The paper introduces boundary operators for Poincaré-Einstein manifolds and proves higher order trace inequalities.
problem Proving higher order trace inequalities on Poincaré-Einstein manifolds.
method Introducing conformally covariant boundary operators and using them to set up higher order Dirichlet problems.
result Sharp higher order Sobolev trace inequalities on the ball are obtained.
In this article, we prove a Sobolev-like inequality for the Dirac operator on closed compact Riemannian spin manifolds with a nearly optimal Sobolev constant. As an application, we give a criterion for the existence of solutions to a nonlinear equation with critical Sobolev exponent involving the Dirac operator. We fin…
The paper defines and analyzes abla-Sobolev spaces and operators on manifolds.
problem Defining and analyzing Sobolev spaces and differential operators on manifolds.
method Coordinate-free approach using connections, proving properties of abla-Sobolev spaces and operators. result Equivalent definitions of abla-Sobolev spaces and operators under certain conditions. Develops local elliptic regularity for geometrically-natural operators with low regularity coefficients.
problem Local elliptic regularity for operators with low regularity coefficients in Sobolev-type spaces.
method Rescaling estimates and multiplication results for function spaces.
result Unified set of interior estimates and regularity inference for operators with Sobolev-type coefficients.
The paper improves CR Sobolev inequalities and classifies minimizers.
problem Higher-order CR Sobolev inequalities on the CR sphere.
method Improvement through vanishing higher order moments of the volume element.
result New direct proof of minimizers' classification and existence of minimizers in C2k(N). We consider the operator algebra generated by pseudodifferential operators on a closed smooth surface and shift operator induced by a Morse--Smale diffeomorphism of this surface. Elements in this algebra are considered as operators in the scale of Sobolev spaces and the aim of this paper is to describe how Fredholm pro…
Optimal multiscale learning of linear operators
problem Statistical and computational limits of learning bounded linear operators between Sobolev spaces
method Reformulate as an infinite-dimensional matrix regression problem with heterogeneous multiscale structure
result Establish minimax rates and construct a finite-resolution blockwise least-squares estimator attaining these rates
Study improves Poincaré-Sobolev inequalities for differential forms.
problem Improving Sobolev space embeddings for differential forms.
method Utilizes Lq,p-cohomology and bi-Lipschitz images to estimate embedding norms. result Estimates for embedding norms in Euclidean balls and their images.
This paper establishes inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane.
problem Establishing higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane.
method Developing factorization theorems and introducing Geller's operators, combining with Helgason-Fourier analysis and kernel estimates.
result Established higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane.
Optimizes learning Hilbert-Schmidt operators between Sobolev spaces.
problem Statistical limits of learning mappings between infinite-dimensional function spaces.
method Minimax optimal regularization and multilevel training.
result Multilevel kernel operator learning achieves optimal learning rate.
The paper finds new inequalities for Laplacian and biharmonic eigenvalues on manifolds.
problem Eigenvalue inequalities for Laplacian and biharmonic operators on submanifolds.
method Using Sobolev inequalities to establish new eigenvalue inequalities.
result Established new inequalities for Laplacian and biharmonic eigenvalues.
The study establishes inequalities for functions on manifolds using Green function estimates.
problem Developing inequalities for functions on manifolds.
method Used integral representations and uniform estimates for Green functions.
result Proved Lp Sobolev-type and Poincaré-type inequalities for functions on real and complex manifolds. Explicit formulas for fractional GJMS operators on hyperbolic spaces and inequalities proved.
problem Proving explicit formulas and inequalities for fractional operators on hyperbolic spaces.
method Scattering theory on hyperbolic space, Helgason-Fourier analysis, and special function analysis.
result Sharp constants in fractional Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities coincide with Euclidean space constants.
Sharp inequalities on Siegel domains and complex hyperbolic spaces established.
problem Establishing inequalities on complex hyperbolic spaces and Siegel domains.
method Helgason-Fourier analysis, Kunze-Stein phenomenon, factorization theorem.
result Sharp Hardy-Adams and Adams type inequalities on Sobolev spaces of any positive fractional order on complex hyperbolic spaces.
The paper introduces new boundary operators and proves higher order CR Sobolev trace inequalities for Siegel domain and complex ball.
problem Establishing higher order CR Sobolev trace inequalities for Siegel domain and complex ball.
method Introducing conformally covariant boundary operators, proving extension theorems, and establishing trace inequalities.
result Generalized CR Sobolev trace inequalities for all γ ∈ (0, n+1) \mathbb{N}.
We prove a uniform Sobolev inequality along the Sasaki-Ricci flow. In the process, we develop the theory of basic Lebesgue and Sobolev function spaces, and prove some general results about the decomposition of the heat kernel for a class of elliptic operators on a Sasaki manifold.
We define a very general "parametric connect sum" construction which can be used to eliminate isolated conical singularities of Riemannian manifolds. We then show that various important analytic and elliptic estimates, formulated in terms of weighted Sobolev spaces, can be obtained independently of the parameters used …
In this paper, we propose a method for estimating the Sobolev type embedding constant on a domain with minimally smooth boundary. We estimate the embedding constant by constructing an extension operator and computing its operator norm. We also present some examples of estimating the embedding constant for certain domai…
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.
New Fredholm criteria for pseudodifferential operators on manifolds.
problem Characterizing the Fredholm property of G-pseudodifferential operators. method General Simonenko principle applied to G-pseudodifferential operators on Sobolev spaces of sections of vector bundles. result Derivation of novel Fredholm criteria and further characterization for finite groups.
Improved CR Sobolev inequalities on CR sphere established.
problem Establishing CR Sobolev inequalities on CR sphere.
method Nice commutator identities involving CR intertwining operators.
result Simpler proof of existence and classification of minimizers.
Continuous family of elliptic operators' projections maintain Cauchy data spaces.
problem Maintaining Cauchy data spaces for a continuous family of elliptic operators.
method Elementary tools and classical results applied to operator graphs, Sobolev spaces, and Green's formula.
result Orthogonalized Calderón projections form a continuous family of projections.
The paper confirms a conjecture about Hardy-Sobolev-Maz'ya inequalities and Green's functions on hyperbolic spaces.
problem Sharp constant in Hardy-Sobolev-Maz'ya inequalities and Green's functions on hyperbolic spaces.
method Fourier analysis techniques on hyperbolic spaces and Green's function estimates.
result The sharp constant in the 2n−1-th order Hardy-Sobolev-Maz'ya inequality in the upper half space of dimension n coincides with the best 2n−1-th order Sobolev constant when n is odd and n≥9. Develops Poisson structures on weak Sobolev loop spaces for integrable systems.
problem Analyzing integrable systems on low regularity loop spaces.
method Extending Mokhov's constructions to weak Sobolev spaces, constructing presymplectic and Poisson structures.
result Valid Poisson structures and deformations for weak Sobolev loops, extending Hamiltonian formalisms.
The paper proves inequalities for twisted differential forms on manifolds.
problem Proving Sobolev-type inequalities for twisted differential forms.
method Integral representations and uniform estimates for Green forms and their differentials.
result Improved L2-estimate of Hörmander on Kähler manifolds. Paper introduces new fractional Dirac operator and Q-curvature.
problem Fractional Dirac operator and Q-curvature in spinors.
method Caffarelli-Silvestre extension, energy inequalities, weighted Sobolev inequality.
result Introduction of conformal fractional Dirac operator and Q-curvature.
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.
Characterizes Schrödinger operator boundedness on weighted Riemannian manifolds.
problem Classifying functions V for bounded Schrödinger operator Δ−V. method Investigates weighted L2-boundedness of Hodge projector. result Characterizes function V for Schrödinger operator boundedness. Unified framework for Sobolev spaces on vector bundles, including explicit integration by parts.
problem Developing a comprehensive theory for Sobolev spaces on vector bundles.
method Explicit higher-order geometric integration by parts formula on arbitrary Riemannian manifolds.
result Direct proofs of classical theorems in Sobolev spaces on vector bundles.
Paper generalizes paracomposition and change of variables for paradifferential operators.
problem Generalizing paracomposition and change of variables for paradifferential operators in low regularity settings.
method Drops diffeomorphism hypothesis, estimates in Sobolev and Zygmund spaces, discusses pull-back of pseudodifferential and paradifferential operators.
result Sharp estimates for composition in Sobolev and Zygmund spaces, change of variables in paradifferential operators.
The paper derives inequalities and formulas for generalized Ricci flow.
problem Understanding and characterizing generalized Ricci flow.
method Using Bochner formula and adapted Malliavin gradient, the paper derives inequalities and characterizes generalized Ricci flow.
result Characterizations of generalized Ricci flow via inequalities for the associated Malliavin gradient.
The paper extends inequalities for convex bodies to higher dimensions and various norms.
problem Extending inequalities for convex bodies to higher dimensions and various norms.
method Developed new operators and inequalities for higher-order Lp norms. result Established mth-order Lp isoperimetric inequalities. We show that some curvature operators are locally invertible, in some weithted sobolev spaces, near the euclidian metric. (Nous montrons que certains opérateurs affines en la courbure de Ricci sont localement inversibles, dans des espaces de Sobolev à poids, au voisinage de la métrique euclidienne.)
Study of conformal logarithmic Laplacian on sphere, connecting Yamabe problems and Sobolev spaces.
problem Yamabe-type problems and Sobolev spaces on the sphere.
method Detailed spectral analysis, conformal invariance, and Hilbert space introduction.
result Established precise connection between sphere and \(\mathbb{R}^N\) logarithmic Laplacian.
Sharp inequalities and eigenvalue problems on Finsler manifolds with nonnegative Ricci curvature.
problem Establishing sharp Morrey-Sobolev inequalities and eigenvalue problems on Finsler manifolds.
method Combining sharp isoperimetric inequality and anisotropic symmetrization argument.
result Existence and multiplicity of solutions for eigenvalue problems and elliptic PDEs.
Quantitative Sobolev extensions lead to Neumann heat kernel bounds.
problem Bounding Neumann heat kernels for domains with integral Ricci curvature.
method Quantitative Sobolev extension operators and Neumann heat kernel estimates.
result Uniform bounds on Neumann heat kernels and eigenvalues.
The paper discusses fractional Sobolev immersions of flat domains into 3D space.
problem Developing C1 regularity and isometric immersions of flat domains with fractional Sobolev regularity. method Analysis of weak Codazzi-Mainardi equations, study of $W^{2, rac2s}$ planar deformations, and properties of the distributional Jacobian determinant.
result Generalization of isometric immersions with local fractional Sobolev regularity.
The paper studies elliptic operators on manifolds with boundary.
problem Characterizing boundary conditions for elliptic operators on manifolds.
method Using Calderón projectors and mixed order Sobolev spaces, the paper describes the space of boundary values and characterizes Fredholm and regular realisations.
result Characterization of boundary conditions for elliptic operators leading to Fredholm and regular realisations.
Geodesic completeness and optimal Sobolev index proven for Minkowski spacetimes.
problem Geodesic completeness and optimal Sobolev index for Minkowski spacetimes.
method Null non-trapping condition and real principal type estimate.
result Optimal Sobolev index proven for asymptotically Minkowski spacetimes.
Unified shape spaces with preserved invariances and regular metrics.
problem Combining shape invariances from Kendall's spaces with regular metrics.
method Defined a Sobolev-type operator to achieve the desired geometry, preserving invariances and regularity.
result Achieved a new landmark shape space with regular metrics and preserved invariances.
Solves Yamabe problem for Sobolev-class asymptotically hyperbolic manifolds.
problem Solving the Yamabe problem for specific types of asymptotically hyperbolic manifolds.
method Introduces new function spaces and uses Fredholm theorems for elliptic operators.
result Solves the Yamabe problem for asymptotically hyperbolic manifolds with Sobolev-class metrics.
We study some basic analytic questions related to differential operators on Lie manifolds, which are manifolds whose large scale geometry can be described by a a Lie algebra of vector fields on a compactification. We extend to Lie manifolds several classical results on Sobolev spaces, elliptic regularity, and mapping p…
Study regularity of Schrödinger eigenfunctions with Coulomb-type potentials.
problem Regularity of eigenfunctions for Schrödinger operators with singular potentials.
method Blow-ups of manifolds with corners and Lie manifolds.
result Proves regularity estimates in weighted Sobolev spaces for eigenfunctions.
Sharp inequality on Siegel domain involving weighted norms and sub-Laplacian.
problem Establishing a Sobolev trace inequality on a specific domain.
method Using weighted norms and fractional powers of sub-Laplacian on Heisenberg group.
result Sharp Sobolev trace inequality on Siegel domain involving weighted norms.
For loop groups (free and based), we compute the exact order of the curvature operator of the Levi-Civita connection depending on a Sobolev space parameter. This extends results of Freed and Maeda-Rosenberg-Torres.
By adapting some ideas of M. Ledoux \cite{ledoux2}, \cite{ledoux-stflour} and \cite{Led} to a sub-Riemannian framework we study Sobolev, Poincaré and isoperimetric inequalities associated to subelliptic diffusion operators that satisfy the generalized curvature dimension inequality that was introduced by F. Baudoin and…
Study Bernstein-Gelfand-Gelfand complexes on Lipschitz domains, computing cohomology and applying to elasticity models.
problem Cohomology of BGG complexes on bounded Lipschitz domains.
method Computes cohomology of conformal deformation and Hessian complexes in Sobolev spaces, allowing multiple input complexes.
result Establishes conformal Korn inequality and proposes generalizations of continuum models with microstructures.
Study on well-posedness of EPDiff equations with pseudo-differential inertia.
problem Analyzing the EPDiff equations with fractional Sobolev metrics.
method Fractional order Sobolev-type metrics on diffeomorphism groups, proving well-posedness.
result Proves local and global well-posedness for EPDiff equations.