Extends strong comparison principle for p-harmonic functions in Carnot-Caratheodory spaces.
problem Proving strong comparison principle for p-harmonic functions in specific geometric settings.
method Extends Bony's propagation of support argument to C^1 solutions of sub-elliptic p-Laplacian.
result Proves strong maximum and comparison principles for p-harmonic functions.
Study strong maximum principles for mean curvature operators on subriemannian manifolds.
problem Investigate strong maximum principles for mean curvature operators on subriemannian manifolds.
method Analyze subriemannian manifolds including Heisenberg groups and cylinders, under Hormander type conditions.
result Show strong maximum principles for horizontal (p-) mean curvature operator and p-(sub)laplacian operator under certain conditions.
Rigidity theorem for curved surfaces, proving uniqueness up to motion.
problem Proving uniqueness of curved surfaces up to rigid motion.
method Employing Hormander's unique continuation principle for elliptic PDEs.
result Proves rigidity of nonnegatively curved surfaces.
New characterizations of partial positivity using Hörmander's L2-estimate.
problem Characterizing partial positivity in complex geometry.
method Using a twisted version of Hörmander's L2-estimate. result New characterizations of partial positivity, including uniform q-positivity and RC-positivity. Recent research connects Hörmander's old work to modern boundary Laplacian analysis.
problem How close is the Dirichlet-to-Neumann map to the boundary Laplacian?
method Investigates techniques from Hörmander's 1950s manuscript to solve modern boundary Laplacian problems.
result Obtained results for DtN maps on non-smooth boundaries, Helmholtz equation, and differential forms.
Establishes uniform Hörmander estimates for flat line bundles on Kähler manifolds.
problem Estimating ∂-operators for flat line bundles. method Uniform L2-estimates for ∂-operators on Kähler manifolds. result Recovers Ueda's lemma for compact Kähler manifolds and generalizes to Ricci-flat manifolds.
Researchers solve p-Laplace equation in Hörmander vector fields.
problem Finding fundamental solution in specific vector field class.
method Used generalized operator in Euclidean space to find solution.
result Computed capacity of annuli centered at singularity.
The paper extends log-Sobolev inequalities to matrix-valued settings using combinatorial methods.
problem Log-Sobolev inequalities for matrix-valued settings.
method Combining noncommutative geometry tools and combinatorial methods.
result Combinatorial methods yield computable lower bounds for matrix-valued log-Sobolev inequalities.
Solves division problem for L. Hörmander's systems.
problem Division problem for L. Hörmander's overdetermined systems.
method Formulates and proves divisibility criterion, coherence theorem.
result Establishes effective divisibility criterion and extends coherence theorem.
New index formula for hypoelliptic operators on manifolds.
problem Index computation for hypoelliptic differential operators.
method Generalized index formula for *-maximally hypoelliptic operators.
result Explicit index computations for Hormander's sum of squares operators.
Eigenvalue estimates for Dirac operators via weighted L2-technique.
problem Lower eigenvalue estimates for Dirac operators under elliptic boundary conditions.
method Hormander's weighted L2-technique and sharp Sobolev inequality. result Lower bounds on eigenvalues in terms of manifold volume.
Extends pseudo-differential operators theory to compact Lie groups.
problem Global pseudo-differential operators on compact Lie groups.
method Develops a subelliptic pseudo-differential calculus for compact Lie groups.
result Establishes subelliptic versions of Fefferman and Calderón-Vaillancourt theorems.
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…
Estimates spectral gap for sub-Laplacian on compact manifolds.
problem Bounding the spectral gap of sub-Laplacian on compact manifolds.
method Lichnerowicz estimate adaptation for sub-Laplacian on compact manifolds.
result A bound for the spectral gap analogous to the Lichnerowicz estimate for the Laplacian.
Paper constructs L2 estimates for flat vector bundles and generalizes Prékopa's theorem.
problem Constructing L2 estimates for flat vector bundles. method Using Hörmander's L2-estimate for the operator d on a flat vector bundle over a p-convex Riemannian manifold. result Generalizes Prékopa's theorem in convex analysis.
Extends elliptic operator regularity to maximally hypoelliptic operators.
problem Maximally hypoelliptic differential operators and their regularity.
method Define a principal symbol for arbitrary differential operators involving vector fields and their commutators.
result Proves the invertibility of the principal symbol is equivalent to maximally hypoellipticity, answering a conjecture.
Kähler-Ricci flows' tangent cones are algebraic varieties.
problem Understanding the structure of Kähler-Ricci flows' tangent cones.
method Analyzing tangent cones as normal affine algebraic varieties and using Hörmander's L2 estimate. result The regular set of tangent cones coincides with the algebraic regular set.
The paper studies boundedness of pseudo-differential operators on smooth manifolds.
problem Boundedness of pseudo-differential operators in Lp-Lq spaces on smooth manifolds. method Using global symbols and extending Hörmander's condition, the paper investigates Lp-boundedness, L∞-BMO estimates, and Lp-Lq boundedness for Fourier multipliers and pseudo-differential operators. result The paper proves Lp-Lq boundedness for the range 1<p≤2≤q<∞. Defines a new algebra for singular foliations, extending Schwartz kernels.
problem Extending Schwartz kernel operators to singular foliations.
method Defines convolution algebra of transverse distributions, proves representation as operators on spaces of functions.
result Generalizes Schwartz kernel operators to singular foliations.
We compute the small time asymptotic of the fundamental solution of Hörmander's type hypoelliptic operators with drift, at a stationary point, x0, 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…
Simplified calculus for manifold operators, proving index theorems.
problem Developing calculus for manifold operators and proving index theorems.
method Introducing a simplified pseudo-differential calculus for zero-order operators on manifolds with a tangent Lie structure.
result Proving index theorems for `h-elliptic' operators on manifolds with a tangent Lie structure.
Researchers compute heat kernel coefficients for 2D diffusion operators.
problem Analyzing heat kernel coefficients for 2D hypoelliptic operators.
method Explicit computation of heat kernel coefficients and interpretation in terms of curvature.
result Interpretation of heat kernel asymptotics for non-sub-Riemannian operators.
Researchers compute Wodzicki residue for pseudo-differential operators on compact Lie groups.
problem Computing the Wodzicki residue for pseudo-differential operators on compact Lie groups.
method Analytic continuation of traces and matrix-valued symbols.
result Main theorem complementary to [2], removing ellipticity hypothesis.
Improved bounds for Carleson-Sjölin operators on manifolds with specific curvature conditions.
problem Bounding Carleson-Sjölin operators on manifolds with special curvature conditions.
method Two different methods: one using distance function conditions and the other using contact orders of oscillatory integral operators.
result Improved Lp bounds for Carleson-Sjölin operators on manifolds with constant sectional curvature and those satisfying Sogge's chaotic curvature condition. Proves curvature positivity of invariant direct images in complex geometry.
problem Curvature positivity of invariant direct images in complex geometry.
method Compact group action and Hörmander's L2 theory of ∂ˉ. result Direct image of Nakano positive vector bundle is Nakano positive.
Extends scaling maps theory to manifolds with boundary.
problem Quantitative study of Carnot-Carathéodory balls on manifolds with boundary.
method Introduction of scaling maps adapted to Carnot-Carathéodory balls and Hörmander vector fields on manifolds with boundary.
result First paper in a series studying maximally subelliptic boundary value problems.
New curvature assumptions prove Nakano positivity for complex vector bundles.
problem Proving Nakano positivity for complex vector bundles under varying curvature assumptions.
method Using a variant of Hörmander's theorem, the authors show Nakano positivity under more general curvature conditions.
result Nakano positivity holds for complex vector bundles under different curvature assumptions.
Study on diffeologies on locally convex spaces and smooth multiplication of distributions.
problem Geometric characterization and smoothness of distribution multiplication.
method Investigation of canonical and c∞-diffeologies on locally convex spaces, proving geometric characterizations, and comparing diffeologies. result Established a framework for nonlinear distribution theory beyond manifolds, realizing microlocally multipliable distributions as a diffeological colimit.
In this paper, we consider the principal eigenvalue problem for Hormander's laplacian on Rn. We also study a related semi-linear sub-elliptic equation in the whole Rn and prove that under a suitable condition, we have infinite many positive solutions of the problem.
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…
The paper bounds Fourier integral operators on Hardy spaces with specific conditions.
problem Bounding Fourier integral operators on Hardy spaces with given conditions.
method Using Hörmander classes and phase conditions, the paper establishes boundedness of Fourier integral operators.
result The Fourier integral operator is bounded from local Hardy space hp to Lp under specified conditions. 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…
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…
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 L2-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…
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…
Improved bounds for eigenfunctions on hyperbolic surfaces found.
problem Establishing improved bounds for eigenfunctions of magnetic Laplacians.
method Using explicit eigenstates called magnetic zonal states.
result Explicit eigenstates called magnetic zonal states found.
We prove that admissible functions for Fubini-Study metrics on the complex projective space PmC, 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 PmC. A similar lower bound holds on some projecti…
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…
Generalizes Connes's tangent groupoid for sub-Riemannian geometry.
problem Calculating tangent cones in sub-Riemannian geometry.
method Constructs a completion of MimesMimesR+imes using sub-Riemannian metric. result Calculates all tangent cones in Gromov-Hausdorff distance.
Given a second order partial differential operator L satisfying the strong Hörmander condition with corresponding heat semigroup Pt, we give two different stochastic representations of dPtf 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…
Novel filter uses deep BSDE for nonlinear density approximation.
problem Nonlinear filtering problem.
method Bayesian filter based on deep BSDE and neural networks.
result Theoretical convergence rate confirmed in numerical examples.
Generalizes mean-value inequality to orbifold setting.
problem Mean-value inequality for orbifold setting.
method Generalizes fundamental results in Kähler geometry to orbifolds.
result Shows mean-value inequality is insensitive to quotient singularities.
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 …
We prove the nonexistence of stable immersed minimal surfaces uniformly conformally equivalent to the complex plane in any complete orientable four-dimensional Riemannian manifold with uniformly positive isotropic curvature. We also generalize the same nonexistence result to higher dimensions provided that the ambient …