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

We show that Sobolev space $W^1_1(\Omega)$ of any planar one-connected domain $\Omega$ has the Bounded Approximation property. The result holds independently from the properties of the boundary of $\Omega$. The prove is based on a new…

Let $\Omega \subset \mathbb{R}^n$ be an open set and $f_k \in W^{s,p}(\Omega;\mathbb{R}^n)$ be a sequence of homeomorphisms weakly converging to $f \in W^{s,p}(\Omega;\mathbb{R}^n)$. It is known that if $s=1$ and $p > n-1$ then $f$ is…

We extend Krylov and R\"{o}ckner's result \cite{KR} to the drift coefficients in critical Lebesgue space, and prove the existence and uniqueness of weak solutions for a class of SDEs. To be more precise, let $b: [0,T]\times{\mathbb…

Let $\Omega\subseteq \mathbb{R}^{d}$ be open and $A$ a complex uniformly strictly accretive $d\times d$ matrix-valued function on $\Omega$ with $L^{\infty}$ coefficients. Consider the divergence-form operator ${\mathscr L}^{A}=-{\rm…

Let $L$ be the Dunkl Laplacian on the Euclidean space $\mathbb{R}^N$ associated with a normalized root system $R$ and a multiplicity function $k(\nu)\geq 0$, $\nu\in R$. We establish a Leibniz-type rule for the fractional powers of $L$ on…

Given a smooth complete Riemannian manifold with bounded geometry $(M,g)$ and a linear connection $\nabla$ on it (not necessarily a metric one), we prove the $L^p$-boundedness of operators belonging to the global pseudo-differential classes…

We derive a discrete $ L^q-L^p$ Sobolev inequality tailored for the Crouzeix--Raviart and discontinuous Crouzeix--Raviart finite element spaces on anisotropic meshes in both two and three dimensions. Subject to a semi-regular mesh…

In this paper we study approximations of functions of Sobolev spaces $W^2_{p,\loc}(\Omega)$, $\Omega\subset\mathbb R^n$, by Lipschitz continuous functions. We prove that if $f\in W^2_{p,\loc}(\Omega)$, $1\leq p<\infty$, then there exists a…

In the first part of the paper the authors study the minimal and maximal extension of a class of weighted pseudodifferential operators in the Fr\'echet space $L^p_{\rm loc}(\Omega)$. In the second one non homogeneous microlocal properties…

This paper deals with a notion of Sobolev space $W^{1,p}$ introduced by J.Bourgain, H.Brezis and P.Mironescu by means of a seminorm involving local averages of finite differences. This seminorm was subsequently used by A.Ponce to obtain a…

We define abstract Sobolev type spaces on $\mathsf{L}^p$-scales, $p\in [1,\infty)$, on Hermitian vector bundles over possibly noncompact manifolds, which are induced by smooth measures and families $\mathfrak{P}$ of linear partial…

Let $\Omega\subset \mathbb{R}^{n}$ be a bounded open set. Given $2\leq m\leq n$, we construct a convex function $\phi :\Omega\to \mathbb{R}$ whose gradient $f= \nabla \phi$ is a H\"older continuous homeomorphism, $f$ is the identity on…

A basilar property and a useful tool in the theory of Sobolev spaces is the density of smooth compactly supported functions in the space $W^{k,p}(\R^n)$ (i.e. the functions with weak derivatives of orders $0$ to $k$ in $L^p$). On Riemannian…

Let $\Omega,\Omega'\subset\mathbb{R}^3$ be Lipschitz domains, let $f_m:\Omega\to\Omega'$ be a sequence of homeomorphisms with prescribed Dirichlet boundary condition and $\sup_m \int_{\Omega}(|Df_m|^2+1/J^2_{f_m})<\infty$. Let $f$ be a weak…

The usual Sobolev inequality in $\mathbb{R}^N$, asserts that $\|\nabla u\|_{L^p(\mathbb{R}^N)} \geq \mathcal{S}\|u\|_{L^{p^*}(\mathbb{R}^N)}$ for $1<p<N$ and $p^*=\frac{pN}{N-p}$, with $\mathcal{S}$ being the sharp constant. Based on a…

In this note, we introduce a variant of Calder\'on and Zygmund's notion of $L^p$-differentiability - an \emph{$L^p$-Taylor approximation}. Our first result is that functions in the Sobolev space $W^{1,p}(\mathbb{R}^N)$ possess a first order…

Let $L^{m,p}(\mathbb{R}^n)$ be the homogeneous Sobolev space for $p \in (n,\infty)$, $\mu$ be a Borel regular measure on $\mathbb{R}^n$, and $L^{m,p}(\mathbb{R}^n) + L^p(d\mu)$ be the space of Borel measurable functions with finite seminorm…

In this work, we introduce a new family of functions spaces, the weighted fractional Sobolev spaces $X^{s,p}_{0,w}(\Omega)$, where $w$ is a weight in the Muckenhoupt class $A_p$. This space is a natural extension of the fractional Sobolev…

Let $\Omega$ be a smooth bounded domain in $\mathbb R^n$ and u be a measurable function on $\Omega$ such that $|u(x)|=1$ almost everywhere in $\Omega$. Assume that u belongs to the $B^s_{p,q}(\Omega)$ Besov space. We investigate whether…

We study a conormal boundary value problem for a class of quasilinear elliptic equations in bounded domain $\Omega$ whose coefficients can be degenerate or singular of the type $\text{dist}(x, \partial \Omega)^\alpha$, where $\partial…