arXiv research

Extends pseudo-differential operators theory to compact Lie groups.

We consider the heat equation associated with a class of second order hypoelliptic Hörmander operators with constant second order term and linear drift. We describe the possible small time heat kernel expansion on the diagonal giving a geometric characterization of the coefficients in terms of the divergence of the dri…

Solves division problem for L. Hörmander's systems.

We find the fundamental solution to the p-Laplace equation in a class of Hörmander vector fields that generate neither a Carnot group nor a Grushin-type space. The singularity occurs at the sub-Riemannian points which naturally corresponds to finding the fundamental solution of a generalized operator in Euclidean space…

New index formula for hypoelliptic operators on manifolds.

New characterizations of partial positivity using Hörmander's $L^2$-estimate.

Simplified calculus for manifold operators, proving index theorems.

Recent research connects Hörmander's old work to modern boundary Laplacian analysis.

Establishes uniform Hörmander estimates for flat line bundles on Kähler manifolds.

Researchers compute Wodzicki residue for pseudo-differential operators on compact Lie groups.

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 o…

The paper extends log-Sobolev inequalities to matrix-valued settings using combinatorial methods.

The paper bounds Fourier integral operators on Hardy spaces with specific conditions.

For a second order operator on a compact manifold satisfying the strong Hörmander condition, we give a bound for the spectral gap analogous to the Lichnerowicz estimate for the Laplacian of a Riemannian manifold. We consider a wide class of such operators which includes horizontal lifts of the Laplacian on Riemannian s…

Paper constructs $L^2$ estimates for flat vector bundles and generalizes Prékopa's theorem.

Extends elliptic operator regularity to maximally hypoelliptic operators.

Kähler-Ricci flows' tangent cones are algebraic varieties.

We prove a subelliptic estimate for systems of complex vector fields under some assumptions that generalize the essential pseudoconcavity for $CR$ manifolds and Hörmander's bracket condition for real vector fields. Applications are given to prove the hypoellipticity of first order systems and second order partial diffe…

The paper studies boundedness of pseudo-differential operators on smooth manifolds.

We compute the small time asymptotic of the fundamental solution of Hörmander's type hypoelliptic operators with drift, at a stationary point, $x_0$, of the drift field. We show that the order of the asymptotic depends on the controllability of an associated control problem and of its approximating system. If the contr…

We consider the Harnack inequality for harmonic functions with respect to three types of infinite dimensional operators. For the infinite dimensional Laplacian, we show no Harnack inequality is possible. We also show that the Harnack inequality fails for a large class of Ornstein-Uhlenbeck processes, although functions…

Improved bounds for Carleson-Sjölin operators on manifolds with specific curvature conditions.

Generalizes mean-value inequality to orbifold setting.

By further developing the generalized $Γ$-calculus for hypoelliptic operators, we prove hypocoercive estimates for a large class of Kolmogorov type operators which are defined on non necessarily totally geodesic Riemannian foliations. We study then in detail the example of the velocity spherical Brownian motion, whose …

Proves curvature positivity of invariant direct images in complex geometry.

Extends scaling maps theory to manifolds with boundary.

New curvature assumptions prove Nakano positivity for complex vector bundles.

Study on diffeologies on locally convex spaces and smooth multiplication of distributions.

In this paper, we consider the principal eigenvalue problem for Hormander's laplacian on $R^n$. We also study a related semi-linear sub-elliptic equation in the whole $R^n$ and prove that under a suitable condition, we have infinite many positive solutions of the problem.

Under various elliptic boundary conditions, we obtain lower eigenvalue estimates for Dirac operators by using Hormander's weighted $L^2$-technique. Lower bounds in terms of the volume of the underlying manifolds are also deduced from the sharp Sobolev inequality due to Li and Zhu(\cite{LZ}).

We discuss positivity properties of `distinguished propagators', i.e. distinguished inverses of operators that frequently occur in scattering theory and wave propagation. We relate this to the work of Duistermaat and Hörmander on distinguished parametrices (approximate inverses), which has played a major role in quantu…

We give a definition of the Maslov fibre bundle for a lagrangian submanifold of the cotangent bundle of a smooth manofold. This definition generelizes the definition given, in homotopic terms, by Arnol'd for lagrangian submanifolds of the cotangent bundle of the euclidean space and coincides with the one of Hörmander i…

We extend Bony's propagation of support argument \cite{Bony} to $C^1$ solutions of the non-homogeneous sub-elliptic $p-$Laplacian associated to a system of smooth vector fields satisfying Hörmander's finite rank condition. As a consequence we prove a strong maximum principle and strong comparison principle that general…

As announced in [12], we develop a calculus of Fourier integral G-operators on any Lie groupoid G. For that purpose, we study convolability and invertibility of Lagrangian conic submanifolds of the symplectic groupoid T * G. We also identify those Lagrangian which correspond to equivariant families parametrized by the …

Associated with a smooth, $d$-closed $(1, 1)$-form $α$ of possibly non-rational De Rham cohomology class on a compact complex manifold $X$ is a sequence of asymptotically holomorphic complex line bundles $L_k$ on $X$ equipped with $(0, 1)$-connections $\bar\partial_k$ for which $\bar\partial_k^2\neq 0$. Their study was…

Several representations of geometric shapes involve quotients of mapping spaces. The projection onto the quotient space defines two sub-bundles of the tangent bundle, called the horizontal and vertical bundle. We investigate in these notes the sub-Riemannian geometries of these bundles. In particular, we show for a sel…

Develops global pseudo-differential calculus on homogeneous vector bundles.

Let $x$ denote a diffusion process defined on a closed compact manifold. In an earlier article, the author introduced a new approach to constructing admissible vector fields on the associated space of paths, under the assumption of ellipticity of $x$. In this article, this method is extended to yield similar results fo…

We prove a conjecture of Gromov's to the effect that manifolds with isotropic curvature bounded below by 1 (after possibly rescaling) are macroscopically 1-dimensional on the scales greater than 1. As a consequence we prove that compact manifolds with positive isotropic curvature have virtually free fundamental groups.…

We prove the classical Nakano vanishing theorem with Hörmander $L^2$-estimates on a compact Kähler manifold using Siu's so called $\partial\dbar$-Bochner-Kodaira method, thereby avoiding the Kähler identities completely. We then introduce singular hermitian metrics on holomorphic vector bundles, and proceed to prove a …

Let Y=G/L be a flag manifold for a reductive G and K a maximal compact subgroup of G. We define an equivariant differential operator on G/(L cap K) playing the role of an equivariant Dolbeault Laplacian when restricted to the complex manifold G/L, using a distribution transverse to the fibers and satisfying the Hormand…

Improved bounds for eigenfunctions on hyperbolic surfaces found.

We prove that admissible functions for Fubini-Study metrics on the complex projective space $P_{m}C$, of complex dimension $m$, invariant by a convenient automorphisms group, are lower bounded by a function going to minus infinity on the boundary of usual charts of $P_{m}C$. A similar lower bound holds on some projecti…

Motivated by the study of Hörmander's sums-of-squares operators and their generalizations, we define the convolution algebra of transverse distributions associated to a singular foliation. We prove that this algebra is represented as continuous linear operators on the spaces of smooth functions and generalized function…

Characterizes polyhomogeneous symbols and applies to Heisenberg calculus.

Generalizes Connes's tangent groupoid for sub-Riemannian geometry.

Given a second order partial differential operator $L$ satisfying the strong Hörmander 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 …

We study the global theory of linear wave equations for sections of vector bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of finite energy sections and show well-posedness of the Cauchy problem in those spaces. These spaces depend in general on the choice of a time function but it turns out tha…