In this paper the characterizations of the Herz-type Besov space onGare obtained

Acta Mathematica Academiae Paedagogicae Ny´ıregyh´aziensis 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. LetG be a locally compact Vilenkin group. In this paper the characterizations of the Herz-type Besov space onGare obtained. And some properties of this space are discussed.

1. Introduction

M. Frazier and B. Jawerth give the characterizations of Besov space onRⁿin [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 onRⁿand 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,Gwill denote a bounded locally compact Vilenkin group, that is, Gis a locally compact Abelian group containing a strictly decreasing se- quence of compact open subgroups {Gn}^∞_n=−∞ such that (a) ∪^∞_n=−∞Gn =Gand

∩^∞_n=−∞Gn = 0; (b) sup{ order (Gn/Gn+1) :n ∈ Z} =B < ∞. We choose Haar measure dxonGso that|G0|= 1, where|A|denotes the measure of a measurable subsetA ofG. Let|Gn|= (mn)⁻¹for eachn∈Z. Since 2mn≤mn+1≤Bmn for eachn∈Z, it follows that

X∞

n=k

(mn)^−α≤c(mk)^−α and

Xk n=−∞

(mn)^α≤c(mk)^α

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 satisfyGk,n∩Gl,n=∅ifk6=land∪^∞_l=0Gl,n=G; moreover, we choosez0,nso that G0,n=Gn. We now define the functiond:G×G→R byd(x, y) = 0 ifx−y= 0 and d(x, y) = (mn)⁻¹ ifx−y ∈Gn\Gn+1, thenddefines a metric on G×Gand the topology onGintroduced by this metric is the same as the original topology on G. Forx∈G, we set|x|=d(x,0). Then|x|= (mn)⁻¹if and only ifx∈Gn\Gn+1.

2000Mathematics Subject Classification. 43A70 43A75.

Key words and phrases. Herz-type Besov space, Vilenkin Group.

The third author was supported by NNSF 10371004.

63

We now briefly recall the definitions of the spaces S(G) of test functions and S⁰(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 suppφ⊂Gk and φis constant on the cosets of the subgroup Gl of G. And the space of all continuous functionals onS(G) will denoted byS⁰(G).

In this paper, the authors characterize the Herz-type homogeneous Besov spaces K˙_q^α,pB˙_β^s(G). In fact, whenα= 0,q=p, ˙K_q^α,pB˙^s_β(G) = ˙B_β^s,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 onRⁿ.

2. Herz-type Homogeneous and Nonhomogeneous Besov Spaces We first recall the definitions of the homogeneous ( non-homogeneous) Herz spaces and Besov spaces onG.

Definition 1. Letα∈R, 0< p, q≤ ∞.

(a) A measurable functionf :G→Rbelongs to the homogeneous Herz space if it satisfies

kfkK˙_q^α,p(G)=

½ X∞ l=−∞

m^−αp_l kf χ_Gl\Gl+1k^p_Lq(G)

¾_1/p

<∞,

a modification will be done ifp=∞orq=∞( That is, ifp=∞,kfkK˙q^α,∞(G)= sup_lm^−α_l kf χGl\Gl+1kL^q(G)).

(b) A measurable function f : G → R belongs to the non-homogeneous Herz space if it satisfies

kfk_K^α,p_{q (G)}=

½

kf χG0k^p_Lq(G)+ X−1

l=−∞

m^−αp_l kf χ_Gl\Gl+1k^p_Lq(G)

¾_1/p

<∞,

a modification will be done if p=∞orq=∞.

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(G) ={ψ∈S(G) : ˆψ(0) = Z

G

ψ(t)dt= 0},

and define convergence inZ(G) like inS(G). LetZ⁰(G) be the space of linear func- tionals on Z(G) with convergence inZ⁰(G) defined like inS⁰(G). If%denotes the set of constant distributions inS⁰(G) thenZ⁰(G) can be identified withS⁰(G)/%in the sense that (i) for each f ∈S⁰(G) its restriction to Z(G) belongs toZ⁰(G); (ii) if f, g∈S⁰(G) and ifg =f +c for some constant c∈S⁰(G) then the restrictions of f andg toZ(G) determine the same element of Z⁰(G); (iii) if ˜f ∈Z⁰(G) then there exists an f ∈S⁰(G) so that its restriction toZ(G) equals ˜f, moreover, mod- ule constants f is determined uniquely by ˜f. We usually disregard the difference between ˜f ∈Z⁰(G) and a corresponding f ∈S⁰(G).

Setχn(x) =χ_Gn\Gn+1(x), ∆n(x) =mnχn(x),ϕn(x) = ∆n(x)−∆n+1(x), and∗ denote convolution operator. Now we give the definition of the Herz-type homoge- neous Besov spaces on G.

Definition 2. Letα∈ R, 0 < p < ∞, 0 < q≤ ∞. The Besov space ˙B_q^α,p(G) is defined as

B˙_q^α,p(G) ={f ∈Z⁰(G)|kfkB˙^α,pq (G):=

³ X^∞

n=−∞

(mn)^αqkf∗ϕnk^q_p

´_1/q

<∞},

with the usual modification ifq=∞.

Definition 3. Letα, s∈R, 0< p, q, β ≤ ∞. Then K˙_q^α,pB˙_β^s(G) ={f ∈Z⁰(G)|kfkK˙^α,p_q B˙^s_β(G):= (

X∞ n=−∞

((mn)^skf∗ϕnkK˙_q^α,p)^β)^1/β<∞},

with the usual modification if β = ∞ (That is, if β = ∞, kfkK˙q^α,pB˙^s_∞(G) = sup_n(mn)^skf∗ϕnkK˙q^α,p).

Definition 4. Letα, s∈R, 0< p, q, β ≤ ∞. Then K˙_q^α,pB_β^s(G) ={f ∈Z⁰(G)|kfkK˙q^α,pB_β^s(G)

:=kf∗∆0kK˙q^α,p+{ X∞ n=1

((mn)^skf∗ϕnkK˙q^α,p)^β}^1/β <∞}, with the usual modification ifβ=∞.

In this section, we first consider the link between the homogeneous and nonho- mogeneous space. In [4, Theorem 3] it was shown that for the Besov space on G we have B_p,β^s =L^p∩B˙_p,β^s whens >0 and 1≤p, β≤ ∞. For Herz-type spaces, we have similar results.

Theorem 1. Let s >0 and1≤β, p, q <∞, Then

K˙_q^α,pB_β^s(G) = ˙K_q^α,pB˙^s_β(G)∩K˙_q^α,p(G).

Proof. Iff ∈K˙_q^α,pB˙^s_β(G)∩K˙_q^α,p(G), thenP_∞

l=−∞m^−αp_l kf χlk^p_q ≤ ∞. Since kf ∗∆0k^p_K˙α,p

q =

X∞ l=−∞

m^−αp_l kf ∗∆0χlk^p_q, and

kf ∗∆0χlk^p_q = ( Z

Gl\Gl+1

|f ∗∆0(x)|^qdx)^pq

≤( Z

G0

( Z

Gl\Gl+1

|f(x−t)|^qdx)^1q|∆0(t)|dt)^p

≤ kf χlk^p_q( Z

G0

|∆0(t)|dt)^p

≤ kf χlk^p_q, therefore,kf∗∆0k^p_K˙α,p

q ≤P_∞

l=−∞m^−αp_l kf χlk^p_q =kfk^p_K˙α,p

q , moreover, kfkK˙_q^α,pB^s_β(G)=kf ∗∆0kK˙_q^α,p+{

X∞ n=1

((mn)^skf∗ϕnkK˙_q^α,p)^β)}^1/β

≤ kfkK˙q^α,p+{ X∞ n=−∞

((mn)^skf∗ϕnkK˙q^α,p)^β)}^1/β

=kfkK˙q^α,p+kfkK˙q^α,pB˙_β^s

<∞.

Conversely, take any f ∈K˙_q^α,pB_β^s, sincef ∈S⁰(G) we have

f =f∗∆0+ X∞ n=1

f∗ϕn

with convergence in S⁰(G). If 1≤β <∞, using the inequalities of Minkowski and H¨older, we have

k X∞ n=1

f∗ϕnkK˙q^α,p≤ X∞ n=1

(mn)^−s(mn)^skf∗ϕnkK˙q^α,p

= ( X∞ n=1

(mn)^−sβ0)^β10( X∞ n=1

(m^s_nkf∗ϕnkK˙_q^α,p)^β)^1β

≤ckfkK˙q^α,pB^s_β,

here β⁰ is the conjugate index ofβ. Thus, we can conclude thatf ∈K˙_q^α,p.

Moreover, as the proof of the first part, through the simple calculation, we have kf∗ϕnχlk^p_q ≤ckf χlk^p_q, consequently,

kfk^β_K˙α,p q B˙_β^s =c(

X0 n=−∞

m^sβ_n kfk^β_K˙α,p

q +

X∞ n=1

m^sβ_n kf∗ϕnk^β_K˙α,p q )

≤ckfk^β_K˙α,p

q B_β^s <∞

This completes the proof of Theorem 1. ¤

In the following,A₁⊂A₂ always means that the topological spaceA₁is contin- uously embedded in the topological spaceA2.

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 <∞andα >−¹_q. (i) If 0< β1≤β2≤ ∞, then

K_q^α,pB_β^s₁(G)⊂K_q^α,pB_β^s₂(G) andK˙_q^α,pB_β^s₁(G)⊂K˙_q^α,pB_β^s₂(G).

(ii) If 0< β1, β2≤ ∞,ε >0, then

K_q^α,pB_β^s+ε₁ (G)⊂K_q^α,pB_β^s₂(G) andK˙_q^α,pB_β^s+ε₁ (G)⊂K˙_q^α,pB_β^s₂(G).

(iii) Ifα1< α2, thenK_q^α2,pB_β^s(G)⊂K_q^α1,pB^s_β(G).

(iv) If b≤c, thenK_q^α,bB_β^s(G)⊂K_q^α,cB_β^s(G)andK˙_q^α,bB^s_β(G)⊂K˙_q^α,cB_β^s(G).

(v) If q1 ≤ q2, then K_q^α,p₂ B_β^s(G) ⊂ K_q^r,p₁ B^s_β(G) and K˙_q^α,p₂ B_β^s(G) ⊂ K˙_q^r,p₁ B_β^s(G) where r=α−(_q¹

1 −_q¹

2).

Theorem 3. Let α = ¹_p − ¹_q, 0 < p ≤ q < ∞, 0 < β ≤ ∞, s ∈ R, then K˙_q^α,pB_β^s(G)⊂B_p,β^s (G).

Proof. LetD_j =G_j\G_j+1. Using the H¨older inequality, we have kfk^p_Lp(G)=

X∞

j=−∞

Z

Dj

|f(x)|^pdx

≤C X∞ j=−∞

|D_j|^1−pq µ Z

Dj

|f(x)|^qdx

¶^p

q

=C X∞ j=−∞

m

p q−1 j

µ Z

Dj

|f(x)|^qdx

¶^p

q

=C X∞ j=−∞

m^−αp_j µ Z

Dj

|f(x)|^qdx

¶^p

q

=Ckfk^p_K˙α,p q (G).

Therefore, ˙K_q^α,pB^s_β(G)⊂B_p,β^s (G). ¤ Theorem 4. Let 0 < p < ∞, 0 < q < ∞, 0 < r ≤ q, 0 < β ≤ ∞, s ∈ R, and 0 < r < p < ∞, α > ¹_r − ¹_q, or 0 < p ≤ r < ∞ and α ≥ ¹_r − ¹_q. Then K_q^α,pB_β^s(G)⊂B_r,β^s (G).

Proof. SupposeD0=G0,Dj =Gj\Gj+1. By the H¨older inequality, we can obtain

kfk^r_Lr(G)= X0 j=−∞

Z

Dj

|f(x)|^pdx

≤C X0 j=−∞

|Dj|^1−rq µ Z

Dj

|f(x)|^qdx

¶^r

q

=C X0 j=−∞

m_j^rq−1 µ Z

Dj

|f(x)|^qdx

¶^r

q

If 0< r < p <∞,α > 1 r−1

q, then

kfk^r_Lr(G)≤C ( ₀

X

j=−∞

m^−αp_j µ Z

Dj

|f(x)|^qdx

¶^p

q

)_r/p

×

½ X₀

j=−∞

m^(1q−1r+α)rp/(p−r) j

¾_1−r/p

≤C

½ X0 j=−∞

m^−αp_j µ Z

Dj

|f(x)|^qdx

¶^p

q¾r/p

≤Ckfk^r_K^α,p_{q (G)}.

We can deduce thatmj≤m0= 1 sincej ≤0, then if 0< p≤r <∞, α≥ ¹_r−¹_q, we have

kfk^r_Lr(G)≤C ( ₀

X

j=−∞

m⁽_j^1q−1r)p µ Z

Dj

|f(x)|^qdx

¶^p

q

)_r/p

≤C ( ₀

X

j=−∞

m^−αp_j µ Z

Dj

|f(x)|^qdx

¶^p

q

)_r/p

≤Ckfk^r_Kq^α,p_(G).

Furthermore,K_q^α,pB^s_β(G)⊂B_r,β^s (G). ¤

3. Atomic Decomposition of the Herz-type Homogeneous Spaces The atomic decomposition of the homogeneous Besov space onGwere 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 functiona:G→Cis an (s,∞) atom,s∈R, if (i)ais supported on a setz+Gn for somez∈Gandn∈Z, (ii)|a(x)| ≤(mn)^s,

(iii)R

Ga(x)dx= 0.

We have the following result.

Theorem 5. Let0< p, β≤ ∞,α >−¹_q,s∈R. Then the following two facts are equivalent.

(a) f ∈K˙_q^α,pB˙_β^s(G),

(b) there exist constants λl,j,l∈Z+ andj ∈Z, and(−(s−α) + 1/q,∞)atoms ai,j withsuppal,j ⊂zl,j+Gj−1 such that

f = X∞ j=−∞

X∞

l=0

λl,jal,j in S⁰(G)/%

Moreover,

kλkp,β := ( X∞ j=−∞

( X∞ l=0

|λl,j|^p)^βp)^1/β≤ckfkK˙q^α,pB˙_β^s.

Proof. (a)⇒(b) For each f ∈S⁰/%we have

f = X∞ n=−∞

f∗ϕn(x)

= X∞ n=−∞

f∗ϕn∗ϕn(x)

= X∞ n=−∞

X∞

l=0

(mn)⁻¹f ∗ϕn(zl,n)ϕn(x−zl,n)

= X∞ n=−∞

X∞ l=0

λl,nal,n

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,nwe have

(i) suppal,n⊂zl,n+Gn−1; (ii)|al,n(x)| ≤(mn−1)^{−(s−α)+1/q}; (iii)R

Gal,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,nwe have

kλkp,β = ( X∞ n=−∞

( X∞

l=0

|(mn−1)^{(s−α)−1/q}f∗ϕn(zl,n)|^p)^βp)^1β

≤c(

X∞ n=−∞

m^sβ_n ( X∞

k=−∞

m^−αp_k kf∗ϕnχkk^p_q)^βp)^β1

=ckfkK˙q^α,pB˙_β^s <∞.

This completes the proof of (a).

(b)⇒(a) Letal,j be an (−(s−α) + 1/q,∞) atom onGwith support inzl,j+Gj. Similar to the proof in [4], we haveal,j∗ϕn= 0 whenj≥nfor eachx∈G. So we only consider the case of j < n.

If k+ 1≤j, thenk(al,j ∗ϕn)χkkq = 0 since suppal,j∗ϕn ⊂Gj. Ifk > j, we have k(al,j∗ϕn)χkkq ≤cm^−(s−α)+_j ^1qm⁻_k^1q.

In the following, we estimate kfkK˙q^α,pB˙^s_β. If f = P_∞

j=−∞

P_∞

l=0λl,jal,j and kλkp,β <∞,

(i) If 0< p≤1, then for eachn∈Z, kf∗ϕnk^p_K˙α,p

q ≤

n−1X

j=−∞

X∞

l=0

|λl,j|^pkal,j∗ϕnk^p_K˙α,p q

Therefore, for each β with 0< β <∞we have kfkK˙q^α,pB˙_β^s ={

X∞ n=−∞

m^sβ_n (

n−1X

j=−∞

X∞ l=0

|λl,j|^pkal,j∗ϕnk^p_K˙α,p q )^βp}^1β

={ X∞ n=−∞

m^sβ_n (

n−1X

j=−∞

X∞ l=0

|λl,j|^p X∞ k=j

m^−αp_k kal,j∗ϕnχkk^p_q)^βp}^β1

≤c{

X∞ n=−∞

m^sβ_n (

n−1X

j=−∞

X∞

l=0

|λl,j|^pm^−sp_j )^βp}^1β

sinceα+ 1/q >0.

If 0< β≤p, so that 0< β/p≤1, then kfkK˙q^α,pB˙^s_β ≤c{

X∞ n=−∞

m^sβ_n

n−1X

j=−∞

m^−sβ_j ( X∞

l=0

|λl,j|^p)^βp}^1β

≤ckλkp,β <∞ here s <0.

Ifp < β <∞, letr=^β_p. Using the H¨older inequality, we can obtain

kfkK˙^α,pq B˙^s_β ≤c{

X∞ n=−∞

m^sβ_n (

n−1X

j=−∞

X∞ l=0

|λl,j|^pm^−sp_j )^βp}^β1

≤c{

X∞ n=−∞

m^sβ_n (

n−1X

j=−∞

m^−sp_j )^rr0

n−1X

j=−∞

m^−sp_j ( X∞ l=0

|λl,j|^p)^βp}^β1

≤c{

X∞ n=−∞

m^sp_n−1

n−1X

j=−∞

m^−sp_j ( X∞ l=0

|λl,j|^p)^βp}^β1

≤ckλkp,β <∞.

Here r⁰ is the conjugate index ofr.

Ifβ =∞, sinces <0 and kfkK˙q^α,pB˙_β^s = sup

n m^s_nkf∗ϕnkK˙q^α,p

≤sup

n m^s_n(

n−1X

j=−∞

X∞

l=0

|λl,j|^pm^−sp_j )^1p moreover,

n−1X

j=−∞

m^−sp_j X∞

l=0

|λl,j|^p≤sup

j ( X∞

l=0

|λl,j|^p)

n−1X

j=−∞

m^−sp_j

≤cm^−sp_n sup

j ( X∞ l=0

|λl,j|^p).

Hence,

kfkK˙q^α,pB˙^s_β ≤csup

n m^s_nm^−s_n sup

j ( X∞

l=0

|λl,j|^p)^1/p=ckλkp,β <∞.

(ii) If 1< p <∞, similar to the proof of Theorem 6 in [4], for eachn∈Z,x∈G and τ with 0< τ <1, we have

kf∗ϕnkK˙^α,pq ≤ck

n−1X

j=−∞

( X∞

l=0

(|λl,j||al,j∗ϕn(x)|^τ)^p)^1/pm^{[−(s−α)+}_j ^1q](1−τ)kK˙^α,pq

≤c

n−1X

j=−∞

m^{[−(s−α)+}_j ^1q](1−τ)k(

X∞ l=0

(|λl,j||al,j∗ϕn(x)|^τ)^p)^1/pkK˙q^α,p

≤c

n−1X

j=−∞

m^{[−(s−α)+}_j ^1q](1−τ)( X∞ k=j

m^−αp_k m^{[−(s−α)+}_j ^{1q]τ p}m⁻_k^pq)^1p( X∞ l=0

|λ_l,j|^p)^1p

≤c

n−1X

j=−∞

m^−s_j ( X∞

l=0

|λl,j|^p)^1p.

Consequently,

kfkK˙q^α,pB˙^s_β ≤c(

X∞ n=−∞

m^sβ_n

n−1X

j=−∞

m^−sβ_j ( X∞

l=0

|λl,j|^p)^βp)^β1

≤c(

X∞ j=−∞

m^−sβ_j ( X∞

l=0

|λl,j|^p)^βp) X∞ n=j+1

m^sβ_n )^1β

≤ckλkp,β.

(iii) Ifp=∞, the proof is simpler. Here we omit it. ¤ References

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Steklova, 60:42–81, 1961.

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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.

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Received October 11, 2005.

Canqin Tang

Department of Mathematics, Dalian Maritime University, Dalian

E-mail address: [email protected] Qingguo Li

Department of Mathematics, Hunan University,

Changsa

E-mail address: [email protected]

Bolin Ma

Department of Mathematics, Hunan University,

Changsa

E-mail address: [email protected]