We consider rough tubes
X
+
i
R
m
⊂
C
m
X+i\mathbb {R}^m\subset \mathbb {C}^m
, where
X
⊂
R
m
X\subset \mathbb {R}^m
is a measurable set, and extend the notion of
C
R
CR
function to the space
L
∞
(
X
,
h
p
(
R
m
)
)
L^\infty (X,h^p(\mathbb {R}^m))
, where
h
p
(
R
m
)
h^p(\mathbb {R}^m)
,
0
>
p
>
∞
0>p>\infty
, is Goldberg’s semilocal Hardy space. We show that if
X
X
is the image of some connected manifold by some
C
1
C^1
map, then all such
C
R
CR
functions can be extended to the convex hull of the tube as
C
R
CR
functions
∈
L
∞
(
c
h
(
X
)
,
h
p
(
R
m
)
)
\in L^\infty (\mathrm {ch}(X),h^p(\mathbb {R}^m))
. This extends previous work of Boggess.