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

Consider a Lipschitz domain $\Omega$ and a measurable function $\mu$ supported in $\overline\Omega$ with $\left\|{\mu}\right\|_{L^\infty}<1$. Then the derivatives of a quasiconformal solution of the Beltrami equation $\overline{\partial} f…

We establish a non-local integral difference quotient representation for symmetric gradient semi-norms in $BD(\Omega)$ and $LD(\Omega)$, which does not require the manipulation of distributional derivatives. Our representation extends the…

In this work, we aim to prove algebra properties for generalized Sobolev spaces $W^{s,p} \cap L^\infty$ on a Riemannian manifold, where $W^{s,p}$ is of Bessel-type $W^{s,p}:=(1+L)^{-s/m}(L^p)$ with an operator $L$ generating a heat…

We prove that given any positive integer $k$, for each open set $\Omega$ and any closed subset $D$ of its closure such that $\Omega$ is locally an (epsilon,delta)-domain near points in the boundary of $\Omega$ not contained in $D$ there…

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

Let $\gamma\in\mathbb{R}\setminus\{0\}$ and $X(\mathbb{R}^n)$ be a ball Banach function space satisfying some extra mild assumptions. Assume that $\Omega=\mathbb{R}^n$ or $\Omega\subset\mathbb{R}^n$ is an $(\varepsilon,\infty)$-domain for…

In this work we investigate the Sobolev space $\mathrm{H}^{1}(\partial\Omega)$ on a strong Lipschitz boundary $\partial\Omega$, i.e., $\Omega$ is a strong Lipschitz domain. In most of the literature this space is defined via charts and…

Bourgain, Brezis & Mironescu showed that (with suitable scaling) the fractional Sobolev $s$-seminorm of a function $f\in W^{1,p}(\rn)$ converges to the Sobolev seminorm of $f$ as $s\to 1^-$. The anisotropic $s$-seminorms of $f$ defined by a…

For each $p>n$ we use local oscillations to give intrinsic characterizations of the trace of the Sobolev space $W^1_p(\Omega)$ to the boundary of an arbitrary domain $\Omega\subset R^n$.

We study the relation between Sobolev inequalities for differential forms on a Riemannian manifold $(M,g)$ and the $L_{q,p}$-cohomology of that manifold. The $L_{q,p}$-cohomology of $(M,g)$ is defined to be the quotient of the space of…

Consider a Lipschitz domain $\Omega$ and the Beurling transform of its characteristic function $\mathcal{B} \chi_\Omega(z)= - {\rm p.v.}\frac1{\pi z^2}*\chi_\Omega (z) $. It is shown that if the outward unit normal vector $N$ of the…

Using the convolution product and weak derivatives, we consider the partial dynamical systems of the locally convex $L^p(\Omega)$ spaces defined by the action of the smooth algebra $\mathscr{K}(\Omega)$ through its nets. Slice analysis is…

For each $m\ge 1$ and $p>2$ we characterize bounded simply connected Sobolev $L^m_p$-extension domains $\Omega\subset R^2$. Our criterion is expressed in terms of certain intrinsic subhyperbolic metrics in $\Omega$. Its proof is based on a…

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…

We consider local weak solutions to the widely degenerate parabolic PDE \[ \partial_{t}u-\mathrm{div}\left((\vert Du\vert-\lambda)_{+}^{p-1}\frac{Du}{\vert Du\vert}\right)=f\qquad\mathrm{in}\ \ \Omega_{T}=\Omega\times(0,T), \] where…

We describe a class of Sobolev $W^k_p$-extension domains $\Omega\subset R^n$ determined by a certain inner subhyperbolic metric in $\Omega$. This enables us to characterize finitely connected Sobolev $W^1_p$-extension domains in $R^2$ for…

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…

Recently, V. Cruz, J. Mateu and J. Orobitg have proved a T(1) theorem for the Beurling transform in the complex plane. It asserts that given $0<s\leq1$, $1<p<\infty$ with $sp>2$ and a Lipschitz domain $\Omega\subset \mathbb{C}$, the…

We characterize the trace of magnetic Sobolev spaces defined in a half-space or in a smooth bounded domain in which the magnetic field $A$ is differentiable and its exterior derivative corresponding to the magnetic field $dA$ is bounded. In…

Let $\Omega\subset\mathbb{C}^n$ be a strictly pseudoconvex Runge domain with $C^2$-smooth defining function, $l\in\mathbb{N},$ $p\in(1,\infty).$ We prove that the holomorphic function $f$ has derivatives of order $l$ in $H^p(\Omega)$ if and…