Let
b
b
denote a special atom,
b
:
[
−
π
,
π
)
→
R
,
b
(
t
)
=
1
/
2
π
b:[ - \pi ,\pi ) \to R,\;b(t) = 1/2\pi
or, for any interval
I
in
[
−
π
,
π
)
I{\text { in }}[ - \pi ,\pi )\;
b
(
t
)
=
−
|
I
|
−
1
/
p
X
R
(
t
)
+
|
I
|
−
1
/
p
X
L
(
t
)
b(t) = - {\left | I \right |^{ - 1/p}}\mathcal {X}R(t) + {\left | I \right |^{ - 1/p}}\mathcal {X}L(t)
L
L
is the left half of
I
I
,
R
R
is the right half,
|
I
|
\left | I \right |
denotes the length of
I
I
and
X
E
\mathcal {X}E
the characteristic function of
E
E
. For
1
/
2
>
p
>
∞
1/2 > p > \infty
, let
(
b
n
)
({b_n})
be special atoms and
(
c
n
)
({c_n})
a sequence of real numbers; then we define the space
\[
B
p
=
{
f
:
[
−
π
,
π
)
→
R
;
f
(
t
)
=
∑
n
=
1
∞
c
n
b
n
(
t
)
,
∑
n
=
1
∞
|
c
n
|
>
∞
}
{B^p} = \left \{ {f:[ - \pi ,\pi ) \to R;f(t) = \sum \limits _{n = 1}^\infty {{c_n}{b_n}(t),} \sum \limits _{n = 1}^\infty {\left | {{c_n}} \right | > \infty } } \right \}
\]
. We endow
B
p
{B^p}
with the norm
‖
f
‖
B
P
=
Inf
∑
n
=
1
∞
|
c
n
|
{\left \| f \right \|_{{B^P}}} = {\text {Inf}}\sum \nolimits _{n = 1}^\infty {\left | {{c_n}} \right |}
, where the infimum is taken over all possible representations of
f
f
. In the early 1960s, the following spaces were introduced, now known as Besov-Bergman-Lipschitz spaces. For
0
>
α
>
1
0 > \alpha > 1
,
1
≤
r
1 \leq r
,
s
≤
∞
s \leq \infty
, let
\[
Λ
(
α
,
r
,
s
)
=
{
f
:
[
−
π
,
π
)
→
R
,
‖
f
‖
Λ
(
α
,
r
,
s
)
=
‖
f
‖
r
+
(
∫
−
π
π
(
‖
f
(
x
+
t
)
−
f
(
x
)
‖
r
)
s
|
t
|
1
+
α
s
d
t
)
1
/
s
>
∞
}
\Lambda (\alpha ,r,s) = \left \{ {f:[ - \pi ,\pi ) \to R,{{\left \| f \right \|}_{\Lambda (\alpha ,r,s)}} = {{\left \| f \right \|}_r} + {{\left ( {\int _{ - \pi }^\pi {\frac {{{{({{\left \| {f(x + t) - f(x)} \right \|}_r})}^s}}}{{{{\left | t \right |}^{1 + \alpha s}}}}dt} } \right )}^{1/s}} > \infty } \right \}
\]
where
|
|
|
|
r
||\;|{|_r}
is the Lebesgue space
L
r
{L^r}
-norm. Now we write down the main theorem of this paper which is as follows. THEOREM
f
∈
B
P
f \in {B^P}
for
1
>
p
>
∞
1 > p > \infty
if and only if
f
∈
Λ
(
1
−
1
/
p
,
1
,
1
)
f \in \Lambda (1 - 1/p,1,1)
. Moreover, there are absolute constants
M
M
and
N
N
such that
\[
N
‖
f
‖
B
p
≤
‖
f
‖
Λ
(
1
−
1
/
p
,
1
,
1
)
≤
M
‖
f
‖
B
p
N{\left \| f \right \|_{{B^p}}} \leq {\left \| f \right \|_{\Lambda (1 - 1/p,1,1)}} \leq M{\left \| f \right \|_{{B^p}}}
\]
.