Related papers: The compactness and the concentration compactness …

Given a connected Riemannian manifold $\mathcal{N}$, an \(m\)--dimensional Riemannian manifold $\mathcal{M}$ which is either compact or the Euclidean space, $p\in [1, +\infty)$ and $s\in (0,1]$, we establish, for the problems of…

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…

Consider a regular domain $\Omega \subset \mathbb{R}^N$ and let $d(x)=\operatorname{dist}(x,\partial\Omega)$. Denote $L^{1,\infty}_a(\Omega)$ the space of functions from $L^{1,\infty}(\Omega)$ having absolutely continuous quasinorms. This…

For a general open set, we characterize the compactness of the embedding $W^{1,p}_0\hookrightarrow L^q$ in terms of the summability of its torsion function. In particular, for $1\le q<p$ we obtain that the embedding is continuous if and…

We study, for $1 \leq p \leq \infty$, the Hardy space $\bm{h}_e^p(\B)$, the elastic analogue of the classical Hardy spaces of harmonic functions in the unit ball of $\mathbb{R}^3$. The space consists of vector-field solutions of the Lam\'e…

This paper establishes a bivariate Hardy-Sobolev inequality. Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be an open domain, $s \in (0,2)$, $\alpha > 1$, $\beta > 1$ with $\alpha + \beta = 2^*(s)$, and $\kappa \in \mathbb{R}$. For any…

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 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…

The notion of $p$-compact sets arises naturally from Grothendieck's characterization of compact sets as those contained in the convex hull of a norm null sequence. The definition, due to Sinha and Karn (2002), leads to the concepts of…

We consider weighted $L^p$-Hardy inequalities involving the distance to the boundary of a domain in the $n$-dimensional Euclidean space with nonempty boundary. Using criticality theory, we give an alternative proof of the following result…

Let $(M, g)$ be a closed Riemannian manifold of dimension $5$. Assume that $(M, g)$ is not conformally equivalent to the round sphere. If the scalar curvature $R_g\geq 0$ and the $Q$-curvature $Q_g\geq 0$ on $M$ with $Q_g(p)>0$ for some…

First of all, we establish compactness of continuous mappings of the Orlicz--Sobolev classes $W^{1,\varphi}_{\rm loc}$ with the Calderon type condition on $\varphi$ and, in particular, of the Sobolev classes $W^{1,p}_{\rm loc}$ for $p>n-1$…

Let $\Omega\subset\mathbb{R}^n$ be an $(\epsilon,\delta,D)$-domain, with $\epsilon\in(0,1]$, $\delta\in(0,\infty]$, and $D\subset \partial \Omega$ being a closed part of $\partial \Omega$, which is a general open connected set when…

In this paper, we generalize the notion of relative $p$-capacity of $K$ with respect to $\Omega$, by replacing the Dirichlet boundary condition with a Robin one. We show that, under volume constraints, our notion of $p$-capacity is minimal…

We obtain inequalities of the form $$\int_C |f(z)|^p |dz| \leq A(p) \int_{\mathbb{T}} |f(z)|^p |dz|, \quad (p>1)$$ where $f$ is harmonic in the unit disk $\mathbb{D}$, $\mathbb{T}$ is the unit circle, and $C$ is any convex curve in…

For a non-empty compact set $E$ in a proper subdomain $\Omega$ of the complex plane, we denote the diameter of $E$ and the distance from $E$ to the boundary of $\Omega$ by $d(E)$ and $d(E,\partial\Omega),$ respectively. The quantity…

We prove an extrapolation of compactness theorem for operators on Banach function spaces satisfying certain convexity and concavity conditions. In particular, we show that the boundedness of an operator $T$ in the weighted Lebesgue scale…

In this article, the class of all Dunford-Pettis $ p $-convergent operators and $ p $-Dunford-Pettis relatively compact property on Banach spaces are investigated. Moreover, we give some conditions on Banach spaces $ X $ and $ Y $ such that…

We prove, using elementary methods of complex analysis, the following generalization of the isoperimetric inequality: if $p\in\re$, $\Omega\subset\re^2$ then the inequality $$…

The exploration of shape metamorphism, surface reconstruction, and image interpolation raises fundamental inquiries concerning the $C^1$ and higher-order regularity of $\infty$-harmonic potentials -- a specialized category of…