The atomic decomposition of Besov-Bergman-Lipschitz spaces
Authors: De Souza Geraldo Soares 
Published: June 2010

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 )\;

<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="b left-parenthesis t right-parenthesis equals minus StartAbsoluteValue upper I EndAbsoluteValue Superscript negative 1 slash p Baseline script upper X upper R left-parenthesis t right-parenthesis plus StartAbsoluteValue upper I EndAbsoluteValue Superscript negative 1 slash p Baseline script upper X upper L left-parenthesis t right-parenthesis">

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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B Superscript p Baseline equals left-brace f colon left-bracket negative pi comma pi right-parenthesis right-arrow upper R semicolon f left-parenthesis t right-parenthesis equals sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts c Subscript n Baseline b Subscript n Baseline left-parenthesis t right-parenthesis comma sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts StartAbsoluteValue c Subscript n Baseline EndAbsoluteValue greater-than normal infinity right-brace">

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
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-vertical-bar f double-vertical-bar Subscript upper B Sub Superscript upper P Subscript Baseline equals Inf sigma-summation Underscript n equals 1 Overscript normal infinity Endscripts StartAbsoluteValue c Subscript n Baseline EndAbsoluteValue">

‖
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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Lamda left-parenthesis alpha comma r comma s right-parenthesis equals left-brace f colon left-bracket negative pi comma pi right-parenthesis right-arrow upper R comma double-vertical-bar f double-vertical-bar Subscript normal upper Lamda left-parenthesis alpha comma r comma s right-parenthesis Baseline equals double-vertical-bar f double-vertical-bar Subscript r Baseline plus left-parenthesis integral Subscript negative pi Superscript pi Baseline StartFraction left-parenthesis double-vertical-bar f left-parenthesis x plus t right-parenthesis minus f left-parenthesis x right-parenthesis double-vertical-bar Subscript r Baseline right-parenthesis Superscript s Baseline Over StartAbsoluteValue t EndAbsoluteValue Superscript 1 plus alpha s Baseline EndFraction d t right-parenthesis Superscript 1 slash s Baseline greater-than normal infinity right-brace">

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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper N double-vertical-bar f double-vertical-bar Subscript upper B Sub Superscript p Baseline less-than-or-equal-to double-vertical-bar f double-vertical-bar Subscript normal upper Lamda left-parenthesis 1 minus 1 slash p comma 1 comma 1 right-parenthesis Baseline less-than-or-equal-to upper M double-vertical-bar f double-vertical-bar Subscript upper B Sub Superscript p">

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}}}

\]
.