Abstract
We study the Banach space
B
H
α
(
α
>
0
) of the harmonic mappings
h
on the open unit disk
D
satisfying the condition
sup
z
∈
D
(
1
-
z
2
)
α
(
h
z
z
+
h
z
¯
z
)
<
∞
,
where
h
z
and
h
z
¯
denote the first complex partial derivatives of
h
. We show that several properties that are valid for the space of analytic functions known as the
α
-Bloch space
extend to
B
H
α
. In particular, we prove that for
α
>
0
the mappings in
B
H
α
can be characterized in terms of a Lipschitz condition relative to the metric defined by
d
H
,
α
(
z
,
w
)
=
sup
{
h
z
-
h
w
:
h
∈
B
H
α
,
h
B
H
α
≤
1
}
. When
α
>
1
, the harmonic
α
-Bloch space can be viewed as the harmonic growth space of order
α
-
1
, while for
0
<
α
<
1
,
B
H
α
is the space of harmonic mappings that are Lipschitz of order
1
-
α
.