Using the Caffarelli–Silvestre extension, we show for a general open set
Ω
⊂
R
n
\Omega \subset \mathbf {R}^n
that a boundary point
x
0
x_0
is regular for the fractional Laplace equation
(
−
Δ
)
s
u
=
0
(-\Delta )^su=0
,
0
>
s
>
1
0>s>1
, if and only if
(
x
0
,
0
)
(x_0,0)
is regular for the extended weighted equation in a subset of
R
n
+
1
\mathbf {R}^{n+1}
. As a consequence, we characterize regular boundary points for
(
−
Δ
)
s
u
=
0
(-\Delta )^su=0
by a Wiener criterion involving a Besov capacity. A decay estimate for the solutions near regular boundary points and the Kellogg property are also obtained.