Let $P$ be a linear partial differential operator with analytic coefficients. We assume that $P$ is of the form “sum of squares”, satisfying Hörmander’s bracket condition. Let $q$ be a characteristic point for $P$. We assume that $q$ lies on a symplectic Poisson stratum of codimension two. General results of Okaji show that $P$ is analytic hypoelliptic at $q$. Hence Okaji has established the validity of Treves’ conjecture in the codimension two case. Our goal here is to give a simple, self-contained proof of...

We prove that every homogeneous Carnot group can be lifted to a free homogeneous Carnot group. Though following the ideas of Rothschild and Stein, we give simple and self-contained arguments, providing a constructive proof, as shown in the examples.

We prove an approximation lemma on (stratified) homogeneous groups that allows one to approximate a function in the non-isotropic Sobolev space ${\dot{NL}}^{1,Q}$ by $L^{\infty }$ functions, generalizing a result of Bourgain–Brezis. We then use this to obtain a Gagliardo–Nirenberg inequality for $$ on the Heisenberg group ${ℍ}^n$.

We propose and analyze numerical schemes for viscosity solutions of time-dependent Hamilton-Jacobi equations on the Heisenberg group. The main idea is to construct a grid compatible with the noncommutative group geometry. Under suitable assumptions on the data, the Hamiltonian and the parameters for the discrete first order scheme, we prove that the error between the viscosity solution computed at the grid nodes and the solution of the discrete problem behaves like $\sqrt{h}$ where h is the mesh step. Such...

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 Hörmander’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...

We construct a degree theory for Vanishing Mean Oscillation functions in metric spaces, following some ideas of Brezis & Nirenberg. The underlying sets of our metric spaces are bounded open subsets of ${ℝ}^N$ and their boundaries. Then, we apply our results in order to analyze the surjectivity properties of the $L$-harmonic extensions of VMO vector-valued functions. The operators $L$ we are dealing with are second order linear differential operators sum of squares of vector fields satisfying the hypoellipticity...

Let $ℒ$ be a sub-laplacian on a stratified Lie group $G$. In this paper we study the Dirichlet problem for $ℒ$ with $L^p$-boundary data, on domains $\Omega$ which are contractible with respect to the natural dilations of $G$. One of the main difficulties we face is the presence of non-regular boundary points for the usual Dirichlet problem for $ℒ$. A potential theory approach is followed. The main results are applied to study a suitable notion of Hardy spaces.

Our purpose is to generalize the dispersive inequalities for the wave equation on the Heisenberg group, obtained in [1], to H-type groups. On those groups we get optimal time decay for solutions to the wave equation (decay as $t^{-p/2}$) and the Schrödinger equation (decay as $t^{(1-p)/2}$), p being the dimension of the center of the group. As a corollary, we obtain the corresponding Strichartz inequalities for the wave equation, and, assuming that p > 1, for the Schrödinger equation.

For a class of non-selfadjoint $h$–pseudodifferential operators with double characteristics, we give a precise description of the spectrum and establish accurate semiclassical resolvent estimates in a neighborhood of the origin. Specifically, assuming that the quadratic approximations of the principal symbol of the operator along the double characteristics enjoy a partial ellipticity property along a suitable subspace of the phase space, namely their singular space, we give a precise description of...

Nous donnons des résultats analytiques sur les propriétés de régularité du laplacien hypoelliptique de Jean-Michel Bismut et plus généralement sur les opérateurs $P$ de type Fokker-Planck géométrique agissant sur le fibré cotangent $\Sigma =T^{*}X$ d’une variété riemannienne compacte $X$. En particulier, nous prouvons un résultat d’hypoellipticité maximale pour $P$, et nous en déduisons des bornes sur la localisation de ses valeurs spectrales.

The sub-Laplacian on the Heisenberg group is first decomposed into twisted Laplacians parametrized by Planck's constant. Using Fourier-Wigner transforms so parametrized, we prove that the twisted Laplacians are globally hypoelliptic in the setting of tempered distributions. This result on global hypoellipticity is then used to obtain Liouville's theorems for harmonic functions for the sub-Laplacian on the Heisenberg group.

Precedenti risultati riguardanti il principio di massimo generalizzato e la valutazione del primo autovalore per operatori uniformemente ellittici di tipo variazionale vengono estesi agli operatori subellittici di tipo Heisenberg non simmetrici e a coefficienti discontinui.

Let $X=(X_1,X_2,\cdots ,X_q)$ be a system of vector fields satisfying the Hörmander condition. We prove $L_X^{2,\lambda }$ local regularity for the gradient $Xu$ of a solution of the following strongly elliptic system $$-X_{\alpha }^{*}\left(a_{ij}^{\alpha \beta }\left(x\right)X_{\beta }{u}^j\right)=g_i-X_{\alpha }^{*}{f}_i^{\alpha }\left(x\right)\forall i=1,2,\cdots ,N,$$ where $a_{ij}^{\alpha \beta }\left(x\right)$ are bounded functions and belong to Vanishing Mean Oscillation space.

On a real hypersurface $M$ in ${ℂ}^{n+1}$ of class $C^{2,\alpha }$ we consider a local CR structure by choosing $n$ complex vector fields $W_j$ in the complex tangent space. Their real and imaginary parts span a $2n$-dimensional subspace of the real tangent space, which has dimension $2n+1.$ If the Levi matrix of $M$ is different from zero at every point, then we can generate the missing direction. Under this assumption we prove interior a priori estimates of Schauder type for solutions of a class of second order partial differential equations...