For
|
a
|
>
1
|a|>1
let
φ
a
\varphi _{a}
be the Möbius transformation defined by
φ
a
(
z
)
=
a
−
z
1
−
a
¯
z
\varphi _{a}(z)=\frac {a-z}{1-\bar az}
, and let
g
(
z
,
a
)
=
log
|
1
−
a
¯
z
z
−
a
|
g(z,a)=\log |\frac {1-\bar az}{z-a}|
be the Green’s function of the unit disk
D
\mathcal {D}
. We construct an analytic function
f
f
belonging to
M
p
#
=
{
f
:
M_{p}^{\#} = \{\, f : {}
f
f
meromorphic in
D
\mathcal {D}
and
sup
a
∈
D
∬
D
(
f
#
(
z
)
)
2
(
1
−
|
φ
a
(
z
)
|
2
)
p
d
A
(
z
)
>
∞
}
\sup _{a\in \mathcal {D}} \iint _{\mathcal {D}}(f^{\#}(z))^{2}(1-|\varphi _{a}(z)|^{2})^{p}\,dA(z)>\infty \, \}
for all
p
p
,
0
>
p
>
∞
0>p>\infty
, but not belonging to
Q
p
#
=
{
f
:
f
Q_{p}^{\#}=\{\,f:f
meromorphic in
D
\mathcal {D}
and
sup
a
∈
D
∬
D
(
f
#
(
z
)
)
2
(
g
(
z
,
a
)
)
p
d
A
(
z
)
>
∞
}
\sup _{a\in \mathcal {D}}\iint _{\mathcal {D}}(f^{\#}(z))^{2}(g(z,a))^{p}\,dA(z)>\infty \,\}
for any
p
p
,
0
>
p
>
∞
0>p>\infty
. This gives a clear difference as compared to the analytic case where the corresponding function spaces (
M
p
M_{p}
and
Q
p
Q_{p}
) are same.