Hormander Condition Research Articles

Global hypoellipticity for a class of overdetermined systems of pseudo-differential operators on the torus

This article studies the global hypoellipticity of a class of overdetermined systems of pseudo-differential operators defined on the torus. The main goal consists in establishing connections between the global hypoellipticity of the system and the global hypoellipticity of its normal form. It is proved that an obstruction of number-theoretical nature appears as a necessary condition to the global hypoellipticity. Conversely, the sufficiency is approached by analyzing three types of hypotheses: a Hormander condition, logarithmic growth and super-logarithmic growth.

Estimates of Dirichlet eigenvalues for a class of sub‐elliptic operators

Let $\Omega$ be a bounded connected open subset in $\mathbb{R}^n$ with smooth boundary $\partial\Omega$. Suppose that we have a system of real smooth vector fields $X=(X_{1},X_{2},$ $\cdots,X_{m})$ defined on a neighborhood of $\overline{\Omega}$ that satisfies the Hormander's condition. Suppose further that $\partial\Omega$ is non-characteristic with respect to $X$. For a self-adjoint sub-elliptic operator $\triangle_{X}= -\sum_{i=1}^{m}X_{i}^{*} X_i$ on $\Omega$, we denote its $k^{th}$ Dirichlet eigenvalue by $\lambda_k$. We will provide an uniform upper bound for the sub-elliptic Dirichlet heat kernel. We will also give an explicit sharp lower bound estimate for $\lambda_{k}$, which has a polynomially growth in $k$ of the order related to the generalized Metivier index. We will establish an explicit asymptotic formula of $\lambda_{k}$ that generalizes the Metivier's results in 1976. Our asymptotic formula shows that under a certain condition, our lower bound estimate for $\lambda_{k}$ is optimal in terms of the growth of $k$. Moreover, the upper bound estimate of the Dirichlet eigenvalues for general sub-elliptic operators will also be given, which, in a certain sense, has the optimal growth order.

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.

Local densities for a class of degenerate diffusions

Dans cet article on etudie une classe de processus de type Markov fort continus a valeurs dans $\mathbb{R}^{d}$ qui sont engendres, seulement localement, par un operateur ultra-parabolique avec des coefficients reguliers par rapport a la geometrie intrinseque induite par l’operateur lui-meme et non par rapport a la geometrie euclidienne. On obtient deux resultats importants. Le premier est une formule locale d’Ito valable pour des fonctions qui ne sont pas deux fois differentiables dans le sens classique, mais seulement intrinsequement par rapport a un ensemble de champs de vecteurs, lies au generateur, satisfaisant a la condition d’Hormander. La deuxieme contribution, qui s’appuie sur la premiere, est un resultat d’existence et de regularite pour la densite de transition locale.

Gaussian lower bounds for non-homogeneous Kolmogorov equations with measurable coefficients

We prove Gaussian upper and lower bounds for the fundamental solutions of a class of degenerate parabolic equations satisfying a weak Hormander condition. The bound is independent of the smoothness of the coefficients and generalizes classical results for uniformly parabolic equations.

Sparse and weighted estimates for generalized Hörmander operators and commutators

In this paper a pointwise sparse domination for generalized Hormander and also for iterated commutators with those operators is provided generalizing the sparse domination result in Lerner et al. (Adv Math 319:153–181, 2017). Relying upon that sparse domination a number of quantitative estimates are derived. Some of them are improvements and complementary results to those contained in a series of papers due to Lorente et al. (Stud Math 195(2):157–192, 2009, J Math Anal Appl 342(2):1399–1425, 2008, J Fourier Anal Appl 11(5):497–509, 2005). Also the quantitative endpoint estimates in Lerner et al. (Adv Math 319:153–181, 2017) are extended to iterated commutators. Other results that are obtained in this work are some local exponential decay estimates for generalized Hormander operators in the spirit of Ortiz-Caraballo et al. (Math Ann 357(4):1217–1243, 2013) and some negative results concerning Coifman–Fefferman estimates for a certain class of kernels satisfying particular generalized Hormander conditions.

Bilinear square spectral multipliers on stratified groups

In this paper, on stratified groups, we give a reasonable definition of bilinear square spectral multiplier and its commutator. Then we study their boundedness on weighted Lebesgue spaces under the assumption of its weak boundedness and a sum-type Hormander condition. Besides, we show that, in Euclidean spaces, the smoothness conditions applied to ensure the boundedness of bilinear square Fourier multiplier can be replaced by the sum-type Hormander condition.

Boundedness of Hausdorff operators on Lebesgue spaces and Hardy spaces

In this paper, we study the boundedness of the Hausdorff operator Hϕ on the real line ℝ. First, we start with an easy case by establishing the boundedness of the Hausdorff operator on the Lebesgue space Lp(ℝ) and the Hardy space H1(ℝ). The key idea is to reformulate Hϕ as a Calderon-Zygmund convolution operator, from which its boundedness is proved by verifying the Hormander condition of the convolution kernel. Secondly, to prove the boundedness on the Hardy space Hp(ℝ) with 0 < p < 1; we rewrite the Hausdorff operator as a singular integral operator with the non-convolution kernel. This novel reformulation, in combination with the Taibleson-Weiss molecular characterization of Hp(ℝ) spaces, enables us to obtain the desired results. Those results significantly extend the known boundedness of the Hausdorff operator on H1(ℝ).

Stochastic Completeness and Gradient Representations for Sub-Riemannian Manifolds

Given a second order partial differential operator L satisfying the strong Hormander condition with corresponding heat semigroup $P_{t}$ , we give two different stochastic representations of $dP_{t} f$ for a bounded smooth function f. We show that the first identity can be used to prove infinite lifetime of a diffusion of $\frac {1}{2} L$ , while the second one is used to find an explicit pointwise bound for the horizontal gradient on a Carnot group. In both cases, the underlying idea is to consider the interplay between sub-Riemannian geometry and connections compatible with this geometry.

The Multilinear Littlewood–Paley Operators with Minimal Regularity Conditions

We investigate the quantitative weighted estimates for a large class of the multilinear Littlewood–Paley square operators. Our kernels satisfy the minimal regularity assumption, called $$L^r$$ -Hormander condition. We respectively establish the pointwise sparse domination for the multilinear square functions and their iterated commutators. Based on them, we obtain the strong type quantitative bounds and endpoint estimates. We recover lots of known weighted inequalities for Littlewood–Paley operators. Significantly, the approach is dyadic, quite elementary and simpler than that presented previously.

Kolmogorov–Fokker–Planck operators in dimension two: heat kernel and curvature

We consider the heat equation associated with a class of hypoelliptic operators of Kolmogorov–Fokker–Planck type in dimension two. We explicitly compute the first meaningful coefficient of the small-time asymptotic expansion of the heat kernel on the diagonal, and we interpret it in terms of curvature-like invariants of the optimal control problem associated with the diffusion. This gives a first example of geometric interpretation of the small-time heat kernel asymptotics of non-homogeneous Hormander operators which are not associated with a sub-Riemannian structure, i.e., whose second-order part does not satisfy the Hormander condition.

Interior regularity for degenerate equations with drift on homogeneous groups

Let \(X_0,X_1,\ldots ,X_m\) be left invariant real smooth vector fields on the homogeneous group \(\mathbb {G}=(\mathbb {R}^N,\circ ,\delta _\lambda )\) satisfying the Hormander condition. Assume that \(X_1,\ldots ,X_m\)\((m\le N-1)\) are homogeneous of degree one and \(X_0\) is homogeneous of degree two with respect to the family of dilations \((\delta _\lambda )_{\lambda >0}\). Consider the second order degenerate equation with drift on \(\mathbb {G}\) $$\begin{aligned} \sum _{i,j=1}^{m}X_i(a_{ij}(x)X_ju)+X_0u=\sum _{j=1}^{m}X_jF_j(x), \end{aligned}$$ where the matrix \(\{a_{ij} (x)\}\) of coefficients is symmetric positive definite on \(\mathbb {R}^m\) belonging to the VMO space defined on \(\mathbb {G}\). We prove a Sobolev type inequality for weak solutions to the equation and then the higher \(L^p\) estimates for the gradients of weak solutions. Interior Morrey estimates and Holder continuity for the weak solutions are derived by using the representation formula of weak solutions and estimates of singular integrals.

Strongly degenerate time inhomogeneous SDEs: Densities and support properties. Application to Hodgkin–Huxley type systems

In this paper, we study the existence of densities for strongly degenerate stochastic differential equations (SDEs) whose coefficients depend on time and are not globally Lipschitz. In these models, neither local ellipticity nor the strong Hormander condition is satisfied. In this general setting, we show that continuous transition densities indeed exist in all neighborhoods of points where the weak Hormander condition is satisfied. We also exhibit regions where these densities remain positive. We then apply these results to stochastic Hodgkin–Huxley models with periodic input as a first step towards the study of ergodicity properties of such systems in the sense of Meyn and Tweedie (Adv. in Appl. Probab. 25 (1993) 487–517; Adv. in Appl. Probab. 25 (1993) 518–548).

On $$L^{p}$$ L p -theory for parabolic and elliptic integro-differential equations with scalable operators in the whole space

Elliptic and parabolic integro-differential model problems are considered in the whole space. By verifying Hormander condition, the existence and uniqueness is proved in \(L_{p}\)-spaces of functions whose regularity is defined by a scalable, possibly nonsymmetric, Levy measure. Some rough probability density function estimates of the associated Levy process are used as well.

Modelling anisotropic covariance using stochastic development and sub-Riemannian frame bundle geometry

We discuss the geometric foundation behind the use of stochastic processes in the frame bundle of a smooth manifold to build stochastic models with applications in statistical analysis of non-linear data. The transition densities for the projection to the manifold of Brownian motions developed in the frame bundle lead to a family of probability distributions on the manifold. We explain how data mean and covariance can be interpreted as points in the frame bundle or, more precisely, in the bundle of symmetric positive definite 2-tensors analogously to the parameters describing Euclidean normal distributions. We discuss a factorization of the frame bundle projection map through this bundle, the natural sub-Riemannian structure of the frame bundle, the effect of holonomy, and the existence of subbundles where the Hormander condition is satisfied such that the Brownian motions have smooth transition densities. We identify the most probable paths for the underlying Euclidean Brownian motion and discuss small time asymptotics of the transition densities on the manifold. The geometric setup yields an intrinsic approach to the estimation of mean and covariance in non-linear spaces.

Tube estimates for diffusion processes under a weak Hörmander condition

On considere une diffusion dont les coefficients satisfont une condition d’Hormander faible locale. On obtient des estimees gaussiennes de la densite en temps court et des bornes inferieures et superieures exponentielles pour la probabilite que la diffusion reste dans un petit tube autour d’une trajectoire deterministe (« squelette »). Ces bornes dependent explicitement du rayon du tube et de l’energie du squelette. On utilise une norme qui prend en compte la structure non isotrope du probleme, dans le sens ou la diffusion se propage dans $\mathbb{R}^{2}$ avec des vitesses differentes dans la direction de $\sigma$ et $[\sigma,b]$. On etablit un lien entre cette norme et la distance de controle standard.

On hypoelliptic bridge

A conditioned hypoelliptic process on a compact manifold, satisfying the strong Hormander’s condition, is a hypoelliptic bridge. If the Markov generator satisfies the two step strong Hormander condition, the drift of the conditioned hypoelliptic bridge is integrable on and the hypoelliptic bridge is a continuous semi-martingale.

A parametrix approach for some degenerate stable driven SDEs

We consider a stable driven degenerate stochastic differential equation, whose coefficients satisfy a kind of weak Hormander condition. Under mild smoothness assumptions we prove the uniqueness of the martingale problem for the associated generator under some dimension constraints. Also, when the driving noise is scalar and tempered, we establish density bounds reflecting the multi-scale behavior of the process.

Study of a degenerate elliptic equation in an optimal consumption problem under partial information

We consider an optimal consumption/investment problem under partial information. By using a known reduction scheme, the problem can be transformed to a standard stochastic control problem. The related Hamilton–Jacobi–Bellman (HJB) equation is, after suitable transformation, a second-order linear degenerate elliptic equation. Although the problem is somewhat simple and classical, and the method of solution is well known, it is not obvious that the related HJB equation has a unique classical solution. Here, we prove the existence, uniqueness, and boundary regularity of the solution to the HJB equation. The main components of the proof are the construction of an appropriate weak solution and the verification of the Hormander condition. Using the classical solution, we also prove the verification theorem for the original optimal consumption/investment problem.

One-Dimensional Level Sets of hc-Differentiable Mappings of Carnot–Carathéodory Spaces

We study continuously hc-differentiable mappings from a Carnot-Caratheodory space M such that dim HgM =d imTgM − 1= N for every g ∈ M to an Euclidean N - dimensional space with the property that the hc-differential of the mapping is surjective. We prove that the level set of such a mapping is a curve with Hausdorff dimension 2 in the sub-Riemannian metric. We obtain area formulas for curves of that kind. Bibliography :1 7titles. In this paper, we study the structure and metric properties of level sets of hc-differentiable mappings from a Carnot-Caratheodory space to the Euclidean space. We consider an equiregular Carnot-Caratheodory space (1) that is a connected smooth Riemannian manifold M in the tangent bundle TM with a distinguished subbundle HM (called horizontal) satisfying the Hormander condition (2). By this condition, any two points of M can be joined by ah orizontal curve, i.e., a piecewise smooth curve such that its tangent vector at each point belongs to the horizontal subbundle HM .M oreover, theCarnot-Caratheodory metric dc(x, y), x, y ∈ M ,i s naturally defined as the infimum of lengths of all horizontal curves joining x and y.