Related papers: On the extension of VMO functions

Given a bounded domain $\Omega \subset {\mathbb R}^{n}$ with $n\ge2$, let $\phi $ is a Young function satisfying the doubling condition with the constant $K_\phi<2^{n}$. If $\Omega$ is a John domain, we show that $\Omega $ supports a…

Wang and Ye conjectured in [22]: Let $\Omega$ be a regular, bounded and convex domain in $\mathbb{R}^{2}$. There exists a finite constant $C({\Omega})>0$ such that \[ \int_{\Omega}e^{\frac{4\pi u^{2}}{H_{d}(u)}}dxdy\le C(\Omega),\;\;\forall…

In the setting of a metric space equipped with a doubling measure supporting a $(1,1)$-Poincar\'e inequality, we study the problem of minimizing the BV-energy in a bounded domain $\Omega$ of functions bounded between two obstacle functions…

We consider Green's function $ G_K $ of the elliptic operator in divergence form $ \mathcal{L}_K=-\text{div}(K(x)\nabla ) $ on a bounded smooth domain $ \Omega\subseteq\mathbb{R}^n (n\geq 2) $ with zero Dirichlet boundary condition, where $…

We show that for a uniformly elliptic divergence form operator $L$, defined in an open set $\Omega$ with Ahlfors-David regular boundary, BMO-solvability implies scale invariant quantitative absolute continuity (the weak-$A_\infty$ property)…

In this paper, we obtain estimates on the quantitative strata of the critical set of non-trivial harmonic functions $u$ which vanish continuously on $V \subset \partial \Omega$, a relatively open subset of the boundary of a convex domain…

Let $\Omega\subset \mathbb{C}$ be a bounded domain with $\mathcal{C}^{1,\omega}$-smooth boundary, where $\omega$ is a Dini-smooth modulus of continuity. We prove that the restricted Beurling transform is bounded on the Campanato space…

We study the Green function for the stationary Stokes system with bounded measurable coefficients in a bounded Lipschitz domain $\Omega\subset \mathbb{R}^n$, $n\ge 3$. We construct the Green function in $\Omega$ under the condition…

We show that if $\Omega\subset\mathbb R^3$ is a two-sided NTA domain with AD-regular boundary such that the logarithm of the Poisson kernel belongs to $\textrm{VMO}(\sigma)$, where $\sigma$ is the surface measure of $\Omega$, then the outer…

Let $\Omega \subset \mathbb{C}^n$ be a bounded domain and let $\mathcal{A} \subset \mathcal{C}(\bar{\Omega})$ be a uniform algebra generated by a set $F$ of holomorphic and pluriharmonic functions. Under natural assumptions on $\Omega$ and…

In this paper, we prove that the oscillation operator, variation operator and maximal differential transform associated with the approximate identities are bounded from ${\rm BMO}({\mathbb R}^n)$ to its subspace ${\rm BLO}({\mathbb R}^n)$.

We consider the zeta function $\zeta_\Omega$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $\Omega$ bounded by a smooth closed curve of perimeter $2\pi$. We prove that $\zeta_\Omega''(0)\ge…

We provide three different characterizations of the space $BV(O,\gamma)$ of the functions of bounded variation with respect to a centred non-degenerate Gaussian measure $ \gamma$ on open domains $O$ in Wiener spaces. Throughout these…

Let B be a p-uniformly convex Banach space, with p >= 2. Let T be a linear operator on B, and let A_n x denote the ergodic average (1 / n) sum_{i< n} T^n x. We prove the following variational inequality in the case where T is power bounded…

It is shown that if $p_n$ is a sequence of continuous, unbounded exponents on a bounded, smooth domain $\Omega\subset {\mathbb R}^n$ with $1<\inf\limits_{x\in \Omega}p_n(x)$ and $p_n\rightarrow \infty$ uniformly, then the sequence $(u_n)$…

A local behavior of closed open discrete mappings of Orlicz--Sobolev classes in ${\Bbb R}^n,$ $n\ge 3,$ is studied. It is proved that, mappings mentioned above have continuous extension to isolated boundary point $x_0$ of a domain…

We consider a domain $\Omega\subseteq\mathbb{R\!}^{\,2}$ with branched fractal boundary $\Gamma^{\infty}$ and parameter $\tau\in[1/2,\tau^{\ast}]$ introduced by Achdou and Tchou \cite{ACH08}, for $\tau^{\ast}\simeq 0.593465$, which acts as…

In this article, we investigate a density problem coming from the linearization of Calder\'on's problem with partial data. More precisely, we prove that the set of products of harmonic functions on a bounded smooth domain $\Omega$ vanishing…

It is shown that the extension of $\R$ by a generic smooth function restricted to the unit cube is o-minimal. The generalization to countably many generic smooth functions is indicated. Possible applications are sketched.

We present an alternative proof of a result of Kenig and Toro, which states that if $\Omega \subset \mathbb{R}^{n+1}$ is a two sided NTA domain, with Ahlfors-David regular boundary, and the $\log$ of the Poisson kernel associated to…