<span class="mathjax-formula">$H^{\bf 1}$</span> and <span class="mathjax-formula">$BMO$</span> for certain locally doubling metric measure spaces

H

¹

and BMO for certain locally doubling metric measure spaces

ANDREACARBONARO, GIANCARLOMAUCERI ANDSTEFANOMEDA

Abstract. Suppose that(M,ρ,µ)is a metric measure space, which possesses two

“geometric” properties, called “isoperimetric” property and approximate mid- point property, and that the measureµ is locally doubling. The isoperimetric property implies that the volume of balls grows at least exponentially with the radius. Hence the measureµis not globally doubling. In this paper we de- fine an atomic Hardy spaceH¹(µ), where atoms are supported only on “small balls”, and a corresponding spaceB M O(µ)of functions of “bounded mean os- cillation”, where the control is only on the oscillation over small balls. We prove that B M O(µ)is the dual of H¹(µ)and that an inequality of John–Nirenberg type on small balls holds for functions inB M O(µ). Furthermore, we show that the L^p(µ)spaces are intermediate spaces between H¹(µ)and B M O(µ), and we develop a theory of singular integral operators acting on function spaces on M. Finally, we show that our theory is strong enough to give H¹(µ)-L¹(µ) andL^∞(µ)-B M O(µ)estimates for various interesting operators on Riemannian manifolds and symmetric spaces which are unbounded onL¹(µ)and onL^∞(µ).

Mathematics Subject Classification (2000):42B20 (primary); 42B30, 46B70, 58C99 (secondary).

1.

Introduction

Suppose that(M, ρ, µ)is a metric measure space. Assume temporarily thatµis a doubling measure; then(M, ρ, µ)is a space of homogeneous type in the sense of Coifman and Weiss. Harmonic analysis on spaces of homogeneous type has been the object of many investigations. In particular, the atomic Hardy space H¹(µ)and the space B M O(µ) of functions of bounded mean oscillation have been defined and studied in this setting. We briefly recall their definitions.

An atomais a function inL¹(µ)supported in a ballBwhich satisfies appropri- ate “size” and cancellation conditions. Then H¹(µ)is the space of all functions in L¹(µ)that admit a decomposition of the form

jλja_j, where thea_j’s are atoms and the sequence of complex numbers{λj}is summable.

Work partially supported by the Progetto Cofinanziato “Analisi Armonica”.

Received October 12, 2008; accepted December 12, 2008

A locally integrable function f is inB M O(µ)if sup

B

1 µ(B)

B

|f − f_B| dµ <∞,

where the supremum is taken over allballs B, and f_B denotes the average of f over B.

These spaces enjoy many of the properties of their Euclidean counterparts. In particular, the topological dual ofH¹(µ)is isomorphic toB M O(µ), an inequality of John–Nirenberg type holds for functions in B M O(µ), the spaces L^p(µ) are intermediate spaces between H¹(µ) and B M O(µ) for the real and the complex interpolation methods. Furthermore, some important operators, which are bounded on L^p(µ)for all pin(1,∞), but otherwise unbounded onL¹(µ)and onL^∞(µ), turn out to be bounded from H¹(µ) toL¹(µ)and from L^∞(µ) to B M O(µ). We remark that the doubling property is key in establishing these results.

There is a huge literature on this subject: we refer the reader to [15,42] and the references therein for further information.

There are interesting cases where µis not doubling; then µmay or may not be locally doubling. An important case in whichµis not even locally doubling is that of nondoubling measures of polynomial growth treated, for instance, in [38,44, 46], where new spaces H¹ andB M O are defined, and a rich theory is developed (see also [34] for more general measures on

R

ⁿ). We also mention recent works of X.T. Duong and L. Yan [19, 20], who define a Hardy space H¹ and a space B M Oof bounded mean oscillation associated to a given operator satisfying suitable estimates. This is done in metric measure spaces with the doubling property, but it is a remarkable fact that the theory works also for “bad domains” in the ambient space, to which the restriction of the measureµmay be nondoubling.

In this paper we consider the case where µ(M) = ∞, and µis a nondou- bling locally doubling measure. By this we roughly mean that for every R in

R

⁺ balls of radius at most Rsatisfy a doubling condition, with doubling constant that may depend on R(see (2.1) in Section 2 for the precise definition). Important ex- amples of this situation are complete Riemannian manifolds with Ricci curvature bounded from below, a class which includes all Riemannian symmetric spaces of the noncompact type and Damek–Ricci spaces. In recent years, analysis on com- plete Riemannian manifolds satisfying the local doubling condition has been the object of many investigations. For instance, see [40] and the references therein for the equivalence between a scale-invariant parabolic Harnack inequality and a scaled Poincar´e inequality, and [2, 16, 39] for recents results on the boundedness of Riesz transforms on such manifolds.

Our approach to the case of locally doubling measures is inspired by a result of A.D. Ionescu [29] on rank one symmetric spaces of the noncompact type and by a recent paper of the second and third named authors concerning the analysis of the Ornstein–Uhlenbeck operator [35].

For each “scale”bin

R

⁺, we define spacesH_b¹(µ)andB M O_b(µ)much as in the case of spaces of homogeneous type, the only difference being that we require

that the balls involved have at most radiusb. So, for instance, an H_b¹(µ)atom is an atom supported in a ball of radius at mostb. We remark that in the case whereMis a symmetric space of the noncompact type and real rank one, the space B M O₁(µ) agrees with the space defined by Ionescu. Ionescu also proved that if pis in(1,2), thenL^p(µ)is an interpolation space betweenL²(µ)andB M O₁(µ)for the complex method of interpolation. In the case where M is a complete noncompact Rieman- nian manifold with locally doubling Riemannian measure and satisfying certain ad- ditional assumptions E. Russ [39] defined a Hardy space that agrees with the space

H₁¹(µ)defined above, but he did not investigate its structural properties.

We prove that under a mild “geometric” assumption, which we call property (AM) (see Section 2), H_b¹(µ) and B M O_b(µ), in fact, do not depend on the pa- rameterbprovided thatbis large enough (see Section 4). Furthermore, we show that B M O_b(µ) is isomorphic to the topological dual of H_b¹(µ) (see Section 6), and that functions in B M O_b(µ)satisfy an inequality of John–Nirenberg type (see Section 5).

As far as interpolation is concerned, there is no reason to believe that in this generalityL^p(µ)spaces with pin(1,∞)are interpolation spaces betweenH_b¹(µ) and B M Ob(µ). However, a remarkable feature of these spaces is that this is true under a simple geometric assumption onM, called property (I). Roughly speaking, M possesses property (I) if a fixed ratio of the measure of any bounded open set is concentrated near its boundary. If M possesses property (I), then a basic relative distributional inequality for the local sharp function and the local Hardy–Littlewood maximal function holds. We prove this in Section 7, by adapting to our setting some ideas of Ionescu [29]. We remark that our approach, which makes use of the dyadic cubes of M. Christ and G. David [14, 18], simplifies considerably the original proof in [29]. As a consequence of the relative distributional inequality we prove an interpolation result for analytic families of operators, analogous to that proved by C. Fefferman and E.M. Stein [23] in the classical setting.

An interesting application of the aforementioned interpolation result is to sin- gular integral operators (Theorem 8.2). We prove that if

T

is a bounded self adjoint operator on L²(µ)and its kernelk is a locally integrable function off the diagonal in M × M and satisfies alocal H¨ormander type condition(i.e. if νk < ∞ and υk < ∞, whereνk andυk are defined in the statement of Theorem 8.2), then

T

extends to a bounded operator onL^p(µ)for all pin(1,∞), from H¹(µ)toL¹(µ) and fromL^∞(µ)to B M O(µ).

It is interesting to speculate about the range of applicability of the theory we develop. In particular, a natural problem is to find conditions (possibly easy to verify) under which a complete Riemannian manifold possesses all the three prop- erties, local doubling, (I), and (AM), needed to prove the results of Sections 2-8.

This problem is considered in Section 9. Suppose thatMis a complete Riemannian manifold with Riemannian distance ρand Riemannian densityµ. A known fact, which is a straightforward consequence of the Bishop–Gromov comparison Theo- rem, is that ifMhas Ricci curvature bounded from below, then(M, ρ, µ)is locally doubling. Furthermore, sinceρis a length distance,(M, ρ, µ)has property (AM).

We shall prove thatMpossesses property (I) if and only if the Cheeger isoperimet-

ric constanth(M)(see (9.1) for the definition) is strictly positive. As a consequence we shall prove that ifMhas Ricci curvature bounded from below, thenMpossesses property (I) if and only if the bottomb(M)of the spectrum ofMis strictly positive.

In Section 10 we apply our theory to obtain endpoint estimates for some opera- tors on Riemannian manifolds and symmetric spaces. We show that, if the manifold has Ricci curvature bounded from below and the bottom of the spectrum is positive, then a class a spectral multipliers of the Laplacian is bounded fromH¹(µ)toL¹(µ) and fromL^∞(µ)to B M O(µ). Similar endpoint estimates hold also for a class of spherical multipliers and for “localized” Riesz transforms on noncompact Rieman- nian symmetric spaces. Our results complement earlier results of M. Taylor [43], J.Ph. Anker [1] and Russ [39]. Similar results on graphs with bounded geometry will appear elsewhere.

To keep the length of this paper reasonable we have considered only the case where µ(M) < ∞. A detailed study of the case whereµ(M) < ∞will appear in [11].

Finally, we would like to mention that, after this paper was completed, M. Tay- lor kindly sent us a preprint in which he develops a theory of “local” Hardy and B M Ospaces on Riemannian manifolds which satisfy bounded curvature assump- tions stronger than those considered in Section 9. Taylor’s theory is a generalization of the theory developed by D. Goldberg in

R

ⁿ[24]. We recall that the definition of the local atomic Hardy space

h

¹(

R

ⁿ⁾of Goldberg is similar to that of the classical atomic Hardy space H¹(

R

ⁿ⁾; the only difference is that atoms supported in balls of radius larger than one do not necessarily have mean value zero. The dual of

h

¹(

R

ⁿ⁾ is the space

bmo

(

R

ⁿ⁾of functions f such that both the mean oscillation on balls of radius at most one and the average of |f| on balls of radius one are bounded.

By imitating Goldberg’s definition, one can define local spaces

h

¹(µ)and

bmo

(µ) also in the context of a locally doubling metric measure space(M, ρ, µ)that satis- fies assumption (AM). It is easy to see that

h

¹(µ)is strictly larger than our H¹(µ) and

bmo

(µ)is strictly smaller than our B M O(µ). For instance, the characteristic function of a ball is in H¹(µ) but not in

h

¹(µ) and the function x → ρ(x,x₀), wherex₀is a fixed point inM, is inB M O(µ)but not in

bmo

_(µ). It is known that in

R

ⁿ there are operators,e.g. the imaginary powers of the Laplacian, which are bounded from the classical space H¹(

R

ⁿ)to L¹(

R

ⁿ)but fail to be bounded from

h

¹(

R

^n)toL1(

R

ⁿ⁾. The same phenomenon occurs in our context: there are metric measure spaces in which “natural” operators are bounded fromH¹(µ)toL¹(µ)but not from

h

¹(µ)to L¹(µ). For instance, this happens for the imaginary powers of the Ornstein-Uhlenbeck operator on Gauss space [35].

2.

Geometric assumptions

Suppose that (M, ρ, µ)is a metric measure space, and denote by

B

the family of all balls on M. We assume thatµ(M) >0 and that every ball has finite measure.

For eachBin

B

we denote byc_Bandr_Bthe centre and the radius ofBrespectively.

Furthermore, we denote byκBthe ball with centrec_B and radiusκr_B. For eachb

in

R

⁺, we denote by

B

bthe family of all balls B in

B

^{such thatr}B ≤b. For any subset AofMand eachκin

R

⁺we denote byA_κ andA^κthe sets

x ∈ A:ρ(x,A^c)≤κ

and

x∈ A:ρ(x,A^c) > κ respectively.

In Sections 2-8 we assume that M isunboundedand possesses the following properties:

(i) local doubling property(LD): for everybin

R

⁺there exists a constantD_bsuch that

µ 2B

≤ D_bµ B

∀ B∈

B

b. (2.1)

This property is often calledlocal doubling conditionin the literature, and we adhere to this terminology. Note that if (2.1) holds and Mis bounded, thenµis doubling.

(ii) isoperimetric property(I): there existκ0andCin

R

⁺such that for every bounded open set A

µ A_κ

≥Cκ µ(A) ∀κ ∈(0, κ0]. (2.2) Suppose that M has property (I). For each t in (0, κ0] we denote by C_t the supremum over all constantsCfor which (2.2) holds for allκin(0,t]. Then we define I_Mby

I_M =sup

C_t :t ∈(0, κ0] .

Note that the functiont →C_t is decreasing on(0, κ0], so that I_M = lim

t→0⁺

C_t; (2.3)

(iii) property (AM) (approximate midpoint property): there exist R₀ in[0,∞)and β in (1/2,1)such that for every pair of pointsx and y in M withρ(x,y) >

R₀ there exists a point z in M such that ρ(x,z) < β ρ(x,y) andρ(y,z) <

β ρ(x,y).

This is clearly equivalent to the requirement that there exists a ball Bcontain- ingxandysuch thatr_B < β ρ(x,y).

Remark 2.1. Observe that the isoperimetric property (I) implies that for every open set Aoffinite measure

µ A_κ

≥Cκ µ(A) ∀κ ∈(0, κ0], whereκ0andC are as in (2.2).

Indeed, suppose that Ais an open set of finite measure. Fix a reference pointo inM and denote byB(o, j)the ball with centreoand radius j, and byA(j)the set

A∩B(o, j). For eachκin(0, κ0]denote by A_j,κ the set x ∈ A(j):ρ

x,B(o,j)^c

≤κ, ρ(x,A^c) > κ .

First we prove that

lim

j→∞µ A_j,κ

=0. (2.4)

Sinceµ(A) <∞, for each >0 there existsJsuch that µ

A∩B(o,J)^c

< .

Now, if j ≥ J +κandxis in A_j,κ, thenx belongs also to A∩ B(o,J)^c, whence µ(A_j,κ) < for all j ≥ J, as required.

Observe thatA_j,κis contained inA(j)_κand thatA_j,κ = A(j)_κ\

B(o,j)∩A_κ . Therefore

µ

B(o, j)∩ A_κ

=µ A(j)_κ

−µ A_j,κ

. (2.5)

Sinceµ(A_κ)=lim_j→∞µ

B(o,j)∩A_κ , µ(A_κ)= lim

j→∞µ A(j)_κ

by (2.5) and (2.4). Since A(j)is a bounded open set, we may conclude that µ(A_κ)≥ lim

j→∞Cκ µ A(j)

=Cκ µ A

, as required.

Remark 2.2. The local doubling property is needed for all the results in this paper, but many results in Sections 2-8 depend only on some but not all the properties (i)- (iii). In particular, Theorem 3.2 and Proposition 3.4 require only the local doubling property, Propositions 3.1 and 3.5, Lemma 7.2 and Theorem 7.3, which are key in proving the interpolation result Theorem 7.4, require property (I), but not property (AM), all the results in Sections 4, 5 and 6 require property (AM) but not property (I). In particular, property (AM) is key to prove the scale invariance of the spaces H¹(µ) and B M O(µ) defined below (Proposition 4.3). Finally, all the properties (i)-(iii) above are needed for the interpolation results in Section 7 and for the results in Section 8.

Remark 2.3. The local doubling property implies that for eachτ ≥2 and for each bin

R

⁺there exists a constantC such that

µ B

≤Cµ(B) (2.6)

for each pair of balls Band B, with B ⊂ B,B in

B

b, andr_B ≤ τr_B. We shall denote by D_τ,b the smallest constant for which (2.6) holds. In particular, if (2.6) holds (with the same constant) for all balls Bin

B

^{, then}µis doubling and we shall denote by D_τ,∞the smallest constant for which (2.6) holds.

Remark 2.4. There are various “structural constants” which appear explicitly in the statements of some of our results. For the reader’s convenience we give here a list of all the relevant constants used is Sections 2-8:

D_τ,b τ ≥2,b∈

R

^{+∪ {∞}}(see Remark 2.3) I_M the isoperimetric constant (see (ii) above) R₀andβ appear in the (AM) property (see (iii) above)

δ,C₁anda₀ appear in the construction of dyadic cubes (see Theorem 3.2).

3.

Preliminary results

Roughly speaking, if M has property (I), then a fixed ratio of the measure of any bounded open set is concentrated near its boundary. The following proposition contains a quantitative version of this statement.

Proposition 3.1. The following hold:

(i) the volume growth of M is at least exponential;

(ii) for every bounded open set A µ(A_t)≥

1−e^−IMt

µ(A) ∀t ∈

R

^+.

Proof. First we prove (i). Denote byoa reference point in M. For everyr > 0 denote byV_rthe measure of the ball with centreoand radiusr. It is straightforward to check that B(o,r)κ ⊂B(o,r)\B(o,r−κ), so that for all sufficiently larger

V_r −V_r−κ ≥µ

B(o,r)_κ

≥CκV_r ∀κ∈(0, κ0] by property (I) (C andκ0are as in (2.2)). Hence

V_r ≥CκV_r+V_r−κ

≥ηV_r−κ ∀κ∈(0, κ0],

whereη=Cκ+1. Denote bynthe positive integer for whichr−nκis in[κ,2κ). Then

V_r ≥ηⁿV_r−nκ

≥η^r/κ−2V_κ, as required.

Now we prove (ii). Suppose that Ais a bounded open subset ofM, and thatt is in(0, κ0]. Note that A^s is a bounded open subset of M. It is straightforward to check that(A^s)t ⊂ A^s\A^s+t. Therefore

µ(A^s+t)−µ(A^s)≤ −µ (A^s)t

≤ −C_ttµ(A^s)

by property (I) (see (2.2) above). Sinces →µ(A^s)is monotonic, it is differentiable almost everywhere. The inequality above and (2.3) imply that for almost everysin

R

⁺

d

dsµ(A^s)≤ −I_Mµ(A^s).

Notice also that lim_s→0+µ(A^s)=µ(A). Therefore µ(A^s)≤e^−IMsµ(A) ∀s∈

R

^+, and finally

µ(A_s)≥

1−e^−IMs

µ(A) ∀s∈

R

^+, as required.

We shall make use of the analogues in our setting of the so-called dyadic cubes Q^k_αintroduced by G. David and M. Christ [14, 18] on spaces of homogeneous type.

It may help to think ofQ^k_αas being essentially a cube of diameterδ^kwith centrez_α^k. Theorem 3.2. There exist a collection of open subsets{Q^k_α : k ∈

Z

^{, α ∈ Ik}and} constantsδin(0,1), a₀, C₁in

R

^{+such that}

(i)

αQ^k_αis a set of full measure in M for each k in

Z

^; (ii) if≥k, then either Q_β ⊂ Q^k_αor Q_β∩Q^k_α= ∅;

(iii) for each(k, α)and each <k there is a uniqueβsuch that Q^k_α⊂ Q_β; (iv) diam(Q^k_α)≤C₁δ^k;

(v) Q^k_αcontains the ball B(z^k_α,a₀δ^k).

Proof. The proof of (ii)–(v) is as in [14]. In fact, the proof depends only on the metric structure of the space and not on the properties of the measureµand is even easier in our case, becauseρis a genuine distance, rather than a quasi-distance.

The proof of (i) is again as in [14]; observe that only a local doubling property is used in the proof.

Note that (iv) and (v) imply that for every integerkand eachαin I_k B(z^k_α,a₀δ^k)⊂ Q^k_α ⊂ B(z^k_α,C₁δ^k).

Remark 3.3. When we use dyadic cubes, we implicitly assume that for eachkin

Z

^{the setM\}_α∈Ik Q^k_αhas been permanently deleted from the space.

We shall denote by

Q

^k the class of all dyadic cubes of “resolution”k,i.e., the family of cubes{Q^k_α:α∈ I_k}, and by

Q

the set of all dyadic cubes. We shall need the following additional properties of dyadic cubes.

Proposition 3.4. Suppose that b is in

R

^{+and thatν is in}

Z

, and let C₁andδare as in Theorem3.2. The following hold:

(i) suppose that Q is in

Q

^{kfor some k}≥ν, and that B is a ball such that c_B ∈Q.

If r_B ≥C₁δ^k, then

µ(B∩Q)=µ(Q); (3.1)

if r_B <C₁δ^k, then

µ(B∩Q)≥D⁻_C¹

1/(a₀δ),a₀δ^νµ(B); (3.2) (ii) suppose thatτ is in[2,∞). For each Q in

Q

^{the spaceQ}, ρ_|Q, µ_|Q

is of homogeneous type. Denote by D_τ,∞^Q its doubling constant (see Remark 2.3 for the definition). Then

sup

D_τ,∞^Q : Q∈ ^∞

k=ν

Q

^k

≤ D_τ,C1δν D_{C1/(a0δ),a0δ}ν;

(iii) for each ball B in

B

b, let k be the integer such thatδ^k ≤r_B < δ^k−1, and and letB denote the ball with centre c_B and radius

1+C₁

r_B. ThenB contains all dyadic cubes in

Q

^kthat intersect B and

µ(B)≤D_1+C1,bµ(B);

(iv) suppose that B is in

B

^b, and that k is an integer such thatδ^k ≤r_B < δ^k−1. Then there are at most D_{(1+C1)/(a0δ),b}dyadic cubes in

Q

^kthat intersect B.

Proof. First we prove (i). Our proof is a version of the proof given by Christ [14, page 613] that keeps track of the various structural constants involved.

First we prove (3.1). By Theorem 3.2 (iv) the diameter of Qis at mostC₁δ^k, so that Q⊂ B, whenceB∩Q= Q, and the required formula is obvious.

To prove (3.2), denote by j the unique integer such that δ^j < r_B

C₁ ≤δ^j−1

and by Q_β^j the unique dyadic cube of resolution j that containsc_B. Then j ≥ k, becauseC₁δ^j < r_B ≤ C₁δ^k. The cubes Q_β^j and Q have nonempty intersection, because, they both containc_B. Thus Q_β^j ⊂ Q. By Theorem 3.2 (iv) the diameter of Q_β^j is at most C₁δ^j, which is < r_B by the definition of j, so that Q_β^j ⊂ B.

ThereforeQ_β^j ⊂ B∩Q, and

µ(B∩Q)≥µ Q_β^j

≥µ

B(z_β^j,a₀δ^j) .

Observe that _a^rB

0δ^j ≤ _a^C₀¹_δ and that B(z_β^j,a₀δ^j)⊂B. Hence µ

B(z_β^j,a₀δ^j)

≥ D_C⁻¹

1/(a₀δ),a₀δ^νµ(B), as required to conclude the proof of (3.2), and of (i).

Next we prove (ii). Suppose that Q is a dyadic cube in

Q

^{k, with k} ≥ ν. Suppose that Band B are balls in

B

^withB ⊂ B such thatc_B andc_B belong to Q andr_B ≤τr_B. We treat the cases whereC₁δ^k ≤r_Bandr_B <C₁δ^k separately.

IfC₁δ^k ≤r_B, thenµ(B ∩Q)=µ(Q)=µ(B∩Q). Ifr_B <C₁δ^k, then

µ(B ∩ Q)≤µ(B)

≤D_τ,C1δkµ(B)

≤D_τ,C1δk D_{C1/(a0δ),a0δ}ν µ(B∩Q),

by the local doubling property ofMand (3.2). ThereforeQis a homogeneous space with doubling constant at most D_τ,C1δk D_{C1/(a0δ),a0δ}ν. Sincek≥ν, these doubling constants are dominated by D_τ,C1δν D_{C1/(a0δ),a0δ}ν, as required.

Now we prove (iii). Denote byQa cube in

Q

^kthat intersectsB. By the triangle inequality and Theorem 3.2 (iv), Qis contained inB. The required estimate of the measure of Bfollows from the local doubling condition (see Remark 2.3).

Finally we prove (iv). Denote by Q₁, . . . ,Q_N the cubes in

Q

^k that intersect B. By (iii) each of these cubes is contained in B. By Theorem 3.2 (v) each cube Q_j contains a ball, B_j say, of radius a₀δ^k, and these balls are pairwise disjoint because they are contained in disjoint dyadic cubes. By the local doubling condition µ(B)≤D_{(1+C1)/(a0δ),b}µ(B_j)for all j. Therefore

Nµ(B)≤ D_{(1+C1)/(a0δ),b} N

j=1

µ(B_j)

= D_{(1+C1)/(a0δ),b}µ _N

j=1

B_j

≤ D_{(1+C1)/(a0δ),b}µ(B), from which the desired estimate follows.

Our next result is a covering property enjoyed by spaces with property (I). It is key in proving relative distributional inequalities for the sharp maximal operator (see Lemma 7.2 below).

Proposition 3.5. Suppose thatνis an integer. For everyκin

R

⁺, every open subset A of M of finite measure and every collection

C

of dyadic cubes of resolution at least νsuch that

Q∈CQ = A, there exist mutually disjoint cubes Q₁, . . . ,Q_kin

C

^such that

(i) k

j=1

µ(Q_j)≥

1−e^−IMκ

µ(A)/2;

(ii) ρ(Q_j,A^c)≤κfor every j in{1, . . . ,k}.

Proof. Denote by

C

the subcollection of all cubes in

C

that intersect A_κ. Clearly the cubes in

C

^{cover A}κ and satisfy (ii).

Next we prove (i). Since two dyadic cubes are either disjoint or contained one in the other, we may consider the sequence{Q_j}of cubes in

C

which are not con- tained in any other cube of

C

. The existence of these “maximal” cubes is guaranteed by the assumption that the resolution of the cubes in

C

is bounded from below. The cubes{Q_j}are mutually disjoint and coverA_κ. Therefore

µ(A)≥

j

µ(Q_j)

≥µ A_κ

≥

1−e^−IMκ µ(A),

(3.3)

where the last inequality holds because M possesses property (I). To conclude the proof of (i) takekso large that_k

j=1µ(Q_j)≥(1/2)_∞

j=1µ(Q_j). Then k

j=1

µ(Q_j)≥

1−e^−IMκ

µ(A)/2, as required.

4.

The Hardy space

H¹

Definition 4.1. Suppose thatr is in(1,∞]. A(1,r)-atom ais a function inL¹(µ) supported in a ball Bin

B

with the following properties:

(i) a_∞ ≤µ(B)⁻¹ifr is equal to∞and 1

µ(B)

B

|a|^r dµ _1/r

≤µ(B)⁻¹ ifr is in(1,∞);

(ii)

B

adµ=0.

Definition 4.2. Suppose thatbis in

R

^{+. The}Hardy space H_b^1,r(µ)is the space of all functionsginL¹(µ)that admit a decomposition of the form

g=^∞

k=1

λka_k, (4.1)

wherea_k is a(1,r)-atomsupported in a ball B of

B

b, and_∞

k=1|λk| < ∞. The norm g_H^1,r

b (µ) ofg is the infimum of _∞

k=1|λk| over all decompositions (4.1) ofg.

Clearly a function in H_c^1,r(µ) is in H_b^1,r(µ)for allc < b. We shall prove in Proposition 4.3 below that, in fact, the reverse inclusion holds whenevercis large enough. Hence forblarge the spaceH_b^1,r(µ)does not depend on the parameterb, and for each pair of sufficiently large parametersbandcthe norms·_H1,r

b (µ) and

·_H1,r

c (µ)are equivalent.

There are cases where H_c^1,r(µ)and H_b^1,r(µ) are isomorphic spaces for each pair of parameterscandb. This happens, for instance, ifMis the upper half plane, ρ the Poincar´e metric andµthe associated Riemannian measure. However, if M is a homogeneous tree of degreeq ≥ 1,ρ denotes the natural distance andµthe counting measure, it is straightforward to check that H₁¹(µ) consists of the null function only, whereas H₂¹(µ)is a much richer space.

Recall that R₀ andβ are the constants which appear in the definition of the (AM) property.

Proposition 4.3. Suppose that r is in(1,∞], b and c are in

R

⁺and satisfy R₀/(1− β) <c<b. The following hold:

(i) there exist a constant C and a nonnegative integer N , depending only on M, b and c, such that for each ball B in

B

b and each(1,r)-atom a supported in B there exist at most N (1,r)-atoms a₁, . . . ,a_N with supports contained in balls B₁, . . . ,B_N in

B

c and N constantsλ1, . . . , λN such thatλj≤C,

a= N

j=1

λja_j and a_Hc^1,r_(µ)≤C N;

(ii) a function f is in H_c^1,r(µ)if and only if f is in H_b^1,r(µ). Furthermore, there exists a constant C such that

f_H^1,r

b (µ)≤ f_H^1,r

c (µ)≤Cf_H^1,r

b (µ) ∀ f ∈ H_c^1,r(µ).

Proof. Chooseβ in(0,1−β)such thatR₀/β <c.

First we prove (i). Suppose thatBis a ball in

B

^{band thatrB} >c, for otherwise there is nothing to prove. Denote by{z₁, . . . ,z_N1}a maximal set of points inBsuch thatρ(z_j,z_k)≥βr_B for all j =kand each point ofBis at distance at mostβr_B from the set{z₁, . . . ,z_N1}. Denote byB_j the ball with centrez_j and radiusβr_B, and by B₀the ball with centrec_B and radiusβr_B. Note that

µ(B)≤D_1/β,bµ(B₀), (4.2)

where D_1/β,bis as in Remark 2.3. We consider the partition of unity{ψ1, . . . ,ψN1} of1

jB_jsubordinated to the covering{B₁, . . . ,B_N1}defined byψj=1_Bj/^N1

k=1

1_Bk. For each j in{1, . . . ,N₁}we define

A_j = 1 µ(B₀)

M

aψjdµ and φj =aψj− A_j1_B0. It is straighforward to check thata=N₁

j=1φj and each functionφj has integral 0 and is supported inB_j ∪B₀. DefineJ andJ by

J = {j :z_j ∈/ B₀} and J = {j :z_j ∈B₀}.

If j is in J , thenz_j is not inB₀, so thatρ(c_B,z_j) > βr_B > βc> R₀. Therefore we may use property (AM) and conclude that there exists a ball B_j containingc_B andz_j with radius< β ρ(c_B,z_j). Denote byB_j the ball centred atc_B

j with radius r_B

j +βr_B. By using the triangle inequality we see that B_j contains B_j ∪ B₀. Observe thatr_B˜

j ≤(β+β)r_B, which is strictly less thanr_B, because we assumed thatβ <1−β.

Next we check that if j is in J , thenφj is a multiple of a(1,r)-atom: we give details in the case where r = 2; the cases wherer ∈ (1,∞)\ {2}may be treated similarly, and the variations needed to treat the case where r = ∞ are straightforward and are omitted. By the triangle inequality

1 µ(B_j)

B_j

φj² dµ 1/2

≤ 1

µ(B_j)

B_j

aψj² dµ 1/2

+A_j 1 µ(Bj)

Bj

1_B0dµ _1/2

≤

µ(B) µ(B_j)

1 µ(B)

B|a|² dµ 1/2

+

µ(B₀) µ(B_j)

1 µ(B₀)

M

aψjdµ

≤

µ(B)

µ(B₀)+ µ(B) µ(B₀)

1 µ(B)

≤ 2D_1/β,b µ(B) .

Observe that

1

µ(B) ≤ 1

µ(B₀) ≤ D_β/β+1,b µ(B_j) ,

becauseB₀is contained both inBandB_j and the ratio between the radii ofB_j and B₀is at mostβ/β +1. Therefore we may conclude that

1 µ(B_j)

B_j

φj² dµ 1/2

≤ 2D_1/β,bD_β/β+1,b µ(B_j) ,

i.e.,φj/(2D_1/β,bD_β/β+1,b)is an atom supported in the ballB_j of radius at most (β+β)r_B.

Now suppose that j is in J . Then B_j ∪ B₀ is contained in 2B₀. Notice that β <1−β <1/2, so thatr_2B0 =2βr_B < (β+β)r_B. By arguing much as above, we see thatφj/(2D_1/β,bD_β/β+1,b)is an atom supported in the ball 2B₀of radius

< (β+β)r_B.

We have writtenaas the sum ofN₁functionsφj, each of which is a multiple of an atom with constant 2D_1/β,bD_β/β+1,b. Thus, we have proved thata_H^1,r

b(β+β) ≤ 2D_1/β,bD_β/β+1,bN₁.

Now, if j is in J , and r_B˜

j < c, then B_j is in

B

c. Similarly, if j is in J , andr_2B0 < c, then 2B₀ is in

B

c. If 2B₀ and all the balls B_j are in

B

c, then the proof is complete. Otherwise either 2B₀or some of the B_j’s is not in

B

c, and we must iterate the construction above. It is clear that after a finite number of steps, depending on the ratiob/c, we end up with the required decomposition.

Next we prove (ii). Obviouslyf_H1,r

b ≤ f_H1,r

c , so we only have to show thatf_H^1,r

c ≤Cf_H^1,r

b

for some constantC depending only onbandcandM.

But this follows directly from (i).

Definition 4.4. Suppose thatris in(1,∞). Then for everybandcin

R

^{+such that} R₀/(1−β) <c<bthe spacesH_b^1,r(µ)andH_c^1,r(µ)are isomorphic (in fact, they contain the same functions) by Proposition 4.3 (ii), and will simply be denoted by

H^1,r(µ).

Later (see Section 6) we shall prove that H^1,r(µ)does not depend on the pa- rameterrin(1,∞), and we shall denoteH^1,r(µ)simply byH¹(µ).

5.

The space

B M O

Suppose thatqis in[1,∞). For each locally integrable function f we defineN_b^q(f) by

N_b^q(f)= sup

B∈Bb

1 µ(B)

B|f − f_B|^q dµ 1/q

,