When I read a parper, there are two lemma in real domain. the lemma
Definition: Let $\Omega $ be an open subset of $R^2$. The space of functions of bounded variation $BV (\Omega ) $is defined as the space of real-valued function $u \in L^1(\Omega ) $ such that the total variation $$\int _\Omega | D_u| := \sup \Bigl\{ \int _\Omega u \operatorname{div}\varphi \ d x : \varphi \in C^1 _c (\Omega ,R^2), \| \varphi \|_{ L^\infty (\Omega ) }\leq 1 \Bigr\} $$ is finite. Then $BV (\Omega )$ is a Banach space with the norm $\| u\|_{ BV (\Omega ) }:= \| u\|_{ L^1(\Omega )} + \int _\Omega | Du|$ .
Lemma 1: Suppose sequence { $u_{n}$} $ \subset BV (\Omega ) $satisfies sup $\| u_n\|_{ BV (\Omega )} < \infty $. Then there exist a subsequence$ \{ u_{n_k}\} $and a function $u \in BV (\Omega )$ such that $u_{n_k} \rightarrow u $ in$\ L^1(\Omega )$ as$\ k \rightarrow \infty$ .
Lemma 2: Assume $\{ u_n\} \subset BV (\Omega ) $ and $ u_n \rightarrow u $in $L^1(\Omega )$. Then $\int _\Omega | D_u| \leq \lim\limits_{n \rightarrow \infty } \inf\int _\Omega | D_{u_n}| $.
I want to know if there are similar results in quaternion domain. Also, some relevant references are needed.