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 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…
Sharp fractional Sobolev inequalities on closed manifolds identified.
problem Critical fractional Sobolev embedding on closed Riemannian manifolds.
method Intrinsic heat-kernel based framework, determining optimal coefficients, proving sharp inequalities.
result Sharp p-power inequality and almost sharp inequality established. We obtain a compact Sobolev embedding for H-invariant functions in compact metric-measure spaces, where H is a subgroup of the measure preserving bijections. In Riemannian manifolds, H is a subgroup of the volume preserving diffeomorphisms: a compact embedding for the critical exponents follows. The results can b…
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.
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.
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 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.
We construct a large class of pathological n-dimensional topological spheres in Rn+1 by showing that for any Cantor set C⊂Rn+1 there is a topological embedding f:Sn→Rn+1 of the Sobolev class W1,n whose image contains the Cantor set C.
Frank and Lieb gave a new, rearrangement-free, proof of the sharp Hardy-Littlewood-Sobolev inequalities by exploiting their conformal covariance. Using this they gave new proofs of sharp Sobolev inequalities for the embeddings Wk,2(Rn)↪Ln−2k2n(Rn). We show that their …
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. 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.
The paper proves isometric embedding equations in low Sobolev regularity.
problem Proving isometric embedding equations in low Sobolev regularity.
method Proving Cartan's and Gauss's equations for C0∩H21 frames and deducing the Gauss equation for C1∩W1+32,3 isometric embeddings. result Gauss equation holds for C1∩W1+32,3 isometric embeddings. 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 inequality derived for Riemannian manifolds with bounded Ricci curvature.
problem Deriving a sharp Sobolev inequality for Riemannian manifolds with bounded Ricci curvature.
method Reduction to functions with small volume support, first order uniform asymptotic expansion of isoperimetric profile, local uniform Sobolev inequality.
result Sharp Sobolev inequality for W1,p(M) into Ln−pnp(M) is derived. Optimizes shapes of curves using Möbius energy gradients.
problem Finding optimal shapes of curves within isotopy classes.
method Gradient-based optimization with Sobolev inner products.
result Significantly more efficient and robust optimization methods.
A complete solution to the quaternionic contact Yamabe problem on the seven dimensional sphere is given. Extremals for the Sobolev inequality on the seven dimensional Hesenberg group are explicitly described and the best constant in the L2 Folland-Stein embedding theorem is determined.
Tensoring p-weak differentiable structures preserves their properties.
problem Tensorization of p-weak differentiable structures. method Proving the product of p-weak charts is a p-weak chart, and showing isometric embeddings. result Tensorization of p-weak differentiable structures is possible under certain conditions. 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 …
The study shows that close hypersurfaces have uniformly bounded inequalities.
problem Bounding inequalities for close hypersurfaces.
method Analyzing families of smooth hypersurfaces close to a fixed one.
result Uniformly bounded constants in Sobolev, Gagliardo-Nirenberg, and geometric Calderón-Zygmund inequalities.
Paper confirms Yau's conjecture about sphere eigenvalues.
problem Yau's conjecture on eigenvalues of minimal hypersurfaces.
method Constructing a minimizing sequence in Sobolev space, using variational principle.
result First non-zero eigenvalue equals hypersurface dimension.
Study embeddings between Barron spaces with various activation functions, focusing on RePU.
problem Understanding the influence of activation functions on infinitely wide neural networks.
method Prove embeddings by constructing push-forward maps on measures representing functions.
result Barron spaces with RePU activation have a hierarchical structure similar to Sobolev spaces.
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.
Given a compact Riemannian Manifold (M,g) of dimension n > 2, a point x_0 in M and s in (0,2). We let 2*(s) = 2(n-s)/(n-2) be the critical Hardy-Sobolev exponent. The Hardy-Sobolev embedding yields the existence of A,B > 0 such that (\int_M|u|^{2*(s)}dv_g)^{2/2*(s)} \leq A\int_M |\nabla u|_g^2 dv_g +B\int_M u^2 dv_g fo…
The paper explores how Finsler manifolds differ from Riemannian ones in functional inequalities.
problem Analytic phenomena on Finsler manifolds differ from Riemannian ones.
method Comparative analysis of Finsler and Riemannian manifolds.
result Functional inequalities (Hardy, uncertainty, CKN) behave differently on Finsler manifolds.
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.
Learning rates for least-squares regression are typically expressed in terms of L2-norms. In this paper we extend these rates to norms stronger than the L2-norm without requiring the regression function to be contained in the hypothesis space. In the special case of Sobolev reproducing kernel Hilbert spaces used …
We study the Yamabe problem on open manifolds of bounded geometry and show that under suitable assumptions there exist Yamabe metrics, i.e. conformal metrics of constant scalar curvature. For that, we use weighted Sobolev embeddings.
A new metric for comparing probability measures on graphs, scalable and negative definite.
problem Optimal transport's high complexity and indefiniteness for kernel machines.
method Sobolev transport metric for graph metrics, closed-form formula, negative definiteness.
result Sobolev transport yields a scalable and negative definite metric.
The paper establishes scattering theory for wave equations on Schwarzschild spacetime.
problem Defocusing semilinear wave equations on Schwarzschild spacetime.
method Combining energy and pointwise decay results with Sobolev embedding, constructing scattering operator.
result Construction of a scattering operator mapping past to future scattering data.
The study characterizes kernel spaces on hyperspheres, impacting cubature algorithms.
problem Characterizing kernel spaces on hyperspheres for cubature algorithms.
method Characterization of Sobolev spaces and reproducing kernel Hilbert spaces over hyperspheres.
result Direct consequences for kernel cubature and worst-case error rates.
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.
We investigate a quantization problem which asks for the construction of an algebra for relative elliptic problems of pseudodifferential type associated to smooth embeddings. Specifically, we study the problem for embeddings in the category of compact manifolds with corners. The construction of a calculus for elliptic …
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.
Investigates energy minimizers and critical points of scale-invariant tangent-point energies for knots.
problem Finding and characterizing minimizers and critical points of scale-invariant tangent-point energies for closed curves.
method Develops convergence and regularity theories based on fractional Sobolev spaces and new energy functionals.
result Minimizing sequences converge to locally critical embeddings in all but finitely many points, and locally critical embeddings are regular.
Study generalizes Möbius energy to non-smooth sets in arbitrary dimensions.
problem Investigate Möbius-invariant energies on non-smooth subsets of arbitrary dimensions.
method Show local finite energy implies embedded Lipschitz submanifold, and low fractional Sobolev regularity guarantees finite energy.
result Local graph structure of low fractional Sobolev regularity on a set is sufficient to guarantee finite energy.
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.
New learning rates for embeddings in RKHSs, even when the target is not Hilbert-Schmidt.
problem Applying conditional mean embeddings to complex ML/RL settings with infinite-dimensional RKHSs.
method Developed novel learning rates using interpolation theory for RKHSs, derived explicit adaptive rates for sample estimator.
result Achieved uniform convergence rates in the output RKHS for certain parameter regimes.
By means of a general gluing and conformal-deformation construction, we prove that any smooth, metrically complete Riemannian manifold with smooth boundary can be realized as a closed domain into a smooth, geodesically complete Riemannan manifold without boundary. Applications to Sobolev spaces, Nash embedding and loca…
New solutions found for Yamabe problem on spheres with foliations.
problem Yamabe problem on spheres with singular Riemannian foliations.
method Variational methods, symmetries from foliations, Sobolev embedding theorem, Principle of Symmetric Criticality.
result Existence of sign-changing and positive solutions with specific symmetries.
Sharp Sobolev theory for scalar elliptic equations on minimal regular manifolds.
problem Well-posedness and regularity for scalar elliptic equations on manifolds of minimal regularity.
method Localization and flat domain techniques combined with Calderón–Zygmund theory and Fredholm alternative.
result Sharp Lp-based Sobolev regularity for scalar elliptic problems on manifolds of minimal regularity. The paper explores how Finsler manifolds differ from Riemannian ones in functional inequalities.
problem Analytic phenomena on Finsler manifolds differ from Riemannian ones.
method Comparative analysis of Finsler and Riemannian manifolds, focusing on Sobolev spaces, Hardy inequalities, and uncertainty principles.
result Functional inequalities (Hardy, uncertainty) break down on Finsler Cartan-Hadamard manifolds, while Caffarelli-Kohn-Nirenberg inequality exhibits a sharp threshold.
This paper analyzes shallow ReLU networks in L^p and Sobolev spaces, focusing on approximation and generalization.
problem Approximation and generalization of shallow ReLU networks in L^p and Sobolev spaces.
method Spherical harmonic analysis and embeddings into spectral Barron spaces for L^p spaces, path-norm control for Sobolev spaces.
result Minimax-optimal rates for nonparametric regression with shallow ReLU networks under path-norm control.
We prove a Poincaré, and a general Sobolev type inequalities for functions with compact support defined on a k-rectifiable varifold V defined on a complete Riemannian manifold with positive injectivity radius and sectional curvature bounded above. Our techniques allow us to consider Riemannian manifolds (Mn,g) w…
We prove the existence of a solution of the Yamabe equation on complete manifolds with finite volume and positive Yamabe invariant. In order to circumvent the standard methods on closed manifolds which heavily rely on global (compact) Sobolev embeddings we approximate the solution by eigenfunctions of certain conformal…
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.
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.