Related papers: An upper gradient approach to weakly differentiabl…

For a function $f$ from the Sobolev space $W^{1,p}(C)$ ($C\subset\mathbb{R}^d$ is an open convex cone), a sharp inequality that estimates $\| f\|_{L_{\infty}}$ via the $L_{p}$-norm of its gradient and a seminorm of the function is obtained.…

We establish a uniform Sobolev inequality for K\"ahler metrics, which only require an entropy bound and no lower bound on the Ricci curvature. We further extend our Sobolev inequality to singular K\"ahler metrics on K\"ahler spaces with…

Consider a BV function on a Riemannian manifold. What is its differential? And what about the Hessian of a convex function? These questions have clear answers in terms of (co)vector/matrix valued measures if the manifold is the Euclidean…

The weak regular coherence is a coarse property of a finitely generated group $\Gamma$. It was introduced by G. Carlsson and this author to play the role of a weakening of Waldhausen's regular coherence as part of computation of the…

Let L be a holomorphic line bundle with a positively curved singular Hermitian metric over a complex manifold X. One can define naturally the sequence of Fubini-Study currents associated to the space of square integrable holomorphic…

The purpose of this note is twofold. First we show that, for weakly differentiable maps between Riemannian manifolds of any dimension, a smallness condition on a Morrey-norm of the gradient is sufficient to guarantee that the pulled-back…

This paper studies smoothing properties of the local fractional maximal operator, which is defined in a proper subdomain of the Euclidean space. We prove new pointwise estimates for the weak gradient of the maximal function, which imply…

We develop the notion of a "filtered cospan" as an algebraic object that stands in the same relation to interlevel persistence modules as filtered chain complexes stand with respect to sublevel persistence modules. This relation is…

We introduce a concept of isoperimetric dimension for magnetic graphs, that is, graphs where every edge is assigned a complex number of modulus one. In analogy with the classical case, we show that isoperimetric inequalities imply Sobolev…

We study linear systems of ordinary differential equations of an arbitrary order on a finite interval with the most general (generic) inhomogeneous boundary conditions in Sobolev spaces. We investigate the character of solvability of…

\textit{Weak moonshine} for a finite group $G$ is the phenomenon where an infinite dimensional graded $G$-module $$V_G=\bigoplus_{n\gg-\infty}V_G(n)$$ has the property that its trace functions, known as McKay-Thompson series, are modular…

The underlying theme of this article is a class of sequences in metric structures satisfying a much weaker kind of Cauchy condition, namely quasi-Cauchy sequences (introduced in \cite{bc}) that has been used to define several new concepts…

The present paper develops two concepts of pointwise differentiability of higher order for arbitrary subsets of Euclidean space defined by comparing their distance functions to those of smooth submanifolds. Results include that…

In this paper, we consider linear elliptic systems from composite materials where the coefficients depend on the shape and might have the discontinuity between the subregions. We derive a function which is related to the gradient of the…

We introduce the notion of weak reduciblity for Dupin submanifolds with arbitrary codimension. We give a complete characterization of all weakly reducible Dupin submanifolds, as a consequence of a general result on a broader class of…

We consider Wasserstein gradient flows of maximum mean discrepancy (MMD) functionals $\text{MMD}_K^2(\cdot, \nu)$ for positive and negative distance kernels $K(x,y) := \pm |x-y|$ and given target measures $\nu$ on $\mathbb{R}$. Since in one…

The challenge of approximating functions in infinite-dimensional spaces from finite samples is widely regarded as formidable. We delve into the challenging problem of the numerical approximation of Sobolev-smooth functions defined on…

We develop a framework for Poisson geometry on loop spaces of low regularity, extending Mokhov's classical constructions from smooth loops to weak Sobolev spaces $W^{s,p}(\mathbb{S^1},\mathbb{R}^m)$ with $o < s \frac{1}{2}$ and $1 < p <…

We introduce and study measures and densities (= geometric measures) on differentiable stacks, using a rather straightforward generalization of Haefliger's approach to leaf spaces and to transverse measures for foliations. In general we…

In this paper, we explore the concept of metric-driven numerical methods as a powerful tool for solving various types of multiscale partial differential equations. Our focus is on computing constrained minimizers of functionals - or,…