Related papers: On limiting relations for capacities

In this paper we consider topological mappings defined by $p$-capacity inequalities in domains of $\mathbb R^n$. In the case $p=n-1$ we prove the Lipschitz continuity of such mappings, that extends the result by F.~W.~Gehring.

In a previous work, we showed that Besov spaces do not enjoy the restriction property unless $q\leq p$. Specifically, we proved that if $p<q$, then it is always possible to construct a function $f\in B_{p,q}^s(\mathbb{R}^N)$ such that…

We study connections between definability in generalized descriptive set theory and large cardinals, under ZFC. We show that if $\kappa$ is a limit of measurables then there is no wellorder of a subset of $P(\kappa)$ of length…

This short note investigates the compact embedding of degenerate matrix weighted Sobolev spaces into weighted Lebesgue spaces. The Sobolev spaces explored are defined as the abstract completion of Lipschitz functions in a bounded domain…

We consider Bernoulli first-passage percolation on the triangular lattice in which sites have 0 and 1 passage times with probability $p$ and $1-p$, respectively. For each $p\in(0,p_c)$, let $\mathcal {B}(p)$ be the limit shape in the…

How large is the Bessel potential, $G_{\alpha,\mu}f$, compared to the Riesz potential, $I_\alpha f$? In this paper, we show that if $I_\alpha f\in L^p$ with $0<\alpha<1$ and $p>1$, then the following interpolation bound holds: \[\Vert…

In 2016, Ellenberg and Gijswijt employed a method of Croot, Lev, and Pach to show that a maximal cap in $AG(n, q)$ has size $O(q^{cn})$ for some $c < 1$. In this paper, we show more generally that if $S$ is a subset of $AG(n, q)$ containing…

Let $q(x)$ and $p(x)$ denote density functions on the $n$-dimensional Euclidean space, and let $p_i(\cdot|y_1,..., y_{i-1},y_{i+1},..., y_n)$ and $Q_i(\cdot|x_1,..., x_{i-1},x_{i+1},..., x_n)$ denote their local specifications. For a class…

In this paper we prove a lower semicontinuity result of Reshetnyak type for a class of functionals which appear in models for small-strain elasto-plasticity coupled with damage. To do so we characterise the limit of measures…

The omega limit sets plays a fundamental role to construct global attractors for topological semi-dynamical systems with continuous time or discrete time. Therefore, it is important to know when omega limit sets become nonempty compact…

We derive new estimates on analytic capacities of finite sequences in the unit disc in Besov spaces with zero smoothness, which sharpen the estimates obtained by N.K.Nikolski in 2005 and, for a range of parameters, are optimal. The work is…

This paper is inspired by the paper of Leonetti, Russo and Somaglia [\textit{Dense lineability and spaceability in certain subsets of $\ell_\infty$.} Bull. London Math. Soc., 55: 2283--2303 (2023)] and the lineability problems raised…

The work is devoted to the relaxation limit in larger Besov spaces for compressible Euler equations, which contains previous results in Sobolev spaces and Besov spaces with critical regularity. Such an extension depends on a revision of…

For any $p\in[1,\infty)$, we prove that the set of simple functions taking at most $k$ different values is proximinal in B\"ochner spaces $L^p(X)$ whenever $X$ is a dual Banach space with $w^*$-sequentially compact unit ball. With…

Properties of Riesz capacity are developed with respect to the kernel exponent $p \in (-\infty,n)$, namely that capacity is monotonic as a function of $p$, that its endpoint limits recover the diameter and volume of the set, and that…

If Bekenstein's conjectured bound on the microcanonical entropy, S < 2 pi E R, is applied to a closed subsystem of maximal linear size R and excitation energy up through E, it can be violated by an arbitrarily large factor by a scalar field…

Let $G$ be a connected unimodular group equipped with a (left and hence right) Haar measure $\mu_G$, and suppose $A, B \subseteq G$ are nonempty and compact. An inequality by Kemperman gives us…

Let $S \subset \mathbb{R}^{n}$ be an arbitrary nonempty compact set such that the $d$-Hausdorff content $\mathcal{H}^{d}_{\infty}(S) > 0$ for some $d \in (0,n]$. For each $p \in (\max\{1,n-d\},n]$, an almost sharp intrinsic description of…

If $X$ is a set, $E$ is an equivalence relation on $X$, and $n \in \omega$, then define $$[X]^n_E = \{(x_0, ..., x_{n - 1}) \in {}^nX : (\forall i,j)(i \neq j \Rightarrow \neg(x_i \ E \ x_j))\}.$$ For $n \in \omega$, a set $X$ has the…

Given a finite point set $P\subset\mathbb{R}^d$, we call a multiset $A$ a one-sided weak $\varepsilon$-approximant for $P$ (with respect to convex sets), if $|P\cap C|/|P|-|A\cap C|/|A|\leq\varepsilon$ for every convex set $C$. We show…