Related papers: On some properties of relative capacity and thinne…

In this paper, we introduce a thinness in sense to a type of relative capacity for weighted variable exponent Sobolev space. Moreover, we reveal some properties of this thinness and consider the relationship with finely open and finely…

In this study, we define double weighted variable exponent Sobolev spaces $W^{1,q(.),p(.)}\left( \Omega ,\vartheta _{0},\vartheta \right) $ with respect to two different weight functions. Also, we investigate the basic properties of this…

The purpose of this article is to introduce the relative $p$-capacity $\Cap_{p,\Omega}$ with respect to an open set $\Omega$ in $\IR^N$. It is a Choquet capacity on the closure of $\Omega$ and extends the classical $p$-capacity $\Cap_p$ in…

In this paper we develop a capacities theory connected with the fractional Sobolev spaces with variable exponents. Two kinds of capacities are studied: Sobolev capacity and relative capacity. Basic properties of capacities, including…

In this paper we present a new characterization of the Sobolev space $W^{1,p}$, $1<p<\infty$ which is a higher dimensional version of a result of Waterman. We also provide a new and simplified proof of a recent result of Alabern, Mateu and…

We characterize the real interpolation space between a weighted $L^p$ space and a weighted Sobolev space in arbitrary bounded domains in $\mathbb{R}^n$, with weights that are positive powers of the distance to the boundary.

We study relations between the variational Sobolev 1-capacity and versions of variational BV-capacity in a complete metric space equipped with a doubling measure and supporting a weak $(1,1)$-Poincar\'e inequality. We prove the equality of…

In this article we study basic properties of the mixed BV-Sobolev capacity with variable exponent p. We give an alternative way to define mixed type BV-Sobolev-space which was originally introduced by Harjulehto, H\"ast\"o, and Latvala. Our…

We prove that weakly differentiable weights $w$ which, together with their reciprocals, satisfy certain local integrability conditions, admit a unique associated first-order $p$-Sobolev space, that is \[H^{1,p}(\mathbb{R}^d,w\,\d…

We study the interpolation property of Sobolev spaces of order 1 denoted by $W^{1}_{p,V}$, arising from Schr\"{o}dinger operators with positive potential. We show that for $1\leq p_1<p<p_2<q_{0}$ with $p>s_0$, $W^{1}_{p,V}$ is a real…

We examine the relations between different capacities in the setting of a metric measure space. First, we prove a comparability result for the Riesz $(\beta,p)$-capacity and the relative Hajlasz $(\beta,p)$-capacity, for $1<p<\infty$ and…

For each $p>n$ we use local oscillations and doubling measures to give intrinsic characterizations of the restriction of the Sobolev space $W_p^1(R^n)$ to an arbitrary closed subset of $R^n$.

We study associate and double associate spaces of two-weighted Sobolev spaces of the first order on real half-line and we show that unlike the notion of duality the associativity is divided into two cases which we call "strong" and "weak"…

Let $\Omega$ be an open set in a metric measure space $X$. Our main result gives an equivalence between the validity of a weighted Hardy-Sobolev inequality in $\Omega$ and quasiadditivity of a weighted capacity with respect to Whitney…

We develop a theory of capacities associated with local Muckenhoupt weights. Fundamental properties of local Muckenhoupt weights will be revisited. Weak type boundedness of nonlinear potential and capacitary strong type inequalities…

Let $p(\cdot):\ \mathbb R^n\to(0,\infty)$ be a variable exponent function and $X$ a ball quasi-Banach function space. In this paper, we first study the relationship between two kinds of variable weights…

This article contains a characterization of when certain weighted Sobolev spaces on $\Bbb R^n$ embed compactly into $L^2(\mathbb R^n, \varphi)$. The characterization is in terms of derivatives of the weight function $\varphi$ and involves…

We characterize one-sided weighted Sobolev spaces $W^{1,p}(\mathbb{R},\omega)$, where $\omega$ is a one-sided Sawyer weight, in terms of a.e.~and weighted $L^p$ limits as $\alpha\to1^-$ of Marchaud fractional derivatives of order $\alpha$.…

We consider different notions of capacity related to the parabolic $p$-Laplace equation. Our focus is on a variational notion, which is consistent in the full range $1<p<\infty$. For such a notion we show some basic properties as well as…

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…