Related papers: Alexander-Taylor's inequality for capacities in co…

In this paper, we apply the so-called Alexandrov-Bakelman-Pucci (ABP) method to establish some geometric inequalities. We first prove a logarithmic Sobolev inequality for closed $n$-dimensional minimal submanifolds $\Sigma$ of $\mathbb…

In this paper, the results of Mei, Wang, Weng and Xia [Math. Z., 2025, MR4911815] on capillary convex bodies are extended to the anisotropic setting. We develop a theory for anisotropic capillary convex bodies in the half-space and…

In the Euclidean space $\mathbb{R}^d$, the sharp classical Sobolev inequality is equivalent by conformal invariance to a Sobolev inequality on the hyperbolic space $\mathbb{H}^d$. This inequality is sharp in dimension $d\geq 4$, but it is…

In the context of Dirichlet type spaces on the unit ball of $\mathbb{C}^d$, also known as Hardy-Sobolev or Besov-Sobolev spaces, we compare two notions of smallness for compact subsets of the unit sphere. We show that the functional…

We study the Hardy identities and inequalities on Cartan-Hadamard manifolds using the notion of a Bessel pair. These Hardy identities offer significantly more information on the existence/nonexistence of the extremal functions of the Hardy…

In this paper we study Hardy-Sobolev inequalities on hypersurfaces of $\mathbb{R}^{n+1}$, all of them involving a mean curvature term and having universal constants independent of the hypersurface. We first consider the celebrated Sobolev…

We study fine properties of quasiplurisubharmonic functions on compact K\"ahler manifolds. We define and study several intrinsic capacities which characterize pluripolar sets and show that locally pluripolar sets are globally…

We present a unified strategy to derive Hardy-Poincar\'e inequalities on bounded and unbounded domains. The approach allows proving a general Hardy-Poincar\'e inequality from which the classical Poincar\'e and Hardy inequalities immediately…

In this short paper we generalize the classical inequality between the norms in Lebesgue spaces of the functions and its derivatives, which in the multidimensional case are called Sobolev's inequalities, on the many popular classes pairs of…

This is the second installment in a series of papers aimed at generalizing symplectic capacities and homologies. We study symmetric versions of symplectic capacities for real symplectic manifolds, and obtain corresponding results for them…

In this article we are mainly concerned with three dimensional compact K\"ahler spaces with log terminal singularities. We establish the orbifold version of the Bogomolov-Gieseker inequality for stable $\mathbb Q$-sheaves.

We consider various types of Hardy-Sobolev inequalities on a Carnot-Carath\'eodory space $(\Om, d)$ associated to a system of smooth vector fields $X=\{X_1, X_2,...,X_m\}$ on $\RR^n$ satisfying the H\"ormander's finite rank condition $rank…

The purpose of this article is twofold. The first is to strengthen fractional Sobolev type inequalities in Besov spaces via the classical Lorentz space. In doing so, we show that the Sobolev inequality in Besov spaces is equivalent to the…

In this paper we study connections between composition operators on Sobolev spaces and mappings defined by $p$-moduli inequalities ($p$-capacity inequalities). We prove that weighted moduli inequalities lead to composition operators on…

Here, the concept of electric capacity on Finsler spaces is introduced and the fundamental conformal invariant property is proved, i.e. the capacity of a compact set on a connected non-compact Finsler manifold is conformal invariant. This…

In this paper, firstly, inspired by Nat\'{a}rio's recent work \cite{Na}, we use the isoperimetric inequality to derive some Alexandrov-Fenchel type inequalities for closed convex hypersurfaces in the hyperbolic space $\H^{n+1}$ and in the…

We give an affirmative answer to the resistance conjecture on characterization of parabolic Harnack inequalities in terms of volume doubling, upper capacity bounds and a Poincar\'e inequalities. The key step is to show that these three…

The space of K\"ahler potentials in a compact K\"ahler manifold, endowed with Mabuchi's metric, is an infinite dimensional Riemannian manifold. We characterize local isometries between spaces of K\"ahler potentials, and prove existence and…

Inequalities for Riesz potentials are well-known to be equivalent to Sobolev inequalities of the same order for domain norms ``far" from $L^1$, but to be weaker otherwise. Recent contributions by Van Schaftingen, by Hernandez, Rai\c{t}\u{a}…

We study eigenvalues of polyharmonic operators on compact Riemannian manifolds with boundary (possibly empty). In particular, we prove a universal inequality for the eigenvalues of the polyharmonic operators on compact domains in a…