New Class of Sums of Squares of Vector Fields in $${\mathcal {R}}^3$$ R 3

This paper is devoted to a substantial generalization of previous work on the analytic hypoellipticity of sums of squares \(P=\sum _1^4X^2_j\) of real vector fields with real analytic coefficient in three variables. For p(x, y) quasi-homogeneous in (x, y), consider the vector fields $$\begin{aligned} X_1 = \frac{\partial }{\partial x}, \quad X_2=-\frac{\partial }{\partial y} + p(x,y)\frac{\partial }{\partial t}, \quad X_3=x^{n_1}\frac{\partial }{\partial t}, \quad X_4=y^{n_2}\frac{\partial }{\partial t}, \end{aligned}$$ \( n_1, n_2 \ne 0\). We show that the operator $$\begin{aligned} P=\sum _1^4 X_j^2, \end{aligned}$$ well known to be \(C^\infty \)-hypoelliptic, is actually analytic hypoelliptic near the origin in \({\mathcal {R}}^3\).

On a new method of proving Gevrey hypoellipticity for certain sums of squares

We consider an operator being a sum of squares of vector fields. It has the form, p,r∈N,P(x,Dx,Dy,Dt)=Dx2+x2(p−1)(Dy−xrDt)2. This type of operator is C∞ hypoelliptic by Hörmander's theorem, [18]. Its analytic or Gevrey hypoellipticity has then been studied by a number of authors and is relevant in relation to the Treves conjecture. The Poisson–Treves stratification of P includes both symplectic and non-symplectic strata.In this paper we show that P is Gevrey (p+r)/p hypoelliptic, by constructing a parametrix whose symbol belongs to some exotic classes. One can also show that this number is optimal.

Wave equation for sums of squares on compact Lie groups

In this paper we investigate the well-posedness of the Cauchy problem for the wave equation for sums of squares of vector fields on compact Lie groups. We obtain the loss of regularity for solutions to the Cauchy problem in local Sobolev spaces depending on the order to which the Hörmander condition is satisfied, but no loss in globally defined spaces. We also establish Gevrey well-posedness for equations with irregular coefficients and/or multiple characteristics. As in the Sobolev spaces, if formulated in local coordinates, we observe well-posedness with the loss of local Gevrey order depending on the order to which the Hörmander condition is satisfied.

Gevrey hypoellipticity for sums of squares of vector fields in $ \R^2 $ with quasi-homogeneous polynomial vanishing

Gevrey hypoellipticity for sums of squares of vector fields in $ \R^2 $ with quasi-homogeneous polynomial vanishing

Wave Front Set of Solutions to Sums of Squares of Vector Fields

The authors study the (micro)hypoanalyticity and the Gevrey hypoellipticity of sums of squares of vector fields in terms of the Poisson-Treves stratification. The FBI transform is used. They prove hypoanalyticity for several classes of sums of squares and show that their method, though not general, includes almost every known hypoanalyticity result. Examples are discussed.

Minimal Microlocal Gevrey Regularity for "Sums of Squares"

A theorem of minimal microlocal Gevrey regularity is proved for operators that are sums of squares of vector fields with real analytic coefficients, thus providing a microlocal version of a well-known theorem of Derridj and Zuily (“Regularite analytique et Gevrey d'operateurs elliptiques degeneres,” Journal de Mathematiques Pures et Appliquees 52 (1973): 65-80).

Analytic Hypoellipticity at Non-Symplectic Poisson-Treves Strata for Certain Sums of Squares of Vector Fields

We consider an operator P which is a sum of squares of vector fields with analytic coefficients. The operator has a non-symplectic characteristic manifold, but the rank of the symplectic form σ is not constant on Char P. Moreover the Hamilton foliation of the non-symplectic stratum of the Poisson-Treves stratification for P consists of closed curves in a ring-shaped open set around the origin. We prove that then P is analytic hypoelliptic on that open set. And we note explicitly that the local Gevrey hypoellipticity for P is Gk+1 and that this is sharp.