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 survey we review Hamilton's entropy and Perelman's entropy, and provide motivations for these concepts. Then we review recent results on the logarithmic Sobolev inequality, the Sobolev inequalities and kappa-noncollapsing estimates along the Ricci flow, including the Ricci flow with surgeries.
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.
Paper proposes new density estimators for high-dimensional data.
problem Prohibitive computational cost and slow convergence rate in high-dimensional density estimation.
method Adaptive hyperbolic cross density estimators in mixed smooth Sobolev spaces.
result Proposed estimators do not suffer curse of dimensionality under Integral Probability Metrics.
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 …
We obtain a local Sobolev constant estimate for integral Ricci curvature, which enables us to extend several important tools such as the maximal principle, the gradient estimate, the heat kernel estimate and the L2 Hessian estimate to manifolds with integral Ricci lower bounds, without the non-collapsing conditions.
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…
This note develops certain sharp inequalities relating the fractional Sobolev capacity of a set to its standard volume and fractional perimeter.
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.
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.
Adapts to estimate functions from noisy ERT data.
problem Estimating functions from noisy Exponential Radon Transform data.
method Locally adaptive kernel type estimator for functions of varying smoothness.
result Achieves minimax optimal rate up to a log(n) factor for Sobolev functions.
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. Optimal geometric estimates for Kähler manifolds with bounded Nash entropy
problem Optimal geometric estimates for compact Kähler manifolds
method Proving Sobolev-type inequality and local volume noncollapsing with optimal exponents
result Uniformly bounded q-Nash entropy The paper extends inequalities to twisted differential forms on Kähler manifolds.
problem Generalizing Sobolev-type inequalities to twisted differential forms.
method Establishing heat kernel estimates for differential forms on Kähler manifolds.
result Proves vanishing theorem and Lq,p-estimates for ∂ˉ-operator. We formulate the notion of minimax estimation under storage or communication constraints, and prove an extension to Pinsker's theorem for nonparametric estimation over Sobolev ellipsoids. Placing limits on the number of bits used to encode any estimator, we give tight lower and upper bounds on the excess risk due to qu…
We extend the validity of a Gromov's dimension comparison estimate for topological hypersurfaces to sufficiently large classes of rectifiable sets, arising from Sobolev mappings. Our tools are a suitably weak exterior differentiation for pullback differential forms and a new low rank property for Sobolev mappings.
The paper proves gradient estimates for a weighted p-Laplacian equation on Riemannian manifolds.
problem Gradient estimates for a weighted p-Laplacian equation on Riemannian manifolds.
method Assumes a Sobolev inequality and integral Ricci bounds, proving local gradient estimates and Liouville type results.
result Proves local gradient estimates and Liouville type results on manifolds with lower bounds of Ricci curvature.
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
Optimal score function estimation via empirical risk minimization
problem Estimating the score function of a probability measure on the flat torus from a sample
method Constraining the hypothesis space to a Sobolev ball
result Minimax estimation rates are achieved
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.
In this paper, we first derive a Sobolev inequality along the harmonic-Ricci flow. We then prove a linear parabolic estimate based on the Sobolev inequality and Moser's iteration. As an application, we will obtain an upper bound estimate for the heat kernel under the flow.
Estimates for harmonic functions in curved spaces.
problem Quantifying harmonic functions in curved spaces.
method Quantitative Sobolev estimates for p-harmonic functions in manifolds with curvature conditions. result Established a quantitative second order Sobolev estimate for p-harmonic functions. 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.
Error estimates for nonlinear PDEs using kernel/GP methods.
problem Error analysis of kernel/GP methods for nonlinear and parametric PDEs.
method Sobolev space error estimates based on minimizing norm property of the solution.
result Dimension-benign convergence rates for smooth solutions.
Gradient estimates for special harmonic functions on manifolds.
problem Estimating gradients of (p,V)-harmonic functions on Riemannian manifolds. method Using Moser iteration method, volume comparison theorem, and Sobolev embedding theorem.
result Explicit global gradient estimates for positive entire (p,V)-harmonic functions. Uniform volume estimate for Kähler metrics in big cohomology classes.
problem Estimating volume for singular Kähler metrics in big cohomology classes.
method Generalized mixed energy estimate for functions in complex Sobolev space to big cohomology classes.
result Uniform non-collapsing volume estimate for local Kähler metrics.
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…
Sharp Sobolev inequalities proved on manifolds with non-negative Ricci curvature.
problem Proving sharp Sobolev inequalities on noncompact Riemannian manifolds with non-negative Ricci curvature.
method Using Optimal Mass Transportation with quadratic distance cost.
result Sharp Lp-Sobolev and Lp-logarithmic Sobolev inequalities established for p>1 and p=1. In this paper, we prove the equivalent of ultracontractive bound of heat semigroup or the uniform upper bound of the heat kernel with the Nash inequality, Log-Sobolev inequalities on graphs. We also show that under the assumption of volume growth and nonnegative curvature CDE′(n,0) the Sobolev inequality, Nash inequa…
The study analyzes weighted manifolds with curvature bounds, proving eigenvalue estimates and inequalities.
problem Analyzing geometric properties of weighted manifolds under Ricci curvature bounds.
method Develops geometric analysis techniques on weighted Riemannian manifolds with lower 0-weighted Ricci curvature bounds. result Proves eigenvalue estimates for Steklov and ABP inequalities on weighted manifolds.
The paper explores how control variates can reduce variance in Monte Carlo simulations, especially for Sobolev functions.
problem Efficiency of control variates in reducing variance for Monte Carlo simulations.
method Study of a specific quadrature rule using nonparametric regression-adjusted control variates.
result A specific quadrature rule can improve the Monte Carlo rate and achieve the minimax optimal rate under sufficient smoothness assumptions.
Motivated by the equation satisfied by the extremals of certain Hardy-Sobolev type inequalities, we show sharp Lq regularity for finite energy solutions of p-laplace equations involving critical exponents and possible singularity on a sub-space of Rn, which imply asymptotic behavior of the solutions at i…
We estimate the rate of change of the best constant in the Sobolev inequality of a Euclidean domain which moves outward. Along the way we prove an inequality which reverses the usual Holder inequality, which may be of independent interest.
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. New method estimates densities using Sobolev regularization, outperforming existing algorithms.
problem Non-parametric density estimation with clear inductive bias.
method Regularizes Sobolev norm of density, approximates kernel via sampling, uses natural gradients for optimization.
result Method ranks second best on ADBench anomaly detection benchmark.
Researchers find second-order estimates for p-Laplacian in RCD spaces.
problem Estimating functions with p-Laplacian in RCD spaces. method Establishing quantitative second-order Sobolev regularity.
result Second-order estimates for p-Laplacian functions in RCD spaces. Lower bounds for eigenvalues on Bakry-Emery manifolds proven.
problem Eigenvalue estimates on Bakry-Emery manifolds.
method Generalised maximum principle and heat kernel estimates.
result Lower bounds for all eigenvalues proven.
Extends mapping results to non-compact Riemannian manifolds with positive reach.
problem Extending mapping results to non-compact Riemannian manifolds with positive reach.
method Using a criterion by A. Petrunin and results by B. Bulanyi and J. Van Schaftingen, the study extends critical Sobolev mappings.
result Extended maps satisfy an exponential weak-type Sobolev-Marcinkiewicz estimate.
In this paper, we prove Poincaré and Sobolev inequalities for differential forms in L1(Rn). The singular integral estimates that it is possible to use for Lp, p>1, are replaced here with inequalities which go back to Bourgain-Brezis.
Study on Kohn Laplacian spectrum on sphere quotients.
problem Determining fundamental group from Kohn Laplacian spectrum.
method Weyl-type theorem, CR manifolds, Sobolev estimates.
result Fundamental group can be determined from Kohn Laplacian spectrum in 3D.
Study nonexistence and gradient estimates for solutions on manifolds with bounded Ricci curvature.
problem Nonexistence and gradient estimates for solutions of a specific quasi-linear equation on manifolds.
method Utilizes Sobolev inequalities and geometric properties to establish results.
result Extends and improves previous results on nonexistence and gradient estimates.
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.
In this paper, we systematically study the heat kernel of the Ricci flows induced by Ricci shrinkers. We develop several estimates which are much sharper than their counterparts in general closed Ricci flows. Many classical results, including the optimal Logarithmic Sobolev constant estimate, the Sobolev constant estim…
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. USD algorithm transports distributions with or without mass conservation.
problem Transporting distributions with different masses.
method Particle descent algorithm using Sobolev-Fisher discrepancy.
result USD converges to target distribution in MMD sense.
The paper provides gradient estimates for solutions on manifolds with integral Ricci bounds.
problem Global regularity estimates for solutions of Δu=f on Riemannian manifolds. method Proves Lp-gradient estimates under integral Ricci bounds and constructs a counterexample. result Optimal constant lower bounds on Ricci curvature are shown in the pointwise sense.
We derive a logarithmic Sobolev inequality along the Ricci flow without any restriction on time, which depends only on the initial metric via rudimentary geometric data, assuming only that a certain first eigenvalue is positive. As a consequence we obtain a uniform Sobolev inequality along the Ricci flow without any re…