*Accepted: 2016*

**Inserted:** 27 apr 2016

**Last Updated:** 28 nov 2016

**Journal:** Annali SNS.

**Year:** 2016

**Doi:** 10.2422/2036-2145.201511_007

**Abstract:**

In this paper we show that every $L^1$-integrable function on $\partial\Omega$ can be obtained as the trace of a function of bounded variation in $\Omega$ whenever $\Omega$ is a domain with regular boundary $\partial\Omega$ in a doubling metric measure space. In particular, the trace class of $BV(\Omega)$ is $L^1(\partial\Omega)$ provided that $\Omega$ supports a $1$-Poincar\'e inequality. We also construct a bounded linear extension from a Besov class of functions on $\partial\Omega$ to $BV(\Omega)$.

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