Trace and extension theorems for functions of bounded variation

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

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