Abstract
We show that, given a homeomorphism
f
:
G
→
Ω
{f:G\rightarrow\Omega}
where G is an open subset of
ℝ
2
{\mathbb{R}^{2}}
and Ω is an open subset of a 2-Ahlfors regular metric measure space supporting a weak
(
1
,
1
)
{(1,1)}
-Poincaré inequality, it holds
f
∈
BV
loc
(
G
,
Ω
)
{f\in{\operatorname{BV_{\mathrm{loc}}}}(G,\Omega)}
if and only if
f
-
1
∈
BV
loc
(
Ω
,
G
)
{f^{-1}\in{\operatorname{BV_{\mathrm{loc}}}}(\Omega,G)}
.
Further, if f satisfies the Luzin N and N
-
1
{{}^{-1}}
conditions, then
f
∈
W
loc
1
,
1
(
G
,
Ω
)
{f\in\operatorname{W_{\mathrm{loc}}^{1,1}}(G,\Omega)}
if and only if
f
-
1
∈
W
loc
1
,
1
(
Ω
,
G
)
{f^{-1}\in\operatorname{W_{\mathrm{loc}}^{1,1}}(\Omega,G)}
.