R E S E A R C H
Open Access
Embedding theorem on RD-spaces
Yanchang Han
**Correspondence:
[email protected] School of Mathematic Sciences, South China Normal University, Guangzhou, 510631, P.R. China
Abstract
An RD-space (X,d,
μ
) is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds. An important class of RD-spaces is provided by Carnot-Carathéodory spaces with a doubling measure. In this article, the author establishes the embedding theorem for Besov and Triebel-Lizorkin spaces on RD-spaces.MSC: Primary 42B25; secondary 46F05; 46E35
Keywords: spaces of homogeneous type; test function space; distributions; Calderón reproducing formula; Besov and Triebel-Lizorkin spaces; embedding
1 Introduction and statement of main results
Spaces of homogeneous type, particularly including metric measure spaces, play a promi-nent role in many fields of mathematics. These spaces constitute natural generalizations of manifolds admitting all kinds of singularities and still providing rich geometric structure; see [, ]. Analysis on spaces of homogenous type has been performed quite intensively; see, for example, [–]. Recently, a theory of Besov and Triebel-Lizorkin spaces on RD-spaces was developed in [, ], which includesn-regular measure spaces.
Let us now recall some notations and definitions. Spaces of homogeneous type were introduced by Coifman and Weiss in the early s, in []. Aquasi-metric don a setXis a functiond:X×X−→[,∞) satisfying (i)d(x,y) =d(y,x)≥ for allx,y∈X; (ii)d(x,y) = if and only ifx=y; and (iii) thequasi-triangle inequality: there is a constantA∈[,∞) such that for allx,y,z∈X,
d(x,y)≤A
d(x,z) +d(z,y). (.)
We define the quasi-metric ball byB(x,r) :={y∈X:d(x,y) <r}forx∈Xandr> . Note that the quasi-metric, in contrast to a metric, may not be Hölder regular and quasi-metric balls may not be open. We say that (X,d,μ) is a space of homogeneous type in sense of Coifman and Weiss ifdis a quasi-metric andμis a nonnegative Borel regular measure on
Xsatisfying thedoubling condition, that is, for allx∈X,r> , then <μ(B(x,r)) <∞and
μB(x, r)≤CμB(x,r), (.)
whereμis assumed to be defined on aσ-algebra which contains all Borel sets and all balls
B(x,r) and the constant <C<∞is independent ofx∈Xandr> .
We point out that the doubling condition (.) implies that there exists a positive con-stantω:=logC(theupper dimensionofμ) such that for allx∈X,λ≥ andr> ,
μB(x,λr)≤CλωμB(x,r). (.)
Macías and Segovia [] showed that the quasi-metricdcan be replaced by another quasi-metricdsuch that the topologies induced onXbydanddcoincide. Moreover,dhas the following regularity property: there exist constantsC> and <θ < such that for all <r<∞and allx,x,y∈X,
d(x,y) –dx,y≤Cdx,xθd(x,y) +dx,y–θ. (.)
Analysis on spaces of homogeneous type has been performed quite intensively in recent years in [, ] and []. For example, Coifman and Weiss introduced atomic Hardy space
Hatp forp∈(, ] in [] and proved that ifTis a Calderón-Zygmund singular integral
oper-ator and is bounded onL, thenTextends a bounded operator fromHptoLpfor suitable p≤. In many applications, the additional assumptions on the measureμare required. For instance, Macías and Segovia in [] provided the maximal function characterization of the Hardy spacesHatp on spaces of homogenous type with additional assumption that
the quasi-metricdsatisfies the regularity condition in (.) and the measureμsatisfies the following property:
μB(x,r)∼r. (.)
Note that property (.) is much stronger than the doubling condition. More precisely, Macías and Segovia provided the maximal function characterization for Hardy spaces
Hp(X) with ( +θ)–<p≤, on spaces of homogeneous type (X,d,μ) that satisfy the prop-erty (.) on the quasi-metricdand the property (.) on the measureμ.
In [], Nagel and Stein developed the productLp ( <p<∞) theory in the setting of
the Carnot-Carathéodory spaces formed by vector fields satisfying Hörmander’s finite rank condition. The particular Carnot-Carathéodory spaces studied in [] are metric spaces with a measureμ satisfying the conditionsμ(B(x,sr))∼sm+μ(B(x,r)) fors≥ andμ(B(x,sr))∼sμ(B(x,r)) fors≤. These conditions on the measure are weaker than property (.) but are still stronger than the original doubling condition (.). In [], moti-vated by the work of Nagel and Stein, Besov and Triebel-Lizorkin spaces were developed on spaces of homogeneous type with a regular quasi-metric and a measure satisfying the reverse doubling condition, that is, there are constantsκ∈(,ω] andc∈(, ] such that
cλκμB(x,r)≤μB(x,λr) (.)
for allx∈X, <r<supx,y∈Xd(x,y)/ and ≤λ<supx,y∈Xd(x,y)/r.
We would like to mention that spaces of homogeneous type encompass several impor-tant examples in harmonic analysis, such as Euclidean spaces withA∞-weights (of the
space of homogeneous type in sense of Coifman and Weiss which has a ‘dimension’ωand satisfies the quasi-metricdsatisfying (.) and the ‘reverse’ doubling property (.). For further developments, including analogous theories of function spaces on RD-spaces, we refer to [, , , ] and [].
On the other hand, embedding theorems are essential tools in many fields for function spaces, especially partial differential equations. For embedding theorems onRn, see [–
] and []. Han, Lin and Yang in [] and [] have proved embedding theorems for Besov and Triebel-Lizorkin spaces on spaces of homogeneous type (X,d,μ), where the quasi-metricdsatisfies (.) and, however, measureμsatisfies (.).
The main purpose in this paper is to establish the embedding theorem for Besov and Triebel-Lizorkin spaces on RD-spaces. We would like to point out that the reverse bling property on the measure played an important role. More precisely, this reverse dou-bling property ensures that
k∈Z:δk≥r μ(B(x,δk))≤
C
μ(B(x,r)),
which is the key to developing the theory of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type. See [, ] for more details. But this reverse doubling property on the measure does not play any role in the proof of the embedding theorem in this paper. How-ever, to achieve the embedding theorem, one needs the density condition on the measure, namely
μ(B(x,r)≥Crω (.)
for anyx∈Xandr> .
Throughout this paper, we useCto denote positive constants, whose value may vary from line to line. Constants with subscripts, such asC, do not change in different occur-rences. ByVr(x) we denote the measure ofB(x,r), the ball centered atxwith radiusr> ;
andV(x,y) denotes the measure ofB(x,y), the ball centered atxwith radiusd(x,y) > . In addition, we use the notationabto mean that there is a constantC> such that
a≤Cb, and the notationa∼b to mean thataba. The implicit constantsC are meant to be independent of other relevant quantities. Also, for two topological spacesA andB,A→Bmeans a linear and continuous embedding. Forp> , letpbe its conjugate index.
Before stating the embedding theorem, we now recall test functions and distributions on RD-spaces (X,d,μ).
Definition .(Test functions, []) Fixx∈X,r> ,γ > andβ∈(,θ). A functionf defined onXis said to be atest function of type(x,r,β,γ)centered at x∈Xiff satisfies the following three conditions.
(i) (Size condition) For allx∈X,
f(x)≤C
Vr(x) +V(x,x)
r r+d(x,x)
γ
(ii) (Regularity condition) For allx,y∈Xwithd(x,y) < (A)–(r+d(x,x)),
f(x) –f(y)≤C d(x,y)
r+d(x,x)
β
Vr(x) +V(x,x) r r+d(x,x)
γ
.
(iii) (Cancelation condition)
f(x)dμ(x) = .
We denote byG(x,r,β,γ) the set of all test functions of type (x,r,β,γ). The norm of
f inG(x,r,β,γ) is defined by
f G(x,r,β,γ):=inf
C> : (i) and (ii) hold.
For each fixedx, letG(β,γ) :=G(x, ,β,γ). It is easy to check that for each fixedx∈X andr> , we haveG(x,r,β,γ) =G(β,γ) with equivalent norms. Furthermore, it is also easy to see thatG(β,γ) is a Banach space with respect to the norm onG(β,γ).
For <β<θ andγ> , letG◦(β,γ) be the completion of the spaceG(θ,γ) in the norm ofG(β,γ). Forf∈G◦(β,γ), define f ◦
G(β,γ):= f G(β,γ).
Definition .(Distributions) Thedistribution space(G◦(β,γ))is defined to be the set of all linear functionalsLfromG◦(β,γ) toCwith the property that there existsC> such that for allf ∈G◦(β,γ),
L(f)≤C f ◦
G(β,γ).
We begin with recalling the definition of approximation to the identity, which plays the same role as the heat kernelH(s,x,y) does in [].
Definition .([]) Letθ be the regularity exponent ofX. A sequence{Sk}k∈Zof linear
operators is said to be an approximation to the identity (in short, ATI) if there exists a constantC,C> such that for allk∈Zand allx,x,y,y∈X,Sk(x,y), the kernel ofSkis a
function fromX×XintoCsatisfying
(i) Sk(x,y) = ifd(x,y)≥C–kand|Sk(x,y)| ≤CV –k(x)+V–k(y);
(ii) |Sk(x,y) –Sk(x,y)| ≤Ckθd(x,x)θV
–k(x)+V–k(y) forρ(x,x
)≤max{C
, }–k;
(iii) |Sk(x,y) –Sk(x,y)| ≤Ckθd(y,y)θV
–k(x)+V–k(y)forρ(y,y
)≤max{C, }–k;
(iv) |[Sk(x,y) –Sk(x,y)] – [Sk(x,y) –Sk(x,y)]| ≤Ckθd(x,x)θd(y,y)θV
–k(x)+V–k(y) for ρ(x,x)≤max{C, }–kandρ(y,y)≤max{C, }–k;
(v) XSk(x,y)dμ(y) = ;
(vi) XSk(x,y)dμ(x) = .
The Besov and Triebel-Lizorkin spaces on RD-spaces are defined as follows.
The Besov space B˙sp,q(X) is the collection of all f ∈(
◦
G(β,γ)) with β,γ ∈(,θ) and max(ωω+θ,ω+ωθ+s) <p≤ ∞and <q≤ ∞such that
f B˙sp,q(X):=
k∈Z
ksqDk(f) q Lp(X)
/q
<∞.
The Triebel-Lizorkin spaceF˙ps,q(X) is the collection of allf∈(
◦
G(β,γ))withβ,γ ∈(,θ) andmax(ωω+θ,ω+ωθ+s) <p<∞,max(ωω+θ,ω+ωθ+s) <q≤ ∞such that
f F˙ps,q(X):=
k∈Z
ksqDk(f) q/q
Lp(X) <∞.
The main result of this paper is the following.
Theorem . Suppose thatμ(B(x,r))≥Crωfor any x∈X and r> and–θ<s
<s<θ. (i) Letmax{ ω
ω+θ, ω
ω+θ+si}<pi≤ ∞, <q≤ ∞,i= , and–θ<s–ω/p=s–ω/p<θ.
Then
˙
Bsp,q→ ˙Bsp,q. (.)
(ii) Letmax{ ω ω+θ,
ω
ω+θ+si}<pi<∞andmax{
ω ω+θ,
ω
ω+θ+si}<qi≤ ∞fori= , ,and –θ<s–ω/p=s–ω/p<θ.Then
˙
Fsp,q→ ˙Fsp,q. (.)
2 The proof of Theorem 1.5
In this section, we will prove Theorem .. Since there is no Fourier transform on spaces of homogeneous type, the proof of Theorem . is quite different from the proof onRnas
given on p. in []. The density property (.) on the measureμplays a crucial role in the proof of Theorem . in this paper.
We first recall the following lemmas, namely the construction provided as an analogue of the grid of Euclidean dyadic cubes on spaces of homogeneous type by Christ in [], the discrete Calderón reproducing formulae and the frame characterizations of Besov and Triebel-Lizorkin spaces on RD-spaces established in [].
Lemma . Let X be a space of homogeneous type in the sense of Coifman and Weiss.Then there exist a collection{Qkα⊂X:k∈Z,α∈Ik}of open subsets,where Ikis some(possible finite)index set,and constantsδ∈(, )and C,C> such that
(i) μ(X\αQk
α) = for each fixedkandQkα∩Qkβ=∅ifα=β;
(ii) for anyα,β,k,lwithl≥k,eitherQl
β⊂QkαorQlβ∩Qkα=∅;
(iii) for each(k,α)and eachl<k,there is a uniqueβsuch thatQk α⊂Qlβ;
(iv) diam(Qk
α)≤Cδk;
(v) eachQkαcontains some ballB(zkα,Cδk),wherezkα∈X.
In fact, we can think ofQk
αas being a dyadic cube with a diameter roughlyδkand
cen-tered atzk
α. In what follows, we always supposeδ= /. See [] for how to remove this
re-striction. Also, in the following, fork∈Z,τ ∈Ik, we will denote byQτk,ν,ν= , . . . ,N(k,τ),
Lemma . Suppose that{Sk}k∈Zis an approximation to the identity as in Definition..
Set Dk =Sk –Sk– for k∈Z. Then, for any fixed M∈ Nlarge enough, there exists a
family of functions {Dk(x,y)}k∈Z such that for any fixed yτk,ν ∈Qkτ,ν, k∈Z, τ ∈Ik and
ν∈ {, . . . ,N(k,τ)}and all f ∈(G◦(β,γ))with <β,γ <θand x∈X,
f(x) =
k∈Z
τ∈Ik
N(k,τ)
ν= μQk,ν
τ Dk
x,yk,ν τ
Dk(f)
yk,ν
τ
, (.)
where the series converges in the norm of(G◦(β,γ))withθ>β>βandθ >γ>γ. More-over,Dk(x,y),k∈Z,the kernel ofDk,satisfy the following estimates:for <<θ,
Dk(x,y)≤C
V–k(x) +V–k(y) +V(x,y)
–k
(–k+d(x,y)); (.) Dk(x,y) –Dk
x,y
≤C d(x,x )
–k+d(x,y)
V–k(x) +V–k(y) +V(x,y)
–k
(–k+d(x,y)) (.)
for d(x,x)≤(–k+d(x,y))/A;
X
Dk(x,y)dμ(y) =
X
Dk(x,y)dμ(x) = (.)
for all k∈Z.
Lemma . Let all the other notation be as in Lemma..Suppose that|s|<θ,ωis the upper dimension of(X,d,μ).
For all f ∈(G◦(β,γ))withβ,γ∈(,θ)andmax(ωω+θ,ω+ωθ+s) <p≤ ∞and <q≤ ∞,then
f B˙sp,q(X)∼
k∈Z
ksq
τ∈Ik
N(k,τ)
ν=
μQkτ,νDk(f)
ykτ,ν
p
q/p/q
.
For all f ∈(G◦(β,γ))withβ,γ ∈(,θ)andmax(ωω+θ,ω+ωθ+s) <p<∞,max(ωω+θ,ω+ωθ+s) <
q≤ ∞,then
f F˙ps,q(X)∼
k∈Z
τ∈Ik
N(k,τ)
ν=
ksqDk(f)
ykτ,ν
q
χQk,ν τ
/q
Lp(X) .
We now prove Theorem ..
Proof of Theorem. To prove (.), letf ∈ ˙Bsp,q(X) with|s|<θ,max(ωω+θ, ω
ω+θ+s) <p≤
∞and <q≤ ∞. Sinces<sands–ω/p=s–ω/p, it follows thatp<p. First, we recall the following known estimates of Lemma . in []: fork,k∈Z,τ∈Ik,τ∈Ikand
ν= , . . . ,N(k,τ),ν= , . . . ,N(k,τ),
DkDk
yk,ν
τ ,y k,ν τ
≤C–|k–k|
V–(k∧k)(yτk,ν) +V(ykτ,ν,yk
,ν τ )
–(k∧k)
(–(k∧k)+d(yk,ν τ ,yk
,ν τ ))
By the discrete Calderón reproducing formula (.) in Lemma ., we can write
Dk(f)
ykτ,ν≤
k∈Z
τ∈Ik
N(k,τ)
ν=
μQτk,νDkDk
ykτ,ν,y
k,ν τ
Dk(f)
ykτ,ν
≤C
k∈Z
τ∈Ik
N(k,τ)
ν=
μQτk,νDk(f)
ykτ,ν–|k–k |
×
V–(k∧k)(yk, ν
τ ) +V(ykτ,ν,yk
,ν τ )
–(k∧k)
(–(k∧k)+d(yτk,ν,yk,ν
τ ))
. (.)
Combining Lemma . and (.), we obtain
f B˙s,q p (X)
k∈Z
ksq
τ∈Ik
N(k,τ)
ν=
μQk,ν τ
/p
k∈Z
τ∈Ik
N(k,τ)
ν=
μQτk,νDk(f)
ykτ,ν
×–|k–k|
V–(k∧k)(yτk,ν) +V(ykτ,ν,yk
,ν
τ )
–(k∧k)
(–(k∧k)+d(yk,ν τ ,yk
,ν
τ ))
pq/p/q
.
(.)
We need to consider two cases. Case :p> .
We choose> and> such that=+,can be taken arbitrarily close to, and using the Hölder inequality, we get
k∈Z
τ∈Ik
N(k,τ)
ν=
μQτk,νDk(f)
ykτ,ν
×–|k–k|
V–(k∧k)(yτk,ν) +V(ykτ,ν,yk
,ν
τ )
–(k∧k)
(–(k∧k)+d(yk,ν τ ,yk
,ν
τ ))
k∈Z
τ∈Ik
N(k,τ)
ν=
μQτk,νDk(f)
ykτ,νp–|k–k|p
V–(k∧k)(ykτ,ν) +V(ykτ,ν,yk
,ν τ )
× –(k∧k
)
(–(k∧k)+d(yk,ν τ ,yk
,ν τ ))
/p
k∈Z
τ∈Ik
N(k,τ)
ν=
μQkτ,ν
× –|k–k |p
V–(k∧k)(yk, ν
τ ) +V(ykτ,ν,yk
,ν τ )
–(k∧k)
(–(k∧k)+d(yk,ν τ ,yk
,ν τ ))
/p
k∈Z
τ∈Ik
N(k,τ)
ν=
μQτk,νDk(f)
ykτ,νp
×–|k–k|p
V–(k∧k)(yτk,ν) +V(ykτ,ν,yk
,ν
τ )
–(k∧k)
(–(k∧k)+d(yk,ν τ ,yk
,ν
τ )) /p
where, since Christ’s construction in [], the dyadic cubes on spaces of homogeneous type are disjoint, the last inequalities follow from the facts
V–(k∧k)(ykτ,ν) +V(ykτ,ν,yk
,ν τ )
–(k∧k)
(–(k∧k)+d(yk,ν τ ,yk
,ν τ ))
∼
V–(k∧k)(ykτ,ν) +V(ykτ,ν,y)
–(k∧k)
(–(k∧k)+d(yk,ν τ ,y))
(.)
for anyy∈Qkτ,ν and
k∈Z
τ∈Ik
N(k,τ)
ν=
μQkτ,ν
–|k–k|p V–(k∧k)(yτk,ν) +V(ykτ,ν,yk
,ν
τ )
–(k∧k)
(–(k∧k)+d(yk,ν τ ,yk
,ν
τ ))
k∈Z
–|k–k|p
X
V–(k∧k)(ykτ,ν) +V(ykτ,ν,y)
–(k∧k)
(–(k∧k)+d(yk,ν τ ,y))
dμ(y)
≤C.
From this, it follows that
f B˙sp,q(X)
k∈Z
ksq
τ∈Ik
N(k,τ)
ν= μQkτ,ν
k∈Z
τ∈Ik
N(k,τ)
ν=
μQτk,νDk(f)
ykτ,νp
×–|k–k|p
V–(k∧k)(yτk,ν) +V(ykτ,ν,yk
,ν τ )
–(k∧k)
(–(k∧k)+d(yk,ν τ ,yk
,ν τ ))
q/p/q
k∈Z
ksq
k∈Z
–|k–k|p τ∈Ik
N(k,τ)
ν=
μQτk,νDk(f)
ykτ,νp
q/p/q
.
Applying thep/p-inequality for pp ≤
j∈Z
|aj|
p p
≤
j∈Z
|aj|
p
p (.)
withaj∈Cfor allj∈Zimplies that the last term above is dominated by
k∈Z
k∈Z
–|k–k|pksp τ∈Ik
N(k,τ)
ν=
μQkτ,ν
p/p Dk(f)
ykτ,νp
q/p/q
.
Applying the density condition (.), it immediately follows that μ(Qk,ν
τ )≥C–k
ω
for any k ∈ Z, τ ∈ Ik and hence we have μ(Qk
,ν
τ )p/p = μ(Q k,ν
τ )p/p–μ(Q k,ν τ ) ≤ C–kp(s–s)μ(Qk,ν
τ ), where we use the facts thatp/p< ands–ω/p=s–ω/p. We
f B˙sp,q(X)
k∈Z
k∈Z
(k–k)sp–|k–k|p τ∈Ik
N(k,τ)
ν= ksp
×μQτk,νDk(f)
ykτ,νp
q/p/q
.
Now we chooses∈(–,), applying the Hölder inequality forq/p> and theq/p -inequality forq/p≤ implies that the last term above is dominated byC f B˙s,q
p (X), which
implies (.) for the case wherep> . Case :p≤.
From (.) and thep-inequality and thep/p-inequality in (.), we deduce that
f B˙sp,q(X)
k∈Z
ksq
k∈Z
–|k–k|p
τ∈Ik
N(k,τ)
ν=
μQτk,νpDk(f)
ykτ,νp
×
τ∈Ik
N(k,τ)
ν= μQkτ,ν
×
V–(k∧k)(ykτ,ν) +V(ykτ,ν,yk
,ν τ )
–(k∧k)
(–(k∧k)+d(yk,ν τ ,yk
,ν τ ))
pq/p/q
k∈Z
ksq
k∈Z
–|k–k|p τ∈Ik
N(k,τ)
ν=
V–(k∧k)
ykτ,ν–p
×μQkτ,ν
p Dk(f)
ykτ,νp
q/p/q
k∈Z
ksq
k∈Z
–|k–k|p
τ∈Ik
N(k,τ)
ν=
V–(k∧k)
ykτ,ν
p/p–
×V–(k∧k)
ykτ,ν–pμQk ,ν
τ p–
μQτk,νDk(f)
ykτ,νp
q/p/q
,
where we used the fact that
τ∈Ik
N(k,τ)
ν= μQkτ,ν
V–(k∧k)(yτk,ν) +V(ykτ,ν,yk
,ν
τ )
–(k∧k)
(–(k∧k)+d(yk,ν τ ,yk
,ν
τ )) p
V–(k∧k)
ykτ,ν–p.
Note that by the doubling property (.),
V–(k∧k)
ykτ,νμByk ,ν
τ ,
–(k∧k)=μByk,ν τ ,
k–(k∧k)–k
[k–(k∧k)]ωμByk,ν τ ,
–k[k–(k∧k)]ωμQk,ν τ
and thus
V–(k∧k)
ykτ,ν
–p
μQkτ,ν
p–
forp≤. The density condition (.) implies that
V–(k∧k)
ykτ,ν
p/p–
–(k∧k)ωp(p–
p)–(k∧k)p(s–s)
forp/p< . Therefore, we further obtain
f B˙sp,q(X)
k∈Z
k∈Z
–|k–k|pksp–ksp[k–(k∧k)]ω(–p)–(k∧k)p(s–s)
×
τ∈Ik
N(k,τ)
ν=
kspμQτk,νDk(f)
ykτ,νp
q/p/q
.
Applying the Hölder inequality forq/p> and theq/p-inequality forq/p≤ implies that the last term above is dominated by C f B˙sp,q(X) when ≥p >p, where the last inequality follows from the facts ifs<andp>ω+ωs+, then
k∈Z
–|k–k|pksp–ksp[k–(k∧k)]ω(–p)–(k∧k)p(s–s)pq∧≤C
and
k∈Z
–|k–k|pksp–ksp[k–(k∧k)]ω(–p)–(k∧k)p(s–s) q p∧≤C.
This completes the proof of (.).
We now show (.). By the homogeneity of the norm · F˙sp,q(X), we may assume f F˙ps,q(X)= without loss of generality. By Lemma . and the estimate in (.), we have
f F˙ps,q(X)
k∈Z
ksq
k∈Z
–|k–k|
τ∈Ik
N(k,τ)
ν=
τ∈Ik
N(k,τ)
ν=
μQτk,νDk(f)
ykτ,ν
×
V–(k∧k)(ykτ,ν) +V(ykτ,ν,yk
,ν τ )
× –(k∧k
)
(–(k∧k)+d(yk,ν τ ,yk
,ν τ ))
χQk,ν τ
q/q
Lp(X)
. (.)
To estimate the last expression in (.), we claim that formax{ωω+,ω+ω+s}<r≤,
τ∈Ik
N(k,τ)
ν=
τ∈Ik
N(k,τ)
ν=
μQτk,νDk(f)
ykτ,ν
×
V–(k∧k)(yτk,ν) +V(ykτ,ν,yk
,ν τ )
–(k∧k)
(–(k∧k)+d(yk,ν τ ,yk
,ν τ ))
χQkτ,ν(x)
≤C–kω(–r)μ(B)r–inf
y∈B
M
τ∈Ik
N(k,τ)
ν=
Dk(f)
ykτ,νχQk,ν τ
r
(y)
r
whereB=B(x, –(k∧k)) andMis the Hardy-Littlewood maximal function. To prove (.), from (.), it follows that
τ∈Ik
N(k,τ)
ν=
τ∈Ik
N(k,τ)
ν=
μQτk,νDk(f)
ykτ,ν
×
V–(k∧k)(ykτ,ν) +V(ykτ,ν,yk
,ν τ )
–(k∧k)
(–(k∧k)+d(yk,ν τ ,yk
,ν τ ))
χQkτ,ν(x)
τ∈Ik
N(k,τ)
ν=
τ∈Ik
N(k,τ)
ν=
μQτk,νDk(f)
ykτ,ν
×
V–(k∧k)(x) +V(x,yk
,ν
τ )
–(k∧k)
(–(k∧k)+d(x,yk,ν τ ))
χQk,ν τ (x).
By Christ’s construction in [], the dyadic cubes on spaces of homogeneous type are dis-joint, and therefore
τ∈Ik
N(k,τ)
ν= χQk,ν
τ (x) = ,
which implies that the last term above is dominated by
τ∈Ik
N(k,τ)
ν=
μQτk,νDk(f)
ykτ,ν
V–(k∧k)(x) +V(x,yk
,ν τ )
–(k∧k)
(–(k∧k)+d(x,yk,ν
τ ))
τ∈Ik
N(k,τ)
ν=
μQkτ,ν
r Dk(f)
ykτ,νr
×
V–(k∧k)(x) +V(x,yk
,ν τ )
–(k∧k)
(–(k∧k)+d(x,yk,ν τ ))
rr
=
τ∈Ik
N(k,τ)
ν=
μQkτ,νr–Dk(f)
ykτ,νr
×
X
V–(k∧k)(x) +V(x,yk
,ν τ )
–(k∧k)
(–(k∧k)+d(x,yk,ν τ ))
r
χ
Qk,ν
τ
(y)dμ(y)
r .
Note that the density condition (.) impliesC–kω≤μ(Qk,ν
τ ). The fact thatr– ≤
yields
τ∈Ik
N(k,τ)
ν=
μQkτ,νr–Dk(f)
ykτ,νr
×
X
V–(k∧k)(x) +V(x,yk
,ν
τ )
–(k∧k)
(–(k∧k)+d(x,yk,ν τ ))
r
χ
Qkτ,ν(y)dμ(y)
–kω(–r)
τ∈Ik
N(k,τ)
ν=
Dk(f)
ykτ,νr
×
X
V–(k∧k)(yτk,ν) +V(x,y)
–(k∧k)
(–(k∧k)+d(x,y)) r
χ
Qk,ν
τ
(y)dμ(y)
r
–kω(–r)
τ∈Ik
N(k,τ)
ν=
B
Dk(f)
ykτ,νr
×
V–(k∧k)(x) +V(x,y)
–(k∧k)
(–(k∧k)+d(x,y)) r
χ
Qk,ν
τ
(y)dμ(y)
r
+ –kω(–r)
τ∈Ik
N(k,τ)
ν=
Dk(f)
ykτ,νr
×
∞
m=
[(m+)B]\[mB]
V–(k∧k)(x) +V(x,y)
–(k∧k)
(–(k∧k)+d(x,y)) r
χ
Qkτ,ν(y)dμ(y)
r
=:H+H.
We first estimate the term ofH, for anyx∈B, we then further have
H–k ω(–
r)
μ(B)–r μ(B)
B
τ∈Ik
N(k,τ)
ν=
Dk(f)
ykτ,νrχQk,ν τ
(y)dμ(y)
r
–kω(–r)μ(B)r–
M
τ∈Ik
N(k,τ)
ν=
Dk(f)
ykτ,νrχQk,ν τ r x r .
Now we estimateH. For anyx∈B, it also follows that
τ∈Ik
N(k,τ)
ν=
∞
m=
[(m+)B]\[mB]
Dk(f)
ykτ,νr
×
V–(k∧k)(x) +V(x,y)
–(k∧k)
(–(k∧k)+d(x,y)) r
χ
Qk,ν
τ
(y)dμ(y)
μmB–r–mr
(m+)B
τ∈Ik
N(k,τ)
ν=
Dk(f)
ykτ,νrχQk,ν τ
(y)dμ(y)
μmB–rμm+B–(m–)rM
τ∈Ik
N(k,τ)
ν=
Dk(f)
ykτ,νχQk,ν τ
r
x
μ(B)–r–m[r–ω(–r)]M
τ∈Ik
N(k,τ)
ν=
Dk(f)
ykτ,νχQk,ν τ
r
x,
where we use the doubling condition (.) which implies thatμ(m+B)–r≤Cmω(–r)× μ(B)–r. Thus, forr> ω
H≤C–k ω(–
r)μ(B)r–
M
τ∈Ik
N(k,τ)
ν=
Dk(f)
ykτ,νχQk,ν τ
r
x
r .
Combining the estimates ofH,Hand the arbitrariness ofx∈Bgives the proof of claim (.).
Applying inequality (.) yields
τ∈Ik
N(k,τ)
ν=
Dk(f)
ykτ,νχQk,ν τ (x)
k∈Z
–|k–k|
τ∈Ik
N(k,τ)
ν=
τ∈Ik
N(k,τ)
ν=
μQτk,νDk(f)
ykτ,ν
×
V–(k∧k)(yτk,ν) +V(ykτ,ν,yk
,ν
τ )
–(k∧k)
(–(k∧k)+d(yk,ν τ ,yk
,ν
τ ))
χ
Qkτ,ν
k∈Z
–|k–k|–kω(–r)μ(B)r–inf
y∈B
M
τ∈Ik
N(k,τ)
ν=
Dk(f)
ykτ,νχQk,ν τ
r
(y)
r
k∈Z
–ks–|k–k|–kω(–r)
×μ(B)r–inf
y∈B
M
τ∈Ik
N(k,τ)
ν=
ksDk(f)
ykτ,νχQk,ν τ
r
(y)
r
for ω
ω+γ <r≤. Choosersatisfying ω
ω+γ <r<max{p,q}andr≤, and denote
Fk(y) =
M
τ∈Ik
N(k,τ)
ν=
ksDk(f)
ykτ,νχQk,ν τ
r
(y)
r .
We now have
N
k=–∞
τ∈Ik
N(k,τ)
ν=
ksqDk(f)
ykτ,νχQkτ,ν(x) q
/q
N
k=–∞ k∈Z
ks–ks–|k–k|–kω(–r)μ(B)r–inf
y∈BFk(y)
q/q
.
The Fefferman-Stein vector-valued maximal function inequality [] and the arbitrariness ofykτ,νfurther yield that
inf
y∈BFk(y)≤Cyinf∈B
k∈Z
Fk(y)
q/q
≤C
μ(B)–
B k∈Z
Fk(x)
qp/q dμ(x)
≤Cμ(B)–/p
k∈Z
Fk(y)
q/q
p
≤Cμ(B)–/p
k∈Z
τ∈Ik
N(k,τ)
ν=
ksDk(f)
ykτ,νχQk,ν τ
q/q
p
≤Cμ(B)–/p,
which implies that
N
k=–∞
τ∈Ik
N(k,τ)
ν=
ksqDk(f)
ykτ,νχQk,ν τ (x)
q
/q
≤C
N
k=–∞ k∈Z
ks–ks–|k–k|–kω(–r)μ(B)r–μ(B)–/p
q/q
.
Sinces–s=pω –pω > , sop>p>+pp. The key point here is to chooser=+pp so that
μ(B)r–μ(B)–/p= . Note thatr<pandr< . We assumeq>r= p
+p for the moment
and this assumption will be removed at the end of the proof. This yields
N
k=–∞
τ∈Ik
N(k,τ)
ν=
ksqDk(f)
ykτ,νχQk,ν τ (x)
q
/q
N
k=–∞ k∈Z
ks–ks–|k–k|–kω(–r)
q/q
N
k=–∞ k∈Z
–|k–k|kpω(k–k)s
q/q
≤C
N
k=–∞
k
k=–∞
–(k–k)kpω(k–k)s
q/q
+C
N
k=–∞
∞
k=k
–(k–k)kpω(k–k)s
q/q
N
k=–∞
kpωq
/q
≤CNpω, (.)
wherecan be taken arbitrarily close toθand satisfies –<s–pω <.
On the other hand, if qq ≤, sinces–s=pω –pω > , applying theq/q-inequality, we have
∞
k=N+
τ∈Ik
N(k,τ)
ν=
ksqDk(f)
ykτ,νχQk,ν τ (x)
q
/q
=
∞
k=N+
k(s–s)q
τ∈Ik
N(k,τ)
ν=
ksqDk(f)
ykτ,νχQkτ,ν(x) q
∞
k=N+
k(s–s)q τ∈Ik
N(k,τ)
ν=
ksqD k(f)
yk,ν
τ χQkτ,ν(x) q
/q
N(pω–
ω
p)
k∈Z
τ∈Ik
N(k,τ)
ν=
ksqDk(f)
ykτ,νχQk,ν τ (x)
q
/q
. (.)
Ifδ:=qq> , using the Hölder inequality also yields that
∞
k=N+
k(s–s)q
τ∈Ik
N(k,τ)
ν=
ksqDk(f)
ykτ,νχQkτ,ν(x) q
/q
∞
k=N+
k(s–s)q
/δ
×
∞
k=N+
k(s–s)q
τ∈Ik
N(k,τ)
ν=
ksqDk(f)
yk,ν
τ χQkτ,ν(x) q
δ/δ/q
∞
k=N+
k(s–s)q
/δ
×
∞
k=N+
k(s–s)q
τ∈Ik
N(k,τ)
ν=
ksqDk(f)
ykτ,νχQk,ν τ (x)
q
/δ/q
N(pω–
ω
p)
k∈Z
τ∈Ik
N(k,τ)
ν=
ksqDk(f)
ykτ,νχQkτ,ν(x) q
/q
. (.)
Combining the estimates in (.) and (.), we obtain
∞
k=N+
τ∈Ik
N(k,τ)
ν=
ksqDk(f)
ykτ,νχQk,ν τ (x)
q
/q
≤CN(pω–
ω
p)
k∈Z
τ∈Ik
N(k,τ)
ν=
ksqDk(f)
ykτ,νχQkτ,ν(x) q
/q
. (.)
Applying (.) and (.) yields
f p˙ Fps,q(X)
=p
∞
tp–μ
x:
∞
k=–∞
τ∈Ik
N(k,τ)
ν=
ksqD k(f)
yk,ν
τ χQkτ,ν(x) q
/q
>t
dt
=p
∞
N=–∞
Cω(N+)/p+/q
CωN/p+/q
tp–
×μ
x:
∞
k=–∞
τ∈Ik
N(k,τ)
ν=
ksqDk(f)
ykτ,ν
q
χQk,ν τ (x)
/q
>t
≤p
∞
N=–∞
Cω(N+)/p+/q
CωN/p+/q
tp–
×μ
x:
N
k=–∞
τ∈Ik
N(k,τ)
ν=
ksqDk(f)
ykτ,ν
q
χ
Qkτ,ν(x) /q
+
∞
k=N+
τ∈Ik
N(k,τ)
ν=
ksqDk(f)
ykτ,ν
q
χQk,ν τ (x)
/q
> –/qt
dt
≤p
∞
N=–∞
Cω(N+)/p+/q
CωN/p+/q t
p– ×μ x: ∞
k=N+
τ∈Ik
N(k,τ)
ν=
ksqDk(f)
ykτ,ν
q
χ
Qkτ,ν(x) /q
> –/qt/
dt
≤p
∞
N=–∞
Cω(N+)/p+/q
CωN/p+/q t
p– ×μ x: ∞
k=–∞
τ∈Ik
N(k,τ)
ν=
ksqDk(f)
ykτ,ν
q
χ
Qkτ,ν(x) /q
>C–N(pω–
ω
p)–/qt/
dt
≤p
∞
N=–∞
Cω(N+)/p+/q
CωN/p+/q
tp–
×μ
x:
∞
k=–∞
τ∈Ik
N(k,τ)
ν=
ksqDk(f)
ykτ,ν
q
χQk,ν τ (x)
/q
>Ctp/p
dt
≤p
∞
tp–
×μ
x:
∞
k=–∞
τ∈Ik
N(k,τ)
ν=
ksqDk(f)
ykτ,ν
q
χQk,ν τ (x)
/q
>Ctp/p
dt
≤p
∞
up–μ
x:
∞
k=–∞
τ∈Ik
N(k,τ)
ν=
ksqDk(f)
yk,ν
τ q
χQk,ν τ (x)
/q
>Cu
du
≤C f p˙ Fps,q(X)
≤C,
where we used the fact thatt≈ωN/p.
This proves (.),i.e.,F˙sp,q → ˙Fsp,q withq>+pp. For anyqwithmax{ωω+θ,ω+θω+s}<
q≤ +pp, it is easy to see thatF˙sp,q→ ˙Fsp,q forq> +pp. The proof of Theorem . is
completed.
Competing interests
