Proves Hölder-type inequality for Lagrangians' distance.
problem Understanding the symplectic geometry of Lagrangians.
method Developed methods from previous works to establish the inequality.
result Established a Hölder-type inequality for the Hausdorff distance between Lagrangians.
Study explores relationship between Hölder and FDPD divergences.
problem Understanding the relationship between Hölder and FDPD divergences.
method Intersection and generalization of divergence families, proving nonnegativity, deriving inequalities.
result Established ξ-Hölder divergence and derived inequalities. Reverse Hölder inequalities on Fano metrics with applications to geodesics and singularities.
problem Establishing reverse Hölder inequalities on Kähler metrics of Fano varieties.
method Using log-concavity and properties of Ricci potentials, the inequality is proven for Fano manifolds with log terminal singularities.
result The inequality holds for Fano varieties with log terminal singularities and the constant depends only on p and the dimension of X.
Estimates how fast the best constant in Sobolev inequality changes as domain expands outward.
problem Rate of change of best constant in Sobolev inequality.
method Analyzes Euclidean domain expansion and proves a reverse Holder inequality.
result Estimates the rate of change of the best constant in Sobolev inequality.
New inequality for eigenfunctions on curved spaces.
problem Eigenfunctions on non-smooth spaces with Ricci curvature.
method Sharp reverse-Hölder inequality for Dirichlet Laplacian eigenfunctions.
result Generalizes classical comparison theorem to curved spaces.
Study on Hölder-equivalence problem for Carnot groups, with partial result.
problem Hölder-equivalence problem for Carnot groups.
method General coarea inequality for packing energies of maps.
result Partial result given for the problem.
New method improves curvature estimates for stable surfaces.
problem Curvature estimates for stable surfaces in Rn+1. method Replacing Young's inequality with Hölder's inequality simplifies and improves curvature estimates.
result The new method yields a strictly smaller constant and a natural extension to CMC settings.
Defines signed quasiregular curves and proves growth theorem.
problem Understanding growth of signed quasiregular curves.
method Proves weak reverse Hölder inequality and uses it to prove growth theorem.
result Proves growth theorem for signed quasiregular curves.
Derivatives of sub-Riemannian geodesics are always Lp-Hölder continuous.
problem Smoothness of sub-Riemannian geodesics
method Proving Lp-Hölder continuity of derivatives result Derivatives of sub-Riemannian geodesics are Lp-Hölder continuous The study proves inequalities for complex operators on curved spaces.
problem Establishing inequalities for nonlocal operators on curved spaces.
method Defining and analyzing nonlocal Pucci operators on manifolds with nonnegative sectional curvatures, proving Harnack inequalities and Holder estimates.
result Harnack inequalities and Holder estimates for nonlocal operators on manifolds with nonnegative sectional curvatures.
The paper proves Hölder continuity for solutions of degenerate parabolic equations in any dimension.
problem Proving Hölder continuity for solutions of degenerate parabolic equations in arbitrary dimensions.
method Establishing Alexandroff-Bakelman-Pucci estimate, Harnack inequality, Hölder regularity, and Schauder estimates for a class of degenerate parabolic equations.
result The paper proves Hölder continuity for solutions of degenerate parabolic equations in all dimensions.
Bounds on treatment effect sensitivity in causal reasoning using Hölder's inequality.
problem Estimating treatment effects in presence of unobserved confounders.
method Using Hölder's inequality, derived bounds on confounding bias based on unmeasured confounding strength.
result Bounds are tight under specific conditions of independence between U and T/Y.
The Heston stochastic volatility process, which is widely used as an asset price model in mathematical finance, is a paradigm for a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this p…
In this article we prove a reverse Hölder inequality for the fundamental eigenfunction of the Dirichlet problem on domains of a compact Riemannian manifold with lower Ricci curvature bounds. We also prove an isoperimetric inequality for the torsional ridigity of such domains.
The paper shows inequality and rigidity for manifolds with integral Ricci curvature.
problem Analyzing structures of manifolds with integral Ricci curvature.
method Using segment inequality and similar methods as in \cite{CC1}, derive almost rigidity structure results.
result Sharp Hölder continuity result holds in the limit space of manifolds with integral Ricci curvature bound.
Develops a deep learning framework for various data types.
problem Handling nonparametric regression and classification across different data types.
method Introduces a general framework with two estimators: NPDNN and SPDNN, based on data satisfying generalized Bernstein-type inequalities.
result Both NPDNN and SPDNN estimators are minimax optimal in many classical settings.
The paper tackles deep learning from dependent data, achieving optimal performance.
problem Deep learning from strongly mixing observations, especially with regularization and optimality.
method Sparse-penalized regularization for deep neural networks, oracle inequality for expected excess risk.
result Deep neural network estimator achieves minimax optimal rate for nonparametric autoregression.
Constructs mappings with restricted Hessians to approximate Hölder functions.
problem Approximating Hölder continuous functions with mappings having singular Hessians.
method Constructs a sequence of mappings with restricted Hessians that converge to a Hölder function in C1,α and weakly in W2,p. result Functions in W2,p with rank inequality can be approximated by mappings with singular Hessians. Sharp Talenti-type comparison theorem for p-Laplacian on RCD(K,N) spaces.
problem Understanding the p-Laplacian on RCD(K,N) spaces.
method Proving a Talenti-type comparison theorem.
result Sharp, rigid and stable Talenti-type comparison theorem.
The paper converts metric bounds to distance function Hölder bounds and proves compactness theorems.
problem Proving geometric stability results with scalar curvature bounds.
method Transforming Lp bounds to Hölder bounds for distance functions. result Compactness theorems and convergence guarantees for Riemannian manifolds.
The paper studies gradient estimates for heat kernels and harmonic functions in metric measure spaces.
problem Gradient estimates for heat kernels and harmonic functions in metric measure spaces.
method Investigation of properties of harmonic functions, heat kernels, and Riesz transforms in metric measure spaces with a Dirichlet form.
result Equivalence of properties (i), (ii), (iii) for p∈(2,∞) and (i), (ii), (iii), (iv) for p=∞. The article proves Talenti's comparison theorem for Poisson equations on Riemannian manifolds with nonnegative Ricci curvature.
problem Proving Talenti's comparison theorem for Poisson equations on Riemannian manifolds with nonnegative Ricci curvature.
method Analyzing complete noncompact Riemannian manifolds with nonnegative Ricci curvature, applying Talenti's comparison theorem to Poisson equations.
result Obtained the Faber-Krahn inequality for the first eigenvalue of Dirichlet Laplacian, L1- and L∞-moment spectrum, and a reverse Hölder inequality for eigenfunctions of Dirichlet Laplacian. Novel oracle-type inequality for logistic loss in DNNs achieves sharp convergence rates.
problem Generalization analysis for binary classification with DNNs and logistic loss.
method Established an oracle-type inequality to handle the boundedness of the target function.
result Optimal convergence rates for fully connected ReLU DNN classifiers trained with logistic loss.
The paper studies Harnack inequalities on Finsler metric measure spaces.
problem Analyzing Harnack inequalities on Finsler metric measure spaces.
method Using weighted Ricci curvature and distortion conditions, the authors derive an elliptic p-Harnack inequality.
result The paper establishes an elliptic p-Harnack inequality and derives Hölder continuity and gradient estimates for positive harmonic functions.
This study analyzes how well GANs approximate distributions from small samples.
problem Understanding how well GANs approximate distributions from limited data.
method Analysis of GANs using integral probability metrics and Hölder classes.
result GANs can adaptively learn low-dimensional structures or Hölder densities.
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.
We derive an exponential inequality for Rényi divergence estimation.
problem Consistent estimation of divergences in machine learning.
method Generalized exponential concentration inequality for Rényi divergence estimation.
result Finite sample exponential inequality convergence bound for Rényi-α divergence estimator. This paper values ESOs considering job termination and stock price jumps.
problem Valuing ESOs with job termination risk and stock price jumps.
method General Lévy stock price dynamics, inhomogeneous partial integro-differential variational inequality (PIDVI), Fourier transform, geometric Brownian motion.
result Higher job termination risk leads to earlier ESO exercise, reducing company costs.
We establish various Lp estimates for the Schrödinger operator −Δ+V on Riemannian manifolds satisfying the doubling property and a Poincaré inequality, where Δ is the Laplace-Beltrami operator and V belongs to a reverse Hölder class. At the end of this paper we apply our result on Lie groups with polynomial …
Study surfaces with nonnegative curvature in spectral sense, proving inequalities and bounds.
problem Closed orientable surfaces with nonnegative curvature in spectral sense.
method Spectral condition and associated conformal metrics to prove inequalities and bounds.
result Isoperimetric inequalities, area growth theorems, and diameter bounds for surfaces.
The study examines sequences of Riemannian manifolds with uniform Sobolev bounds and their convergence properties.
problem Analyzing sequences of compact Riemannian manifolds with uniform Sobolev bounds and their convergence.
method Establishing a general trace inequality on Riemannian manifolds and proving convergence properties using Hölder and Intrinsic Flat metrics.
result Sequences of compact Riemannian manifolds with uniform Hölder bounds on their distance functions have subsequences converging in the Gromov--Hausdorff sense.
Study of area minimizing surfaces in homotopy classes of maps.
problem Existence and regularity of area minimizing surfaces in metric spaces.
method Introducing relative 1-homotopy type for Sobolev maps, using local quadratic isoperimetric inequality, and analog for closed surfaces.
result Existence and local Hölder regularity of area minimizing surfaces in proper geodesic metric spaces.
Sharp L∞ estimates for fully non-linear elliptic equations on compact complex manifolds.
problem Sharp L∞ estimates for fully non-linear elliptic equations on compact complex manifolds.
method Comparison with auxiliary complex Monge-Ampère equations, Hölder-Young inequality, and De Giorgi iteration lemma.
result Improved L∞ estimates for fully non-linear elliptic equations on Kähler and Hermitian manifolds.
Establishes interior regularity results for a broad class of two-dimensional nonlinear elliptic systems using a unified abstract framework.
problem Interior regularity results for two-dimensional nonlinear elliptic systems
method A unified abstract framework built around a Campanato-type discrete iteration scheme coupled with a Caccioppoli-type estimate
result Local Hölder continuity of the map u is established, with an explicit Hölder exponent that optimally attains the classical Morrey--Campanato threshold dictated by the Lebesgue integrability of the source term f Solves area minimizing surface problem in metric spaces with bounded genus.
problem Finding area minimizing surfaces of bounded genus in metric spaces.
method Solves Plateau-Douglas problem in proper metric spaces with local quadratic isoperimetric inequality.
result Generalizes results from Riemannian manifolds to proper metric spaces.
Analyzes surfaces with bounded curvature, proving properties and continuity.
problem Analyzes surfaces with locally bounded integral curvature.
method Analyzes surfaces as metric measure spaces, proving infinitesimal Hilbertianity, local doubling, and Poincaré inequality.
result Proves existence of jointly Hölder continuous heat kernel for Cheeger Laplacian.
Analyzes surfaces with bounded curvature, proving properties and existence of heat kernels.
problem Analyzes surfaces with locally bounded integral curvature.
method Analyzes surfaces as metric measure spaces, proving infinitesimal Hilbertianity, local doubling, and Poincaré inequality.
result Existence of jointly Hölder continuous heat kernel for Cheeger Laplacian.
Develops analysis of Hölder continuous mappings on Heisenberg groups.
problem Analyzing Hölder continuous mappings on Heisenberg groups.
method Theory of distributional Jacobians and pullbacks of differential forms.
result Simple proof of a generalization of the Gromov non-embedding theorem and new results about Hölder homotopy groups.
Proves Hölder continuity of complex Monge-Ampère solutions.
problem Global Hölder continuity of solutions to complex Monge-Ampère equation.
method Analyzes Dirichlet problem on strictly pseudoconvex domains or Hermitian manifolds.
result Proves global Hölder continuity of solutions under given conditions.
Proves rigidity of maps between balls with Hölder boundary continuity.
problem Rigidity of proper holomorphic maps between unit balls with Hölder boundary continuity.
method Proves rigidity for maps with symmetries and Hölder boundary continuity.
result Proves rigidity for maps with Hölder exponent > 1/2 on the boundary.
Study finds Holder solutions for complex geometry equations.
problem Finding solutions to complex geometry equations on compact manifolds.
method Analyzes measures and functions on compact Hermitian manifolds.
result Positive measures on compact Hermitian manifolds admit Holder continuous solutions to the Monge-Ampere equation.
This paper proves Hölder continuity for complex Monge-Ampère equations on Kähler varieties.
problem Establishing Hölder estimates on singular Kähler varieties.
method Geometric regularization based on partial C0 estimate. result Uniform Hölder continuity for complex Monge-Ampère equations on Kähler varieties.
Theoretical limits on verifying self-improving systems without risking unbounded utility.
problem Formalizing and proving the limits of safety verification for self-improving systems.
method Developed dual conditions and used Holder's inequality, NP counting method, and Lipschitz bounds to establish impossibility and ceiling results.
result A classifier-based safety gate cannot simultaneously permit unbounded beneficial self-modification and bounded cumulative risk.
Study nonparametric density estimation via measure transport, achieving optimal rates.
problem Nonparametric density estimation with optimal rates.
method Measure transport, penalized maximum likelihood, and sieved wavelet estimators.
result Achieve minimax optimal convergence rates over Hölder classes of densities.
Solves complex Monge-Ampère equation with Hölder continuous boundary data.
problem Complex Monge-Ampère equation with Hölder continuous boundary data.
method Solves the Dirichlet problem for the complex Monge-Ampère equation.
result The solution is Hölder continuous if the boundary data is Hölder continuous.
We solve the classical problem of Plateau in the setting of proper metric spaces. Precisely, we prove that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadra…
The study explores geodesics and KL-divergence on Hölder equilibrium probabilities.
problem Finding the probability that minimizes KL-divergence from a fixed probability in a convex set of probabilities.
method Analyzes geodesics paths on the manifold of Hölder equilibrium probabilities and uses KL-divergence as a metric.
result Explicit equations for the solution of the minimization problem are derived.
Adapts Hölder smoothness with normalized gradients.
problem Improving smoothness adaptation methods.
method Black-box adaptation of Levy's method using normalized gradients.
result Bound depends on local Hölder smoothness.