Related papers: Sharply estimating hyperbolic capacities

In this paper, we obtain the sharp $k$-th order Sobolev inequalities in the hyperbolic space ${\H}^n$ for all $k=1,2,3,\cdots$. This gives an answer to an open question raised by Aubin in [5, p.$\;$176-177] for $W^{k,2}({\H}^n)$ with $k>1$.…

In this paper, we study the sharp Poincar\'e inequality and the Sobolev inequalities in the higher order Lorentz--Sobolev spaces in the hyperbolic spaces. These results generalize the ones obtained in \cite{Nguyen2020a} to the higher order…

In this paper, new sharp bounds for circular functions are proved. We provide some improvements of previous results by using infinite products, power series expansions and a generalisation of the so-called Bernoulli inequality. New proofs,…

In this article, we introduce and study capacities related to nonlocal Sobolev spaces, with focus on spaces corresponding to zero-order nonlocal operators. In particular, we prove Hardy-type inequalities to obtain Sobolev embeddings and use…

Let $W^1L^{p,q}(\mathbb H^n)$, $1\leq q,p < \infty$ denote the Lorentz-Sobolev spaces of order one in the hyperbolic spaces $\mathbb H^n$. Our aim in this paper is three-fold. First of all, we establish a sharp Poincar\'e inequality in…

We establish sharp Adams type inequalities on Sobolev spaces $W^{\alpha, n/\alpha}(X)$ of any fractional order $\alpha< n$ on Riemannian symmetric space $X$ of noncompact type with dimension $n$ and of arbitrary rank. We also establish…

We prove sharp inequalities of Hardy type for functions in the Sobolev space $W^{1,p}$ on the unit sphere $\mathbb{S}^{n-1}$ in $\mathbb{R}^{n}$. We achieve this in both the subcritical and critical cases. The method we use to show…

This is a short survey on finite-volume hyperbolic four-manifolds. We describe some general theorems and focus on the concrete examples that we found in the literature. The paper contains no new result.

We derive various sharp upper bounds for the $p$-capacity of a smooth compact set $K$ in the hyperbolic space $\mathbb{H}^n$ and the Euclidean space $\mathbb{R}^n$. Firstly, using the inverse mean curvature flow, for the mean convex and…

In this article we compute the best Sobolev constants for various Hardy-Sobolev inequalities with sharp Hardy term. This is carried out in three different environments: interior point singularity in Euclidean space, interior point…

We give formulas for the numbers of type II and type IV hyperbolic components in the space of quadratic rational maps, for all fixed periods of attractive cycles.

This article simply presents several coordinate systems for 2 and 3-dimensional hyperbolic spaces, describing the general solutions of Helmholtz equation in each one of these systems.

This paper is addressed to establishing an internal observability estimate for some linear stochastic hyperbolic equations. The key is to establish a new global Carleman estimate for forward stochastic hyperbolic equations in the…

Using some harmonic extensions on the upper-half plane, and probabilistic representations, and curvature-dimension inequalities with some negative dimensions, we obtain some new opimal functional inequalities of the Beckner type for the…

Using a four points inequality for the boundary of CAT(-1)-spaces, we study the relation between Gromov hyperbolic spaces and CAT(-1)-spaces.

We prove capacitary strong type inequalities for functions belonging to Orlicz-Sobolev spaces. As an application we consider capacitary averages and their limits.

We study the growth of hyperbolic type distances in starlike domains. We derive estimates for various hyperbolic type distances and consider the asymptotic sharpness of the estimates.

We construct examples of inhomogeneous isoparametric real hypersurfaces in complex hyperbolic spaces.

This note develops certain sharp inequalities relating the fractional Sobolev capacity of a set to its standard volume and fractional perimeter.

We suggest a new approach to the study of relatively hyperbolic groups based on relative isoperimetric inequalities. Various geometric, algebraic, and algorithmic properties are discussed.