Vietnam Journal of Mathematics 33:4 (2005) 369–379
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A Charact erization of M orrey Type B esov
and Trieb el-Lizorkin Space s*
Jingshi X u
Department of Mathematics, Hunan Norm al University,
Changsha, 410081, China
Received Sept emb er 25, 2003
Revised J une 1, 2005
A bs t ract . In t his pap er t he aut hor gives a maximal funct ion charact erizat ion of t he
Morrey-type Besov and Trieb el-Lizorkin spaces, M B s , β
p , q (Rn)and M F s , β
p , q (Rn),which
are t h e generalizat ions of t h e well-known Morrey-t yp e spaces and t he inh omogeneous
Besov and Trieb el-Lizorkin spaces.
1. Int ro d uct ion
In recent years, t he Morr ey-t yp e space cont inues t o at t ract t he at t ent ion of
many au t hors. Many problem s of p art ia l differ ent ia l equ at ion based on Mor rey
space and Morrey t yp e Besov space have been considered in [1 - 6, 11, 16].
Ma ny result s ob t ained parallel wit h t he t heory of st a ndard Besov a nd Trieb el-
Lizorkin spaces an d new a pplica t ions h ave also b een given. Act ually, in [7]
Mazzuat o est ablished some decomp osit ions of Morrey typ e Besov spaces (in [7],
t hey were called B esov-Morr ey sp aces) in t er ms of sm oot h wavelet s, m olecu les
concent rat ed on dyadic cub es, and at oms supp ort ed on dyadic cub es. In [10],
Ta ng Lin a nd t he a ut h or ob t a in ed som e pr op er t ies in clu din g lift p rop er t ies a nd
a Fourier mult ip lier t h eor em on Morrey t yp e Besov and Trieb el-Lizorkin sp aces,
and a discret e ch aract er iza t ion of t hese spa ces. Mor eover, in [10] t h e a ut hors
a lso con sid er ed t h e b ou n ded ness of a cla ss p seu d o-d iffer ent ia l op er at or s on t h ese
spaces.
∗
T he pr oj ect was support ed by t he NN SF(60474070) of China.
370 Jingshi Xu
For readers int er est in g in st a ndard Besov and Tr ieb el-Lizorkin spaces and
t h eir a pp lica t ion s, we r ecom m en d t h em T rieb el’s b ooks [12 - 15].
Mot ivat ed by [8], our purp ose is t o give a m aximal funct ion inequa lit y on
Mor rey-t yp e B esov an d Tr ieb el-Lizor kin sp aces, wh ich is a cha r act er iza t ion of
Mor rey-t yp e B esov a nd Tr ieb el-Lizor kin sp aces. Before st at ing it , we r ecall som e
not at ions and t he definit ion of Morrey-typ e Besov a nd Trieb el-Lizorkin spaces
(see, e.g., [10]).
Let Rnb e t he n-dimensional real E uclidean space. Let S(Rn) b e t h e Schwa rt z
space of all com plex-valued rapid ly d ecreasin g infinit ely different iable funct ions
on Rn.Let S′(Rn) b e t he set of all t he t em pered dist ribut ion on Rn.If ϕ∈S(Rn),
t hen ϕdenot es t h e Fourier t r ansform of ϕ , and ϕ∨denot es t he inver se Fourier
t ra nsfor m of ϕ .
D efinit io n 1. If 0< qp< ∞and f∈Lq
L o c (Rn),wesayf∈Mp
q(Rn)
provided that, for any ball BR , x cen t ered at xwith radius R,
fMp
q= : sup
x∈Rn, R > 0
Rn(1/ p −1/ q )
BR , x
|f(y)|qdy
1/ q
<∞.
Morr ey sp aces can b e seen as a com plem ent t o Lpspaces. In fact , Mp
q≡Lp
and Lp⊂Mp
q.
F o r j∈Nwe put ϕj(x)= 2
n j ϕ(2jx),x∈Rn.Let fun ct ions A , θ ∈S(Rn)
sat isfy t he following condit ions:
|
A(ξ)|>0on{ |ξ|<2},supp
A⊂{|ξ|<4},
|
θ(ξ)|>0on{1/2<|ξ|<2},supp
θ⊂{1/4<|ξ|<4}.
Now t h e M orrey t yp e Besov a nd Tr ieb el-Lizor kin sp aces can b e defin ed as
follows.
D efinit io n 2 . Let − ∞ < s< ∞,0< qp< ∞,0< β ∞,andA , θ be as
above, then we define
(i) T he Morrey type Besov spaces as
M B s , β
p , q (Rn)=
f∈S′(Rn): fM B s , β
p , q
=A∗fMp
q+
{2s j θj∗f}∞
1
ℓβ(Mp
q)<∞
.
(ii) T he Morrey type Triebel-Lizorkin spaces as
M F s , β
p , q (Rn)=
f∈S′(Rn): fM F s , β
p , q
=A∗fMp
q+
2s j θj∗f
∞
1
Mp
q(ℓβ)<∞
.
Obviously, for s∈R,0< p=q< ∞,and0β∞,then M B s , β
q,q =Bs
q,β
and M F s , β
q,q =Fs
q,β , st a n da rd B esov a nd Tr ieb el-Lizorkin spaces resp ect ively; see
[22].
A Characterization of Morrey Type Besov and Triebel-Lizorkin Spaces 371
To m ake t hese space m ea ningful, t he key p oint is t o show t hat Defin it ion 2
is in dep endent of t he choice of funct ions Aand θ. Act u ally, by t he m et hod of
Trieb el’s b ook [12] we had proved this in a m odified d efinit ion in [10]. In t his
pa per, we will con sid er t his by using maxim al fun ct ion a ga in. T h e following
T h eorem 1 is st ronger t h an wha t we obt ained in [10].
Let Ψ ,ψ ∈S(Rn),ǫ > 0,a n int eger S≥ −1 b e su ch t h at
|
Ψ(ξ)|>0on{ |ξ|<2ǫ},(1)
|
ψ(ξ)|>0on{ǫ / 2<|ξ|<2ǫ},
and Dτ
ψ(0) = 0 for all |τ|S. (2)
Here (1) are Taub eria n condit ion s, while (2) expresses m om ent con dit ion s on ψ.
F o r a n y a> 0,f ∈S
′(Rn),and x∈Rn,we int roduce m axim al funct ions,
Ψ∗
af(x)= sup
y∈Rn
|Ψ∗f(y)|
(1 + |x−y|)a,(3)
and
ψ∗
j , a f(x)= sup
y∈Rn
|ψj∗f(y)|
(1 + 2j|x−y|)a.(3′)
In wh at follows, by writ ing A1A2we mean t ha t A1C A 2,C is a p osit ive
con st ant in d ep en dent of f∈S
′(Rn).
T heorem 1.
(i) Let s< S+ 1,0< β ∞,0< q,p∞,a > n/ q. T hen for all f∈S
′(Rn)
Ψ∗
afMp
q+ { 2s j ψ∗
j , a f}∞
1ℓβ(Mp
q)fMp
qBs
β
Ψ∗fMp
q+ { 2j s ψj∗f}∞
1ℓβ(Mp
q).(4)
(ii) Let s< S+ 1,0< β ∞,0< q,p< ∞,a > n/ min(q, β ).T hen for all
f∈S
′
Ψ∗
afMp
q+ { 2s j ψ∗
j , a f}∞
1Mp
q(ℓβ)fMp
qFs
β
Ψ∗fMp
q+ { 2j s ψj∗f}∞
1Mp
q(ℓβ).(5)
T h e r em ainder of t h e pap er is t o give t he proof of T heorem 1. To d o t his,
we need some lemm as, which will be given in Sec. 2. T he com plet e proof will
b e given in Sec. 3. F in ally, we p oint t h at let t er Cwill d enot e va rious p osit ive
const a nt s. T he const ant s m ay in gener al d ep en d on a ll fixed pa ramet ers, an d
som et im es we sh ow t h is d ep en den ce exp licit ly by wr it in g, e.g., CN.In t h e sequel,
for convenience we om it t h e ra nge of int egr at ion when it is Rn.
372 Jingshi Xu
2. Som e Lem m as
Lem m a 1. (see [8]) Let μ , ν ∈S(Rn),M ≥ −1integer,
Dτ
μ(0) = 0 for all |τ|M .
T hen for any N > 0there is a constant CNsuch that
sup
z∈Rn
|μt∗ν(z)|(1 + |z|)N
CNtM+ 1.
T h e followin g Lem ma 2 is ea sy t o ob t ain. For it s proof one can a lso see [8].
Lem m a 2. Let 0< β ∞,δ > 0.For any sequence {gj}∞
0of nonnegative
measurable functions on Rn,put
Gj(x)=
∞
k= 0
2− |k−j|δ
gk(x),x∈Rn.
T hen
{ Gj(x)}∞
0ℓβ
C { gj(x)}∞
0ℓβ(6)
holds, where Cis a constant only dependent on β , δ.
Lem m a 3. Let 0< p, q,β ∞,δ>0.For any sequence {gj}∞
0of nonnegative
measurable functions on Rn,set
Gj(x)=
∞
k= 0
2− |k−j|δ
gk(x),x∈Rn.
T hen
{ Gj}∞
0Mp
q(ℓβ)C1 { gj}∞
0Mp
q(ℓβ),(7)
and
{ Gj}∞
0ℓβ(Mp
q)C2 { gj}∞
0ℓβ(Mp
q)(8)
hold with some constants C1=C1(β, δ)and C2=C2(p, q, β , δ).
Proof. By Lem ma 2, (7) follows im mediat ely from (6). Now we pr ove (8) by
considerin g t wo cases.
Case 1. q≥1.Since · Mp
qis a nor m , by Minkowski’s inequ a lity, we have
GjMp
q
∞
k= 0
2− |k−j|δgkMp
q.
Hence (8) follows from Lem ma 2.
Case 2. q1.B y D efinit ion 1
A Characterization of Morrey Type Besov and Triebel-Lizorkin Spaces 373
Gjq
Mp
q
= sup
x∈Rn, R > 0
Rn q (1/ p −1/ q )
BR , x
|Gj(y)|qdy
sup
x∈Rn, R > 0
Rn q (1/ p −1/ q )
∞
k= 0
2−q|k−j|δ
BR , x
|gk(y)|qdy
∞
k= 0
2−q|k−j|δsup
x∈Rn, R > 0
Rn q (1/ p −1/ q )
BR , x
|gk(y)|qdy
=
∞
k= 0
2− |k−j|qδ gkq
Mp
q
.
By Lemm a 2 wit h β , and δreplaced by β / q and qδ resp ect ively, we h ave
Gjq
Mp
q
ℓβ / q
C
gjq
Mp
q
ℓβ / q
.
R a isin g t h e a b ove in equ alit y t o p ower 1/ q, we obt ain (8).
T his com plet es t he proof of Lemm a 3.
Lem m a 4. (see [10]) Let 1< β < ∞and 1< qp< ∞.If{fj}∞
j= 0 is a
sequ en ce of local in t egral f un ct ion s on Rn,then
(
∞
j= 0
|M fj|β)
1
βMp
q
C
∞
j= 0
|fj|β
1
βMp
q,
where the constant Cis independent of {fj}∞
j= 0 and Mdenotes standard Hardy-
L i t t lewood m axi m al operat or.
Lem m a 5. (see [8]) Let 0< r 1,and let {bj}∞
0,{dj}∞
0be t wo sequ en ces t aki n g
values in (0,+∞]and (0,+∞)respectively. A ssum e that for some N0>0
dj=O(2j N 0),j→∞,
and that for any N > 0,and j∈N0=N∪{0}, there exists a constant CN
independent of jsuch that
djCN
∞
k=j
2(j−k)Nbkd1−r
k.
T hen for any N > 0and j∈N0,
dr
jCN
∞
k=j
2(j−k)N r bk
hold with the sam e constants CNas above.
3. P ro of of T heore m 1
T he id ea of t h e p roof is from R ych kov [8]. In fact , we will u se t h e m et hod in [8]
wit h Lem ma 3 a nd Lem ma 4. To do t h e end, we give t he proof in t hree st eps.