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 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…
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.
Solves complex Monge-Ampère equations with Hölder continuous solutions in Kähler manifolds.
problem Finding Hölder continuous solutions to complex Monge-Ampère equations.
method Analyzes the complex Monge-Ampère equation in Kähler manifolds using Sobolev spaces and Hölder continuity.
result Hölder continuity of solutions is equivalent to the measure's Hölder continuity in a complex Sobolev space.
Continuity of polynomial roots shown for varying coefficients.
problem Continuity of polynomial roots under varying coefficients.
method Uniform bounds and Sobolev space analysis.
result Solution map is continuous for Cd coefficients. Ancient Ricci flows with bounded Nash entropy have uniform Sobolev inequalities.
problem Bounding Nash entropy in ancient Ricci flows.
method Uniformly bounded Nash entropy implies uniform bounds on the ν-functional, leading to uniform logarithmic and Sobolev inequalities.
result Uniform logarithmic and Sobolev inequalities on ancient Ricci flows with bounded Nash entropy.
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.
We concerns here with the continuity on the geometry of the second Riemannian L^p-Sobolev best constant B_0(p,g) associated to the AB program. Precisely, for 1 <= p <= 2, we prove that B_0(p,g) depends continuously on g in the C^2-topology. Moreover, this topology is sharp for p = 2. From this discussion, we deduce som…
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.
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…
Study compactness and continuity in Sobolev wave front set spaces for smooth vector bundles.
problem Compactness and continuity in Sobolev wave front set spaces for smooth vector bundles.
method Introduced a locally convex topology, extended compactness theorem, studied pseudo-differential operators, and applied to microlocal defect measures.
result Extended microlocal defect measures and compensated compactness theorem to Sobolev wave front set spaces.
Characterizes nodal volumes of Gaussian fields on manifolds, extending previous work.
problem Understanding the law and regularity of nodal volumes for Gaussian fields on manifolds.
method Gaussian measures, Morse theory, Malliavin-Sobolev spaces, ray absolute continuity.
result Extension and generalization of previous work on stationary fields to arbitrary dimensions.
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.
The paper proves lifting theorems for complex representations of finite groups.
problem Proving lifting theorems for complex representations of finite groups.
method Analyzing continuous maps and their invariants to establish Sobolev regularity.
result Continuous maps are locally of Sobolev class W1,p for all 1≤p<d/(d−1). The paper provides approximation guarantees for neural networks trained with gradient flow.
problem Approximating neural networks trained with gradient flow in continuous L2(Sd−1)-norm. method NTK argument for non-convex second but last layer, under-parametrized regime.
result Gradient flow convergence guarantees for neural networks under Sobolev smoothness assumptions.
Develops topological concepts for Morrey-Sobolev bundles in high dimensions.
problem Lack of continuity in transition maps for Morrey-Sobolev bundles.
method Introduces topological isomorphism classes and uses connection-oriented approach.
result Derives approximability results for bundles and connections in Morrey-Sobolev setting.
Lipschitz maps on metric surfaces are rigid if they preserve area.
problem Understanding the rigidity of Lipschitz maps on metric surfaces.
method Established a coarea inequality for continuous Sobolev functions on metric surfaces.
result Proved that 1-Lipschitz maps from a closed metric surface to a closed Riemannian surface preserving area are isometries.
LMC algorithm receives first convergence guarantees under weak smoothness conditions.
problem Convergence guarantees for LMC under weak smoothness conditions.
method Using Latała--Oleszkiewicz or modified log-Sobolev inequalities.
result First convergence guarantees for LMC under weak smoothness conditions.
We develop a comprehensive study on sharp potential type Riemannian Sobolev inequalities of order 2 by means of a local geometric Sobolev inequality of same kind and suitable De Giorgi-Nash-Moser estimates. In particular we discuss questions like continuous dependence of optimal constants and existence and compactness …
Study shows limitations of Lie bracket commutation for nonsmooth vector fields.
problem Limitations of Lie bracket commutation for nonsmooth vector fields.
method Analysis of nonsmooth vector fields, focusing on commutation of flows and Lie bracket conditions.
result Lie bracket commutation cannot be extended to general a.e. differentiable vector fields, but holds for certain Sobolev regular fields.
The transition maps for a Sobolev G-bundle are not continuous in the critical dimension and thus the usual notion of topology does not make sense. In this work, we show that if such a bundle P is equipped with a Sobolev connection A, then one can associate a topological isomorphism class to the pair $\left( P, A\…
Gradient descent achieves optimal learning for elliptic PDEs via Sobolev norms.
problem Learning elliptic PDEs from noisy data.
method Gradient descent on Sobolev norm objective functions.
result Gradient descent achieves statistical optimality for elliptic PDEs.
The paper proves continuity of Morse index for Ricci shrinkers.
problem Lower and upper semi-continuity of the Morse index for gradient Ricci shrinkers.
method Adapting and refining recent arguments on CMC hypersurfaces and polynomially weighted Sobolev spaces, with techniques for non-compact shrinkers.
result Identifies a condition ensuring the Morse index of asymptotically conical shrinkers is bounded below by the f-index of their asymptotic cone.
We consider rough metrics on smooth manifolds and corresponding Laplacians induced by such metrics. We demonstrate that globally continuous heat kernels exist and are Hölder continuous locally in space and time. This is done via local parabolic Harnack estimates for weak solutions of operators in divergence form with b…
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.
We continue our study of geometric analysis on (possibly non-reversible) Finsler manifolds, based on the Bochner inequality established by the author and Sturm. Following the approach of the Γ-calculus a la Bakry et al, we show the dimensional versions of the Poincare--Lichnerowicz inequality, the logarithmic Sobolev…
We show that the complex Monge-Ampere equation on a compact Kaehler manifold (X,ω) of dimension n admits a Holder continuous omega-psh solution if and only if its right-hand side is a positive measure with Holder continuous super-potential. This property is true in particular when the measure has locally Holder continu…
Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds.
problem Characterizing and proving rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds.
method Analysis of Riemannian manifolds and metric measure spaces with synthetic lower Ricci curvature bounds, using concentration compactness and Polya-Szego inequalities.
result Closed Riemannian manifolds with optimal Sobolev constant are isometric to the sphere, and almost equality implies close measure Gromov-Hausdorff convergence to a spherical suspension.
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.
The article examines entropy-information inequalities for continuous-time Markov chains under curvature-dimension conditions.
problem Proving Li-Yau inequalities and modified logarithmic Sobolev inequalities for reversible Markov chains.
method Introducing the CDΥ(κ,F) condition and deriving entropy-information inequalities. result Derives functional inequalities relating entropy to Fisher information.
We extend the well-known result that any f∈W1,n(Ω,Rn), Ω⊂Rn with strictly positive Jacobian is actually continuous: it is also true for fractional Sobolev spaces Ws,sn(Ω) for any s≥n+1n, where the sign condition on the Jacobian is understood in a distr…
Study geodesics on flat tori, focusing on convex bodies.
problem Analyze geodesics orthogonal to convex subsets on flat tori.
method Define anisotropic Sobolev spaces and study properties of geodesics.
result Compute residues of geometric Epstein function in terms of intrinsic volumes.
Kernel interpolation is inconsistent for norms with smoothness above a constant.
problem Inconsistency of kernel interpolation in reproducing kernel Hilbert spaces.
method Lower bounds for generalization error in Sobolev norms.
result Kernel interpolation is always inconsistent for norms with smoothness above a constant.
Develops analysis of weak immersions with bounded second fundamental forms in critical Sobolev space.
problem Analyzing weak immersions with bounded second fundamental forms in a critical Sobolev space.
method Develops analysis of Lipschitz immersions with bounded second fundamental forms in W2n−1,2 space. result Proves existence of C1 differential structure from weak immersions with bounded second fundamental forms. Paper proposes new Langevin samplers for sampling from log-concave distributions with superlinear gradient growth.
problem Sampling from log-concave distributions with superlinear gradient growth.
method Proposes two novel discretizations of kinetic Langevin SDEs, showing contractivity and log-Sobolev inequality.
result Establishes non-asymptotic bounds in 2-Wasserstein distance between sampled distributions and target measures.
Paper shows deep neural networks can approximate Korobov functions nearly optimally.
problem Approximating Korobov functions with deep neural networks.
method Used deep neural networks and measured approximation rates with Lp and H1 norms. result Achieved a super-convergence rate, outperforming traditional methods.
We prove a positive mass theorem for continuous Riemannian metrics in the Sobolev space Wloc2,n/2(M). We argue that this is the largest class of metrics with scalar curvature a positive a.c. measure for which the positive mass theorem may be proved by our methods.
The paper proves LOO CV is reliable under estimator stability.
problem Ensuring the reliability of leave-one-out cross validation.
method Using concentration inequalities based on logarithmic Sobolev inequality.
result LOO CV is a valid procedure under estimator stability.
Gradient flow of curve length on Sobolev metrics preserves convexity.
problem Optimal low-regularity gradient flow of curve length.
method Explicit gradient formula, Picard-Lindelöf theorem, time-reparametrisation.
result Exponential decay of length and preservation of convexity.
SBS uses SVGD to optimize continuous functions globally.
problem Global optimization of continuous Sobolev functions.
method Stein Boltzmann Sampling (SBS) with SVGD.
result SBS and its variants are highly competitive in global optimization.
Let $f : U\subset\Rm \to \calQ_Q(\ell_2)$ be of Sobolev class W1,p, 1<p<∞. If f almost minimizes its p Dirichlet energy then f is Hölder continuous. If p=2 and f is squeeze and squash stationary then f is in VMO.
Optimal rates for vector-valued regression on various norms.
problem Optimal rates for vector-valued ridge regression on continuous norms.
method Combining standard capacity assumptions with tensor product constructions of vector-valued interpolation spaces.
result Optimal rates for vector-valued ridge regression, independent of output space dimension.
The aim of the present paper is to define a notion of weakly differentiable cochain in the generality of metric measure spaces and to study basic properties of such cochains. Our cochains are (sub-)linear functionals on a subspace of chains, and a suitable notion of chains in metric spaces is given by Ambrosio-Kirchhei…
Transformer networks approximate Hölder and Sobolev functions with fixed-depth networks.
problem Nonparametric regression with dependent observations.
method Established novel upper bounds for Transformer networks approximating Hölder and Sobolev functions under various β-mixing data assumptions. result Explicit convergence rates for nonparametric regression problems under β-mixing data assumptions. New Sobolev inequalities on Kähler manifolds with positive Ricci curvature.
problem Proving Sobolev inequalities on Kähler manifolds.
method Using a classical Bidaut-Véron and Véron approach.
result Proved new Sobolev inequalities on compact Kähler manifolds with positive Ricci curvature.
Sharp inequality found on three-balls for fourth order Sobolev traces.
problem Fourth order Sobolev trace inequality on three-balls.
method Established through equivalence to a third order Sobolev inequality on two-spheres.
result Sharp fourth order Sobolev trace inequality on three-balls.
Continuity of roots of hyperbolic polynomials with smooth coefficients.
problem Continuity of the solution map for hyperbolic polynomials.
method Proving continuity of the solution map from hyperbolic polynomials of degree d with C^d coefficients to their increasingly ordered roots.
result Continuity of the solution map for hyperbolic polynomials with C^d coefficients.
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.