*Published Paper*

**Inserted:** 16 nov 2012

**Last Updated:** 16 nov 2012

**Journal:** in "Around the research of Vladimir Maz'ya. {I}"

**Volume:** 11

**Year:** 2010

**Doi:** 10.1007/978-1-4419-1341-8_11}

**Abstract:**

A domain $\Omega\subset R^2$ is called a $BV$-extension domain if there is a constant $c$ and an extension operator $T:BV(\Omega)\to BV(R^2)$, not necessarily linear, so that $Tu_{

\Omega}=u$ and $\

Tu\

_{BV(R^2)}\leq c \

u\

_{BV(\Omega)}$. The main result of the paper is the following:

Let $\Omega\subset R^2$ be a bounded, simply connected domain. Then $\Omega$ is a $BV$-extension domain if and only if the complement of $\Omega$ is a quasidisk, i.e. for every $x,y\in R^2 \setminus \Omega$ there is a rectifiable curve $\gamma\subset R^2\setminus \Omega$ connecting $x$ and $y$ with $length(γ)\leq C

x−y

$.