Suppose
f
f
is a holomorphic function on the open unit ball
B
n
{B_n}
of
C
n
{{\mathbf {C}}^n}
. For
1
⩽
p
>
∞
1 \leqslant p > \infty
and
m
>
0
m > 0
an integer, we show that
f
f
is in
L
p
(
B
n
,
d
V
)
{L^p}({B_n},\,dV)
(with
d
V
dV
the volume measure) iff all the functions
∂
m
f
/
∂
z
α
(
|
α
|
=
m
)
{\partial ^m}f/\partial {z^{\alpha \,}}\;(|\alpha |\, = m)
are in
L
p
(
B
n
,
d
V
)
{L^p}({B_n},\,dV)
. We also prove that
f
f
is in the Bloch space of
B
n
{B_n}
iff all the functions
∂
m
f
/
∂
z
α
(
|
α
|
=
m
)
{\partial ^m}f/\partial {z^\alpha }\;(|\alpha |\, = m)
are bounded on
B
n
{B_n}
. The corresponding result for the little Bloch space of
B
n
{B_n}
is established as well. We will solve Gleason’s problem for the Bergman spaces and the Bloch space of
B
n
{B_n}
before proving the results stated above. The approach here is functional analytic. We make extensive use of the reproducing kernels of
B
n
{B_n}
. The corresponding results for the polydisc in
C
n
{{\mathbf {C}}^n}
are indicated without detailed proof.