Related papers: A Bourgain-Brezis-Mironescu -type characterization…

Bourgain et al.(2001) proved that for $p>1$ and smooth bounded domain $\Omega\subseteq\mathbb{R}^N$, \begin{equation*} \lim\limits_{s\to1}(1-s)\iint \limits_{\Omega \times \Omega}\frac{\lvert f(x)-f(y) \rvert^p}{\lvert x-y \rvert^{N+sp}}dx…

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and $1 < p < \infty$. We characterize $(1,p)$-extension domains in terms of inequalities of Bourgain--Brezis--Mironescu type. More precisely, we show that $\Omega$ is a $(1,p)$-extension…

In this paper we connect Calder\'on and Zygmund's notion of $L^p$\- -differentiability with some recent characterizations of Sobolev spaces via the asymptotics of non-local functionals due to Bourgain, Brezis, and Mironescu. We show how the…

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 $\alpha$.…

Under certain restrictions on $s,p,q$, the Triebel-Lizorkin spaces can be viewed as generalised fractional Sobolev spaces $W^{s,p}_q$. In this article, we show that the Bourgain-Brezis-Mironescu formula holds for $W^{s,p}_q$-seminorms in…

The $L^p$-spaces, with $p \not = \infty$, form a partial algebra $(L^p(\Omega), \Gamma, \cdot)$ with pointwise multiplication of functions. The Sobolev spaces $W^{k,p}(\Omega)$, delineated by weak derivatives as subspaces of $L^p$-spaces is…

A Bourgain--Brezis--Mironescu-type theorem for fractional Sobolev spaces with variable exponents is established for sufficiently regular functions. We prove, however, that a limiting embedding theorem for these spaces fails to hold in…

We collect and extend results on the limit of $\sigma^{1-k}(1-\sigma)^k |v|_{l+\sigma,p,\Omega}^p$ as $\sigma$ tends to $0^+$ or $1^-$, where $\Omega$ is $\mathbb{R}^n$ or a smooth bounded domain, $k$ is 0 or 1, $l$ is a nonnegative…

We show that in a bounded Gromov hyperbolic domain $\Omega$ smooth functions with bounded derivatives $C^\infty(\Omega)\cap W^{k,\infty}(\Omega)$ are dense in the homogeneous Sobolev spaces $L^{k,p}(\Omega)$.

Let $d\geq 2$ be an integer, $1\leq l\leq d-1$ and $\varphi$ be a differential $l$-form on ${\mathbb R}^d$ with $\dot{W}^{1,d}$ coefficients. It was proved by Bourgain and Brezis (\cite[Theorem 5]{MR2293957}) that there exists a…

We study a class of nonlocal functionals in the spirit of the recent characterization of the Sobolev spaces $W^{1,p}$ derived by Bourgain, Brezis and Mironescu. We show that it provides a common roof to the description of the…

Let $\Omega$ be a bounded domain in R n with a Sobolev extension property around the complement of a closed part D of its boundary. We prove that a function u $\in$ W 1,p ($\Omega$) vanishes on D in the sense of an interior trace if and…

Let $\Omega\subset\mathbb{R}^n$ be a bounded $(\varepsilon,\infty)$-domain with $\varepsilon\in(0,1]$, $X(\mathbb{R}^n)$ a ball Banach function space satisfying some extra mild assumptions, and $\{\rho_\nu\}_{\nu\in(0,\nu_0)}$ with…

We obtain asymptotically sharp identification of fractional Sobolev spaces $ W^{s}_{p,q}$, extension spaces $E^{s}_{p,q}$, and Triebel-Lizorkin spaces $\dot{F}^s_{p,q}$. In particular we obtain for $W^{s}_{p,q}$ and $E^{s}_{p,q}$ a…

We consider a nonlinear pseudo-differential equation driven by the fractional $p$-Laplacian $(-\Delta)^s_p$ with $s\in(0,1)$ and $p\ge 2$ (degenerate case), under Dirichlet type conditions in a smooth domain $\Omega$. We prove that local…

We provide an intrinsic atomic characterization for 2-microlocal Besov and Triebel-Lizorkin spaces with variable integrability on domains, $B_{\p,\q}^{\bm{w}}(\Omega)$ and $F_{\p,\q}^{\bm{w}}(\Omega)$, where $\Omega$ is a regular domain. We…

We obtain a Bourgain-Br\'ezis-Mironescu formula on the limit behaviour of a modified fractional Sobolev seminorm when $s\nearrow 1$, which is valid in arbitrary bounded domains. In the case of extension domains, we recover the classical…

We introduce a large class of concentrated $p$-L\'{e}vy integrable functions approximating the unity, which serves as the core tool from which we provide a nonlocal characterization of Sobolev spaces and the space of functions of bounded…

We establish a pointwise limit theorem for a broad class of pa\-ra\-me\-ter-\-de\-pen\-dent BMO-type seminorms as the parameter tends to zero. By introducing novel BMO-type seminorms, we provide a unified framework that extends several…

We prove density of smooth functions in subspaces of Sobolev- and higher order $BV$-spaces of kind $W^{m,p}(\Omega)\cap L^q(\Omega-D)$ and $BV^m(\Omega)\cap L^q(\Omega-D)$, respectively, where $\Omega\subset\mathbb{R}^n$ ($n\in\mathbb{N}$)…