Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
22 (2006), 63–71
www.emis.de/journals
ISSN 1786-0091
HERZ-TYPE BESOV SPACES ON LOCALLY COMPACT
VILENKIN GROUPS
CANQIN TANG, QINGGUO LI AND BOLIN MA
Abstract. Let G be a locally compact Vilenkin group. In this paper the
characterizations of the Herz-type Besov space on G are obtained. And some
properties of this space are discussed.
1. Introduction
M. Frazier and B. Jawerth give the characterizations of Besov space on Rn in [3].
For additional results, see [4], the atomic decomposition of Besov spaces on locally
compact Vilenkin groups G is obtained by C.W. Onneweer and Su Weiyi. Xu
([7]) introduced the Herz-type Besov spaces on Rn and give an unified approach for
Herz-type Besov spaces and Triebel–Lizorkin spaces. We can also find the associate
results in [5], [2], [6]. These papers were the motivation for the present paper in
which we consider the characterizations of this Herz-type Space on locally compact
Vilenkin groups G.
Throughout this paper, G will denote a bounded locally compact Vilenkin group,
that is, G is a locally compact Abelian group containing a strictly decreasing se∞
quence of compact open subgroups {Gn }∞
n=−∞ such that (a) ∪n=−∞ Gn = G and
∞
∩n=−∞ Gn = 0; (b) sup{ order (Gn /Gn+1 ) : n ∈ Z} = B < ∞. We choose Haar
measure dx on G so that |G0 | = 1, where |A| denotes the measure of a measurable
subset A of G. Let |Gn | = (mn )−1 for each n ∈ Z. Since 2mn ≤ mn+1 ≤ Bmn for
each n ∈ Z, it follows that
∞
X
(mn )−α ≤ c(mk )−α
n=k
and
k
X
(mn )α ≤ c(mk )α
n=−∞
for any α > 0, k ∈ Z, where c is a constant independent of k. For each n ∈ Z
we choose elements zl,n ∈ G(l ∈ Z+ ) so that the subsets Gl,n := zl,n + Gn of G
satisfy Gk,n ∩ Gl,n = ∅ if k 6= l and ∪∞
l=0 Gl,n = G; moreover, we choose z0,n so that
G0,n = Gn . We now define the function d : G × G → R by d(x, y) = 0 if x − y = 0
and d(x, y) = (mn )−1 if x − y ∈ Gn \Gn+1 , then d defines a metric on G × G and
the topology on G introduced by this metric is the same as the original topology on
G. For x ∈ G, we set |x| = d(x, 0). Then |x| = (mn )−1 if and only if x ∈ Gn \Gn+1 .
2000 Mathematics Subject Classification. 43A70 43A75.
Key words and phrases. Herz-type Besov space, Vilenkin Group.
The third author was supported by NNSF 10371004.
63
64
CANQIN TANG, QINGGUO LI AND BOLIN MA
We now briefly recall the definitions of the spaces S(G) of test functions and
S 0 (G) of distributions; for more details, see [6]. A function φ : G → C belongs
to S(G) if there exist integers k, l, depending on φ, so that supp φ ⊂ Gk and φ is
constant on the cosets of the subgroup Gl of G. And the space of all continuous
functionals on S(G) will denoted by S 0 (G).
In this paper, the authors characterize the Herz-type homogeneous Besov spaces
K̇qα,p Ḃβs (G). In fact, when α = 0, q = p, K̇qα,p Ḃβs (G) = Ḃβs,p (G). then it turns to be
the case which was discussed in [4]. According to the definition of locally compact
Vilenkin groups and its topological structure, we can give the atomic decomposition
of this Herz-type space on Rn .
2. Herz-type Homogeneous and Nonhomogeneous Besov Spaces
We first recall the definitions of the homogeneous ( non-homogeneous) Herz
spaces and Besov spaces on G.
Definition 1. Let α ∈ R, 0 < p, q ≤ ∞.
(a) A measurable function f : G → R belongs to the homogeneous Herz space if
it satisfies
½ X
¾1/p
∞
−αp
p
kf kK̇qα,p (G) =
ml kf χGl \Gl+1 kLq (G)
< ∞,
l=−∞
a modification will be done if p = ∞ or q = ∞ ( That is, if p = ∞, kf kK̇qα,∞ (G) =
supl m−α
l kf χGl \Gl+1 kLq (G) ).
(b) A measurable function f : G → R belongs to the non-homogeneous Herz
space if it satisfies
½
¾1/p
−1
X
p
−αp
p
kf kKqα,p (G) = kf χG0 kLq (G) +
ml kf χGl \Gl+1 kLq (G)
< ∞,
l=−∞
a modification will be done if p = ∞ or q = ∞.
Before giving the definition of the Herz-type Besov spaces on G, we give the
notes of a second space of test functions and distributions( [4]). Let
Z
Z(G) = {ψ ∈ S(G) : ψ̂(0) =
ψ(t)dt = 0},
G
and define convergence in Z(G) like in S(G). Let Z 0 (G) be the space of linear functionals on Z(G) with convergence in Z 0 (G) defined like in S 0 (G). If % denotes the
set of constant distributions in S 0 (G) then Z 0 (G) can be identified with S 0 (G)/% in
the sense that (i) for each f ∈ S 0 (G) its restriction to Z(G) belongs to Z 0 (G); (ii)
if f, g ∈ S 0 (G) and if g = f + c for some constant c ∈ S 0 (G) then the restrictions
of f and g to Z(G) determine the same element of Z 0 (G); (iii) if f˜ ∈ Z 0 (G) then
there exists an f ∈ S 0 (G) so that its restriction to Z(G) equals f˜, moreover, module constants f is determined uniquely by f˜. We usually disregard the difference
between f˜ ∈ Z 0 (G) and a corresponding f ∈ S 0 (G).
Set χn (x) = χGn \Gn+1 (x), ∆n (x) = mn χn (x), ϕn (x) = ∆n (x) − ∆n+1 (x), and ∗
denote convolution operator. Now we give the definition of the Herz-type homogeneous Besov spaces on G.
Definition 2. Let α ∈ R, 0 < p < ∞, 0 < q ≤ ∞. The Besov space Ḃqα,p (G) is
defined as
∞
³ X
´1/q
Ḃqα,p (G) = {f ∈ Z 0 (G)|kf kḂqα,p (G) :=
(mn )αq kf ∗ ϕn kqp
< ∞},
n=−∞
HERZ-TYPE BESOV SPACES
65
with the usual modification if q = ∞.
Definition 3. Let α, s ∈ R, 0 < p, q, β ≤ ∞. Then
K̇qα,p Ḃβs (G) = {f ∈ Z 0 (G)|kf kK̇qα,p Ḃ s (G) := (
β
∞
X
((mn )s kf ∗ϕn kK̇qα,p )β )1/β < ∞},
n=−∞
with the usual modification if β = ∞ (That is, if β = ∞, kf kK̇qα,p Ḃ s (G) =
∞
supn (mn )s kf ∗ ϕn kK̇qα,p ).
Definition 4. Let α, s ∈ R, 0 < p, q, β ≤ ∞. Then
K̇qα,p Bβs (G) = {f ∈ Z 0 (G)|kf kK̇qα,p B s (G)
β
:= kf ∗ ∆0 kK̇qα,p + {
∞
X
((mn )s kf ∗ ϕn kK̇qα,p )β }1/β < ∞},
n=1
with the usual modification if β = ∞.
In this section, we first consider the link between the homogeneous and nonhomogeneous space. In [4, Theorem 3] it was shown that for the Besov space on G
s
s
we have Bp,β
= Lp ∩ Ḃp,β
when s > 0 and 1 ≤ p, β ≤ ∞. For Herz-type spaces, we
have similar results.
Theorem 1. Let s > 0 and 1 ≤ β, p, q < ∞, Then
K̇qα,p Bβs (G) = K̇qα,p Ḃβs (G) ∩ K̇qα,p (G).
P∞
kf χl kpq ≤ ∞. Since
Proof. If f ∈ K̇qα,p Ḃβs (G) ∩ K̇qα,p (G), then l=−∞ m−αp
l
∞
X
p
kf ∗ ∆0 kK̇
α,p =
q
and
m−αp
kf ∗ ∆0 χl kpq ,
l
l=−∞
Z
kf ∗
∆0 χl kpq
p
|f ∗ ∆0 (x)|q dx) q
=(
Gl \Gl+1
Z
≤(
G0
Z
(
1
|f (x − t)|q dx) q |∆0 (t)|dt)p
Gl \Gl+1
Z
≤ kf χl kpq (
|∆0 (t)|dt)p
G0
therefore, kf ∗ ∆0 kpK̇ α,p
q
≤ kf χl kpq ,
P∞
kf χl kpq = kf kpK̇ α,p , moreover,
≤ l=−∞ m−αp
l
q
kf kK̇qα,p B s (G) = kf ∗ ∆0 kK̇qα,p + {
∞
X
β
((mn )s kf ∗ ϕn kK̇qα,p )β )}1/β
n=1
≤ kf kK̇qα,p + {
∞
X
((mn )s kf ∗ ϕn kK̇qα,p )β )}1/β
n=−∞
= kf kK̇qα,p + kf kK̇qα,p Ḃ s
β
< ∞.
Conversely, take any f ∈ K̇qα,p Bβs , since f ∈ S 0 (G) we have
f = f ∗ ∆0 +
∞
X
n=1
f ∗ ϕn
66
CANQIN TANG, QINGGUO LI AND BOLIN MA
with convergence in S 0 (G). If 1 ≤ β < ∞, using the inequalities of Minkowski and
Hölder, we have
∞
∞
X
X
k
f ∗ ϕn kK̇qα,p ≤
(mn )−s (mn )s kf ∗ ϕn kK̇qα,p
n=1
n=1
∞
X
=(
1
0
(mn )−sβ ) β0 (
∞
X
1
(msn kf ∗ ϕn kK̇qα,p )β ) β
n=1
n=1
≤ ckf kK̇qα,p B s ,
β
0
here β is the conjugate index of β. Thus, we can conclude that f ∈ K̇qα,p .
Moreover, as the proof of the first part, through the simple calculation, we have
kf ∗ ϕn χl kpq ≤ ckf χl kpq , consequently,
kf kβK̇ α,p Ḃ s = c(
q
β
0
X
n=−∞
β
msβ
n kf kK̇ α,p +
q
∞
X
n=1
β
msβ
n kf ∗ ϕn kK̇ α,p )
q
≤ ckf kβK̇ α,p B s < ∞
q
β
This completes the proof of Theorem 1.
¤
In the following, A1 ⊂ A2 always means that the topological space A1 is continuously embedded in the topological space A2 .
Using the proposition 2.2.1 in [8] and the embedding properties of Herz space(see
[1]), Theorem 2 is easy to be proved. Here we omit the proof.
Theorem 2. Let −∞ < s < ∞, 0 < p, q < ∞ and α > − 1q .
(i) If 0 < β1 ≤ β2 ≤ ∞, then
Kqα,p Bβs1 (G) ⊂ Kqα,p Bβs2 (G) and K̇qα,p Bβs1 (G) ⊂ K̇qα,p Bβs2 (G).
(ii) If 0 < β1 , β2 ≤ ∞, ε > 0, then
Kqα,p Bβs+ε
(G) ⊂ Kqα,p Bβs2 (G) and K̇qα,p Bβs+ε
(G) ⊂ K̇qα,p Bβs2 (G).
1
1
(iii) If α1 < α2 , then Kqα2 ,p Bβs (G) ⊂ Kqα1 ,p Bβs (G).
(iv) If b ≤ c, then Kqα,b Bβs (G) ⊂ Kqα,c Bβs (G) and K̇qα,b Bβs (G) ⊂ K̇qα,c Bβs (G).
(v) If q1 ≤ q2 , then Kqα,p
Bβs (G) ⊂ Kqr,p
Bβs (G) and K̇qα,p
Bβs (G) ⊂ K̇qr,p
Bβs (G)
2
1
2
1
1
1
where r = α − ( q1 − q2 ).
Theorem 3. Let α =
1
p
− 1q , 0 < p ≤ q < ∞, 0 < β ≤ ∞, s ∈ R, then
s
K̇qα,p Bβs (G) ⊂ Bp,β
(G).
Proof. Let Dj = Gj \Gj+1 . Using the Hölder inequality, we have
∞ Z
X
p
|f (x)|p dx
kf kLp (G) =
j=−∞
≤C
Dj
∞
X
p
=C
p
q −1
=C
=
µZ
mj
j=−∞
∞
X
¶ pq
|f (x)|q dx
Dj
j=−∞
∞
X
µZ
|Dj |1− q
¶ pq
|f (x)| dx
q
Dj
µZ
m−αp
j
j=−∞
Ckf kpK̇ α,p (G) .
q
¶ pq
|f (x)| dx
q
Dj
HERZ-TYPE BESOV SPACES
67
s
Therefore, K̇qα,p Bβs (G) ⊂ Bp,β
(G).
¤
Theorem 4. Let 0 < p < ∞, 0 < q < ∞, 0 < r ≤ q, 0 < β ≤ ∞, s ∈ R,
and 0 < r < p < ∞, α > 1r − 1q , or 0 < p ≤ r < ∞ and α ≥ 1r − 1q . Then
s
Kqα,p Bβs (G) ⊂ Br,β
(G).
Proof. Suppose D0 = G0 , Dj = Gj \Gj+1 . By the Hölder inequality, we can obtain
Z
0
X
r
kf kLr (G) =
|f (x)|p dx
Dj
j=−∞
µZ
0
X
≤C
|Dj |
1− rq
|f (x)| dx
Dj
j=−∞
0
X
=C
¶ rq
q
r
q −1
µZ
q
mj
¶ rq
|f (x)| dx
Dj
j=−∞
1 1
− , then
r
q
( 0
µZ
X
r
m−αp
kf kLr (G) ≤ C
j
If 0 < r < p < ∞, α >
½ X
0
¶ pq )r/p
|f (x)| dx
Dj
j=−∞
×
q
( 1 − r1 +α)rp/(p−r)
¾1−r/p
mj q
j=−∞
≤C
≤
½ X
0
µZ
m−αp
j
j=−∞
Ckf krKqα,p (G) .
¶ pq ¾r/p
|f (x)|q dx
Dj
We can deduce that mj ≤ m0 = 1 since j ≤ 0, then if 0 < p ≤ r < ∞, α ≥
we have
( 0
µZ
¶ pq )r/p
X
( q1 − r1 )p
r
q
mj
kf kLr (G) ≤ C
|f (x)| dx
(
≤C
≤
− 1q ,
Dj
j=−∞
0
X
1
r
¶ pq )r/p
|f (x)| dx
µZ
m−αp
j
j=−∞
Ckf krKqα,p (G) .
q
Dj
s
Furthermore, Kqα,p Bβs (G) ⊂ Br,β
(G).
¤
3. Atomic Decomposition of the Herz-type Homogeneous Spaces
The atomic decomposition of the homogeneous Besov space on G were obtained
by Onneweer and Su [4, Theorem 6]. Motivated by their work, we will give the
atomic decomposition of the Herz-type Homogeneous Besov space.
Definition 5. A function a : G → C is an (s, ∞) atom, s ∈ R, if
(i) a is supported on a set z + Gn for some z ∈ G and n ∈ Z,
(ii) |a(x)|
≤ (mn )s ,
R
(iii) G a(x)dx = 0.
We have the following result.
68
CANQIN TANG, QINGGUO LI AND BOLIN MA
Theorem 5. Let 0 < p, β ≤ ∞, α > − 1q , s ∈ R. Then the following two facts are
equivalent.
(a) f ∈ K̇qα,p Ḃβs (G),
(b) there exist constants λl,j , l ∈ Z+ and j ∈ Z, and (−(s − α) + 1/q, ∞) atoms
ai,j with supp al,j ⊂ zl,j + Gj−1 such that
∞ X
∞
X
f=
λl,j al,j
in
S 0 (G)/%
j=−∞ l=0
Moreover,
kλkp,β
∞
∞
X
X
β
:= (
(
|λl,j |p ) p )1/β ≤ ckf kK̇qα,p Ḃ s .
β
j=−∞ l=0
Proof. (a) ⇒ (b) For each f ∈ S 0 /% we have
∞
X
f=
=
=
=
n=−∞
∞
X
f ∗ ϕn (x)
f ∗ ϕn ∗ ϕn (x)
n=−∞
∞ X
∞
X
n=−∞ l=0
∞ X
∞
X
(mn )−1 f ∗ ϕn (zl,n )ϕn (x − zl,n )
λl,n al,n
n=−∞ l=0
where λl,n = (mn−1 )(s−α)−1/q f ∗ ϕn (zl,n ) and
al,n = (mn−1 )−(s−α)+1/q m−1
n ϕn (x − zl,n ).
For each al,n we have
(i) supp al,n ⊂ zl,n + Gn−1 ;
−(s−α)+1/q
(ii) |a
;
R l,n (x)| ≤ (mn−1 )
(iii) G al,n (x)dx = 0.
Thus al,n is an (−(s − α) + 1/q, ∞) atom on G with support in zl,n + Gn−1 .
Moreover, for the λl,n we have
kλkp,β = (
≤
∞
∞
X
X
β 1
(
|(mn−1 )(s−α)−1/q f ∗ ϕn (zl,n )|p ) p ) β
n=−∞ l=0
∞
∞
X
X
m−αp
kf
msβ
(
c(
n
k
n=−∞
k=−∞
β
1
∗ ϕn χk kpq ) p ) β
= ckf kK̇qα,p Ḃ s < ∞.
β
This completes the proof of (a).
(b) ⇒ (a) Let al,j be an (−(s − α) + 1/q, ∞) atom on G with support in zl,j + Gj .
Similar to the proof in [4], we have al,j ∗ ϕn = 0 when j ≥ n for each x ∈ G. So we
only consider the case of j < n.
If k + 1 ≤ j, then k(al,j ∗ ϕn )χk kq = 0 since supp al,j ∗ ϕn ⊂ Gj . If k > j, we
−(s−α)+ 1
−1
q
mk q .
have k(al,j ∗ ϕn )χk kq ≤ cmj
In the following, we estimate kf kK̇qα,p Ḃ s .
β
kλkp,β < ∞,
If f =
P∞
j=−∞
P∞
l=0
λl,j al,j and
HERZ-TYPE BESOV SPACES
69
(i) If 0 < p ≤ 1, then for each n ∈ Z,
n−1
X
kf ∗ ϕn kpK̇ α,p ≤
q
∞
X
j=−∞ l=0
|λl,j |p kal,j ∗ ϕn kpK̇ α,p
q
Therefore, for each β with 0 < β < ∞ we have
∞
X
kf kK̇qα,p Ḃ s = {
β
n=−∞
∞
X
={
∞
X
j=−∞ l=0
n−1
X
msβ
n (
n=−∞
∞
X
β
n−1
X
msβ
n (
n=−∞
1
|λl,j |p kal,j ∗ ϕn kpK̇ α,p ) p } β
q
∞
X
|λl,j |p
j=−∞ l=0
∞
X
≤ c{
n−1
X
msβ
n (
β
k=j
∞
X
β
1
|λl,j |p m−sp
)p }β
j
j=−∞ l=0
since α + 1/q > 0.
If 0 < β ≤ p, so that 0 < β/p ≤ 1, then
kf kK̇qα,p Ḃ s ≤ c{
β
∞
X
n−1
X
msβ
n
n=−∞
m−sβ
(
j
j=−∞
∞
X
β
1
|λl,j |p ) p } β
l=0
≤ ckλkp,β < ∞
here s < 0.
If p < β < ∞, let r =
Using the Hölder inequality, we can obtain
∞
X
kf kK̇qα,p Ḃ s ≤ c{
β
β
p.
msβ
n (
n=−∞
≤ c{
∞
X
n=−∞
≤ c{
∞
X
β
1
|λl,j |p m−sp
)p }β
j
j=−∞ l=0
msβ
n (
∞
X
n−1
X
n−1
X
j=−∞
n=−∞
∞
X
β
1
m−sp
(
|λl,j |p ) p } β
j
j=−∞
n−1
X
msp
n−1
n−1
X
r
m−sp
) r0
j
m−sp
(
j
j=−∞
∞
X
l=0
β
1
|λl,j |p ) p } β
l=0
≤ ckλkp,β < ∞.
Here r0 is the conjugate index of r.
If β = ∞, since s < 0 and
kf kK̇qα,p Ḃ s = sup msn kf ∗ ϕn kK̇qα,p
β
n
≤ sup msn (
n
n−1
X
∞
X
1
|λl,j |p m−sp
)p
j
j=−∞ l=0
moreover,
n−1
X
j=−∞
m−sp
j
∞
X
∞
n−1
X
X
|λl,j |p ≤ sup(
|λl,j |p )
m−sp
j
j
l=0
j=−∞
l=0
≤ cm−sp
sup(
n
j
∞
X
|λl,j |p ).
l=0
Hence,
kf kK̇qα,p Ḃ s ≤ c sup msn m−s
n sup(
β
n
j
∞
X
l=0
1
m−αp
kal,j ∗ ϕn χk kpq ) p } β
k
|λl,j |p )1/p = ckλkp,β < ∞.
70
CANQIN TANG, QINGGUO LI AND BOLIN MA
(ii) If 1 < p < ∞, similar to the proof of Theorem 6 in [4], for each n ∈ Z, x ∈ G
and τ with 0 < τ < 1, we have
kf ∗ ϕn kK̇qα,p ≤ ck
n−1
X
∞
X
[−(s−α)+ q1 ](1−τ )
( (|λl,j ||al,j ∗ ϕn (x)|τ )p )1/p mj
kK̇qα,p
j=−∞ l=0
≤c
n−1
X
[−(s−α)+ q1 ](1−τ )
mj
∞
X
k( (|λl,j ||al,j ∗ ϕn (x)|τ )p )1/p kK̇qα,p
j=−∞
≤c
n−1
X
l=0
[−(s−α)+ q1 ](1−τ )
mj
(
j=−∞
≤c
n−1
X
j=−∞
∞
X
[−(s−α)+ q1 ]τ p
m−αp
mj
k
−p
1
mk q ) p (
k=j
∞
X
m−s
j (
∞
X
1
|λl,j |p ) p
l=0
1
|λl,j |p ) p .
l=0
Consequently,
kf kK̇qα,p Ḃ s ≤ c(
β
∞
X
msβ
n
n−1
X
∞
X
β 1
m−sβ
(
|λl,j |p ) p ) β
j
n=−∞
≤
j=−∞
l=0
∞
∞
X
X
β
1
p p
β
)
msβ
c(
m−sβ
(
|λ
|
)
l,j
n )
j
n=j+1
j=−∞
l=0
∞
X
≤ ckλkp,β .
(iii) If p = ∞, the proof is simpler. Here we omit it.
¤
References
[1] A. Baernstein and E. Sawyer. Embedding and multiplier theorems for H p (Rn ). Mem. Am.
Math. Soc., 59(318), 1985.
[2] O. Besov. On embedding and extension theorems for some function classes. Tr. Mat. Inst.
Steklova, 60:42–81, 1961.
[3] M. Frazier and B. Jawerth. Decomposition of Besov spaces. Indiana Univ. Math. J., 34:777–
799, 1985.
[4] C. Onneweer and W. Su. Homogeneous Besov spaces on locally compact Vilenkin groups. Stud.
Math., 93(1):17–39, 1989.
[5] B. H. Qui. On Besov, Hardy and Triebel spaces for 0 < p ≤ 1. Ark. Mat., 21:169–184, 1983.
[6] M. Taibleson. Fourier analysis on local fields. Mathematical Notes. Vol. 15. Princeton, N. J.:
Princeton University Press and University of Tokyo Press., 1975.
[7] J. Xu. Equivalent norms of Herz-type Besov and Triebel-Lizorkin spaces. J. Funct. Spaces
Appl., 3(1):17–31, 2005.
s (K ) and F s (K ) spaces. Approximation
[8] G. Zhou and W. Su. Elementary aspects of Bp,q
n
n
p,q
Theory Appl., 8(2):11–28, 1992.
Received October 11, 2005.
Canqin Tang
Department of Mathematics,
Dalian Maritime University,
Dalian
E-mail address: tangcq2000@yahoo.com.cn
Qingguo Li
Department of Mathematics,
Hunan University,
Changsa
E-mail address: liqingguoli@yahoo.com.cn
HERZ-TYPE BESOV SPACES
Bolin Ma
Department of Mathematics,
Hunan University,
Changsa
E-mail address: blma@hnu.net.cn
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