Let
B
B
be the open unit ball of
C
n
{C^n}
, and set
S
=
∂
B
S = \partial B
. It is shown that if
φ
∈
L
p
(
S
)
,
φ
>
0
\varphi \in {L^p}\left ( S \right ),\varphi > 0
, is a lower semicontinuous function on
S
S
and
1
/
q
>
1
+
1
/
p
1/q > 1 + 1/p
, then, for a given
ε
>
0
\varepsilon > 0
, there exists a function
f
∈
H
p
(
B
)
f \in {H^p}\left ( B \right )
with
f
(
0
)
=
0
f\left ( 0 \right ) = 0
such that
|
f
∗
|
=
φ
\left | {{f^ * }} \right | = \varphi
almost everywhere on
S
S
and
∫
B
|
∇
f
|
q
d
V
>
ε
\int _B {{{\left | {\nabla f} \right |}^q}dV > \varepsilon }
where
V
V
denotes the normalized volume measure on
B
B
.