arXiv research

New neural network rates for unbounded domains with weighted Sobolev spaces.

A new statistical model uses Orlicz-Sobolev spaces with Gaussian weight.

The paper defines and analyzes $abla$-Sobolev spaces and operators on manifolds.

In this paper, we prove a Sobolev and isoperimetric inequalities for submanifold in weighted manifold. Our results generalize the Hoffman-Spruck's inequalities.

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 …

Characterizes Schrödinger operator boundedness on weighted Riemannian manifolds.

Maximum principle proves positivity of forward rates in stochastic models.

We study some basic analytic questions related to differential operators on Lie manifolds, which are manifolds whose large scale geometry can be described by a a Lie algebra of vector fields on a compactification. We extend to Lie manifolds several classical results on Sobolev spaces, elliptic regularity, and mapping p…

Estimates for harmonic functions in curved spaces.

We study in detail Hodge-Helmholtz decompositions in non-smooth exterior domains filled with inhomogeneous and anisotropic media. We show decompositions of alternating differential forms belonging to weighted Sobolev spaces into irrotational and solenoidal forms. These decompositions are essential tools, for example, i…

Study extends neural network approximation to time-varying PDEs using Fourier-Lebesgue 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…

Sharp inequalities for radial functions on hyperbolic spaces without boundary conditions.

Study shows rates for Laplacian-eigenmap methods in nonparametric regression.

The study improves Poincaré and log-Sobolev inequalities on hyperbolic spaces.

The paper shows neural networks can approximate functions over non-compact domains with non-polynomial activation.

Study fine Pólya-Szegő inequalities in metric spaces with applications.

We show in this note that the Sobolev Discrepancy introduced in Mroueh et al in the context of generative adversarial networks, is actually the weighted negative Sobolev norm $||.||_{\dot{H}^{-1}(ν_q)}$, that is known to linearize the Wasserstein $W_2$ distance and plays a fundamental role in the dynamic formulation of…

Sharp inequality on Siegel domain involving weighted norms and sub-Laplacian.

We study the existence and regularity of solutions to the Cauchy problem for the inhomogeneous heat equation on compact Riemannian manifolds with conical singularities. We introduce weighted Hölder and Sobolev spaces with discrete asymptotics and we prove existence and maximal regularity of solutions to the Cauchy prob…

The paper proves gradient estimates for a weighted p-Laplacian equation on Riemannian manifolds.

We compute the index of the real Cauchy-Riemann operator defined in FJRW theory in case of the smooth metric. For the cylindrical metric, we study the relation between the index of the linearized operator of Witten map and weights in weighted Sobolev space.

We derive weighted log-Sobolev inequalities from a class of super Poincaré inequalities. As an application, the Talagrand inequality with larger distances are obtained. In particular, on a complete connected Riemannian manifold, we prove that the $\log^\dd$-Sobolev inequality with $\dd\in (1,2)$ implies the $L^{2/(2-\d…

We prove a weighted Sobolev inequality and a Hardy inequality on manifolds with nonnegative Ricci curvature satisfying an inverse doubling volume condition. It enables us to obtain rigidity results for Ricci flat manifolds, generalizing earlier work of Bando, Kasue and Nakajima.

In this note we prove a new ε-regularity theorem for the Ricci flow. Let (M^n,g(t)) with t\in [-T,0] be a Ricci flow and H_{x} the conjugate heat kernel centered at a point (x,0) in the final time slice. Substituting H_{x} into Perelman's W-functional produces a monotone function W_{x}(s) of s \in [-T,0], the pointed e…

Novel algorithm speeds up computation of Sobolev IPM for graph-based probability measures.

The study analyzes weighted manifolds with curvature bounds, proving eigenvalue estimates and inequalities.

Modified Bakry-Émery criterion inequality for Tsallis entropy monotonicity.

We develop a solution theory for a generalized electro-magneto static Maxwell system in an exterior domain with anisotropic coefficients converging at infinity with a certain rate towards the identity. Our main goal is to treat right hand side data from some polynomially weighted Sobolev spaces and obtain solutions whi…

A new metric for comparing probability measures on graphs, scalable and negative definite.

Study regularity of Schrödinger eigenfunctions with Coulomb-type potentials.

Neural networks cannot approximate certain functions in Sobolev spaces, leading to unbounded parameter growth.

Given an elliptic operator $P$ on a non-compact manifold (with proper asymptotic conditions), there is a discrete set of numbers called indicial roots. It's known that $P$ is Fredholm between weighted Sobolev spaces if and only if the weight is not indicial. We show that an elliptic theory exists even when the weight i…

This paper is a self-contained presentation of certain aspects of the theory of weighted Sobolev spaces and elliptic operators on non-compact Riemannian manifolds. Specifically, we discuss (i) the standard and weighted Sobolev Embedding Theorems for general manifolds and (ii) Fredholm results for elliptic operators on …

We investigate the rigidity problem for the logarithmic Sobolev inequality on weighted Riemannian manifolds satisfying $\mathrm{Ric}_{\infty} \ge K>0$. Assuming equality holds, we show that the $1$-dimensional Gaussian space is necessarily split off, similarly to the rigidity results of Cheng--Zhou on the spectral gap …

The main result of this paper is the following: any `weighted' Riemannian manifold $(M,g,μ)$ - i.e. endowed with a generic non-negative Radon measure $μ$ - is `infinitesimally Hilbertian', which means that its associated Sobolev space $W^{1,2}(M,g,μ)$ is a Hilbert space. We actually prove a stronger result: the abstrac…

Here shape space is either the manifold of simple closed smooth unparameterized curves in $\mathbb R^2$ or is the orbifold of immersions from $S^1$ to $\mathbb R^2$ modulo the group of diffeomorphisms of $S^1$. We investige several Riemannian metrics on shape space: $L^2$-metrics weighted by expressions in length and c…

This paper presents a convergence analysis of kernel-based quadrature rules in misspecified settings, focusing on deterministic quadrature in Sobolev spaces. In particular, we deal with misspecified settings where a test integrand is less smooth than a Sobolev RKHS based on which a quadrature rule is constructed. We pr…

The paper proves local rigidity theorems for scalar curvature and related inequalities.

The paper proves continuity of Morse index for Ricci shrinkers.

Generalizes Sobolev IPM for graph-based measures using Orlicz geometric structure.

Develops a new method for neural network significance testing without strict constraints.

Proves sub-Riemannian manifolds are infinitesimally Hilbertian with arbitrary measures.

We propose a new Integral Probability Metric (IPM) between distributions: the Sobolev IPM. The Sobolev IPM compares the mean discrepancy of two distributions for functions (critic) restricted to a Sobolev ball defined with respect to a dominant measure $μ$. We show that the Sobolev IPM compares two distributions in hig…

Upper bounds on constants for Brownian motion with sticky boundary.

Paper introduces a new regularization method for kernel gradient descent learning.

Finite time for subsolutions on Riemannian manifolds proved.

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…