Study continuity of complex Sobolev functions, with applications to Kaehler metrics.
problem Continuity of functions in complex Sobolev spaces.
method Analysis of function regularity in Sobolev spaces, with applications to Kaehler metrics.
result Hermitian generalizations of recent results on Kaehler metrics.
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.
Completeness of surface metrics established for Sobolev spaces.
problem Ensuring completeness of reparametrization-invariant Sobolev metrics on surface spaces.
method Recasting completeness criteria for infinite-dimensional Riemannian manifolds and applying geometric estimates based on the Michael--Simon--Sobolev inequality.
result Established metric and geodesic completeness for specific Sobolev metrics on immersed surfaces, validating Mumford's conjecture.
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. New Sobolev inequality found for mean convex spacelike submanifolds in Minkowski space.
problem Finding a Sobolev inequality for mean convex spacelike submanifolds in Minkowski space.
method Applying the ABP estimate method to spacelike submanifolds in Rn,1. result Obtained a Sobolev inequality without a mean curvature term for mean convex hypersurfaces.
We study completeness properties of Sobolev metrics on the space of immersed curves and on the shape space of unparametrized curves. We show that Sobolev metrics of order n≥2 are metrically complete on the space In(S1,Rd) of Sobolev immersions of the same regularity and that any two curves i…
Sharp log-Sobolev inequalities proved for CD(0,N) spaces.
problem Proving log-Sobolev inequalities in noncompact metric measure spaces.
method Sharp isoperimetric inequality, symmetrisation, scaling argument, Hamilton-Jacobi inequality, Sobolev regularity.
result Sharp log-Sobolev inequalities established in CD(0,N) spaces. This paper introduces first order Sobolev spaces on certain rectifiable varifolds. These complete locally convex spaces are contained in the generally nonlinear class of generalised weakly differentiable functions and share key functional analytic properties with their Euclidean counterparts. Assuming the varifold to s…
We prove a sharp logarithmic Sobolev inequality which holds for submanifolds in Euclidean space of arbitrary dimension and codimension. Like the Michael-Simon Sobolev inequality, this inequality includes a term involving the mean curvature.
New tensorization theorem for Sobolev spaces on product spaces.
problem Characterize Sobolev spaces on product metric measure spaces.
method Showed two descriptions of Sobolev space on product spaces coincide.
result Norm equivalence and density results for Sobolev spaces.
Several possible notions of Hardy-Sobolev spaces on a Riemannian manifold with a doubling measure are considered. Under the assumption of a Poincaré inequality, the space $\Mone$, defined by Hajłasz, is identified with a Hardy-Sobolev space defined in terms of atoms. Decomposition results are proved for both the homoge…
We study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and higher is globally well-posed for smooth initial data as well as initial data in certain Sobolev spaces. Thus the space of closed plane…
Study on completeness of Sobolev metrics on manifold-valued curves.
problem Completeness of Sobolev metrics on spaces of manifold-valued curves.
method Analysis of reparametrization invariant Sobolev metrics of order n≥2. result Sobolev immersions are metrically and geodesically complete for several important cases of metrics.
Logarithmic Sobolev inequality proven for non-compact self-shrinkers.
problem Establishing a logarithmic Sobolev inequality for non-compact self-shrinkers.
method Using Alexandrov-Bakelman-Pucci (ABP) method to prove the inequality for Euclidean space, then applying this method to non-compact self-shrinkers.
result Optimal logarithmic Sobolev inequality for complete, non-compact, properly embedded self-shrinkers.
Uniform Sobolev inequality for Kähler metrics with entropy bound.
problem Establishing Sobolev inequalities for Kähler metrics with entropy bound.
method Uniform Sobolev inequality for Kähler metrics with entropy bound and no lower Ricci curvature bound.
result Derive various geometric estimates for Kähler-Einstein currents.
In this article we compute the best Sobolev constants for various Hardy-Sobolev inequalities with sharp Hardy term. This is carried out in three different environments: interior point singularity in Euclidean space, interior point singularity in hyperbolic space and boundary point singularity in Euclidean domains.
Stability of Dehn functions proven for ultralimits of Sobolev maps.
problem Stability of Dehn functions under ultralimit of Sobolev maps.
method Using ultralimits of Sobolev maps and properties of ultralimits of Lipschitz maps.
result Stability of Dehn functions under ultraconvergence of pointed length spaces.
Survey on smooth function and form density in Riemannian Sobolev spaces.
problem Density of smooth functions and forms in Sobolev spaces on Riemannian manifolds.
method Careful examination of weak covariant derivatives and partial derivatives.
result Equivalence of weak covariant derivatives to weak partial derivatives.
The paper examines stability of Sobolev inequalities on manifolds with Ricci curvature bounds.
problem Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds.
method Generalized Lions' concentration compactness and rigidity results of Sobolev inequalities on singular spaces.
result Almost extremal functions are close to extremal functions on the round sphere and Euclidean Sobolev inequality.
The paper shows how Sobolev maps affect currents in metric spaces.
problem Understanding how Sobolev maps affect currents in metric spaces.
method Proving that a Sobolev map pushes almost every compactly supported integral current to an Ambrosio-Kirchheim integral current.
result The paper proves an isoperimetric inequality for Sobolev mappings relative to bounded, closed, and additive cochains.
New neural network rates for unbounded domains with weighted Sobolev spaces.
problem Improving neural network approximation rates for unbounded domains.
method Embedding results for weighted Fourier-Lebesgue spaces in weighted Sobolev spaces, followed by asymptotic approximation rates.
result Asymptotic approximation rates for shallow neural networks without curse of dimensionality for unbounded domains and Muckenhoupt weights.
A new statistical model uses Orlicz-Sobolev spaces with Gaussian weight.
problem Statistical modeling of infinite-dimensional probability measures.
method Affine statistical bundle on Gaussian Orlicz-Sobolev space.
result Provides tools for solving infinite-dimensional evolution problems.
New Sobolev inequalities found for curved spaces.
problem Sobolev inequalities in curved spaces with specific decay conditions.
method Used ABP method developed by Cabré and Brendle.
result Established Sobolev inequalities for compact domains and submanifolds.
Study noncommutative Sobolev inequalities using quantum state metrics.
problem Establishing Sobolev inequalities in noncommutative settings.
method Generalizing monotone metrics in quantum states.
result Developed new matrix-valued Beckner inequalities.
We study reparametrization invariant Sobolev metrics on spaces of regular curves. We discuss their completeness properties and the resulting usability for applications in shape analysis. In particular, we will argue, that the development of efficient numerical methods for higher order Sobolev type metrics is an extreme…
Study on functional inequalities on simple edge spaces.
problem Whether classical functional inequalities hold in simple edge spaces.
method Analyzing Sobolev and Poincaré inequalities, proving optimality of Sobolev constant.
result Optimality result concerning the B-constant of the Sobolev inequality.
Holomorphic solutions vary in Sobolev spaces for Beltrami equations.
problem Analyzing the behavior of solutions to Beltrami equations in Sobolev spaces.
method Holomorphic dependence of solutions in Sobolev spaces of varying Beltrami differentials.
result Solutions vary holomorphically in Sobolev spaces of higher order.
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.
New heat semigroup characterizes Sobolev and BV spaces in Carnot groups.
problem Lack of explicit representations and symmetry in heat kernels in sub-Riemannian geometry.
method Establishes a new heat semigroup characterisation using integral decoupling property.
result Characterizes Sobolev and BV spaces in Carnot groups.
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.
Study shows Sobolev functions on non-compact manifolds can't be approximated by smooth compactly supported functions.
problem Sobolev functions on non-compact manifolds cannot be approximated by smooth compactly supported functions.
method Analysis of Sobolev spaces on non-compact manifolds.
result Proves the failure of the density of smooth compactly supported functions in Sobolev spaces on non-compact manifolds.
Study shows harmful overfitting in Sobolev spaces even as training data grows.
problem Harmful overfitting in Sobolev spaces under noisy conditions.
method Geometric argument using Sobolev inequalities.
result Approximately norm-minimizing interpolators exhibit harmful overfitting.
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 …
Study on fractional Sobolev metrics on curves, proving completeness and geodesic properties.
problem Investigating geometric properties of immersed curves with fractional Sobolev metrics.
method Analyzing Riemannian metrics on spaces of immersed curves, proving completeness and geodesic properties.
result Fractional Sobolev metrics are geodesically complete for q>3/2. Sharp inequalities for radial functions on hyperbolic spaces without boundary conditions.
problem Establishing inequalities for radial functions on hyperbolic spaces without zero boundary conditions.
method Novel approach considering both bounded and unbounded domains, focusing on weighted Sobolev and Adams-Trudinger-Moser embeddings.
result Theorems 1.2, 1.3, and 1.4 for weighted Sobolev embedding theorems, and Theorems 1.5 and 1.6 for Adams-Trudinger-Moser type embedding theorems.
In this paper, we introduce the anisotropic Sobolev capacity with fractional order and develop some basic properties for this new object. Applications to the theory of anisotropic fractional Sobolev spaces are provided. In particular, we give geometric characterizations for a nonnegative Radon measure μ that naturall…
Study on infinite-dimensional Heisenberg groups using hypoelliptic heat kernels.
problem Properties of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
method Construction and study of hypoelliptic heat kernels on infinite-dimensional reduced Heisenberg groups.
result Hypoelliptic logarithmic Sobolev inequalities on the space.
Study of geometric analysis on asymmetric metric spaces, including heat flow and Sobolev spaces.
problem Analysis of geometric properties on asymmetric metric measure spaces.
method Introduction of upper gradients, q-Laplacian, and q-heat flow in asymmetric settings. result Extension of concepts from symmetric to asymmetric metric measure spaces.
In this article we prove completeness results for Sobolev metrics with nonconstant coefficients on the space of immersed curves and on the space of unparametrized curves. We provide necessary as well as sufficient conditions for the coefficients of the Riemannian metric for the metric to be metrically complete and we c…
Paper connects Fenchel-Willmore and Sobolev inequalities for submanifolds in curved spaces.
problem Developing inequalities for submanifolds in curved spaces.
method Connecting Fenchel-Willmore and logarithmic Sobolev inequalities for mean-convex submanifolds.
result Established extensions of Fenchel-Willmore inequality and derived new Sobolev-type inequalities.
In this paper, we obtain the sharp k-th order Sobolev inequalities in the hyperbolic space ${\H}^n$ for all k=1,2,3,⋯. This gives an answer to an open question raised by Aubin in [5, p.176-177] for $W^{k,2}({\H}^n)$ with k>1. In addition, we prove that the associated Sobolev constants are optimal.
Paper uses ABP method to prove logarithmic Sobolev inequalities on curved spaces.
problem Proving logarithmic Sobolev inequalities on manifolds with nonnegative curvature.
method Employing the ABP method developed by Brendle.
result Sharp L2 and Lp logarithmic Sobolev inequalities established. 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.
Geodesically complete spaces with curvature bounded above have maps with finite energy that are Lipschitz.
problem Analyzing the properties of geodesically complete spaces with curvature constraints.
method Geometric perturbations of geodesics to curves with zero length on singular sets.
result Every Sobolev map in W1,∞ space has a Lipschitz representative with the same Lipschitz constant as its infinity energy. Researchers prove rigidity for log-Sobolev inequality on specific metric spaces.
problem Proving rigidity for the logarithmic Sobolev inequality on metric measure spaces.
method Using a new approach to prove the rigidity result.
result Proved that if equality holds in the log-Sobolev inequality, the space must split into a product of a manifold and the Gaussian shrinking soliton.
Deep ReLU networks can efficiently approximate Sobolev and Besov functions.
problem Approximating functions in Sobolev and Besov spaces using deep neural networks.
method Used deep ReLU neural networks with varied width and depth to approximate functions in Sobolev and Besov spaces.
result Generalized the approximation rate to hold under the Sobolev embedding condition.
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.
Novel algorithm speeds up computation of Sobolev IPM for graph-based probability measures.
problem Efficient computation of Sobolev IPM for graph-based probability measures.
method Established relation between Sobolev norm and weighted Lp-norm, proposed novel regularization, leveraged graph structure. result Proposed regularized Sobolev IPM provides closed-form expression for fast computation.