VECTOR VALUED MEASURES OF BOUNDED MEAN OSCILLATION

The duality between H and BMO, the space of functions of bounded mean oscillation (see [JN]), was first proved by C. Fefferman (see [F], [FS]) and then other proofs of it were obtained. Using the atomic decomposition approach ([C], [L]) the author studied the problem of characterizing the dual space of H of vector-valued functions . In [B2] the author showed, for the case Ω = {|z| = 1} , that the expected duality result H-BMO holds in the vector valued setting if and only if X∗ has the Radon-Nikodym property. If we want to get a duality result valid for all Banach spaces we may consider vector valued measures (see [BT], where the vector valued Lp case is treated, for an explanation) and therefore to deal with the general case it was necessary to consider a new space of vector valued measures closely related to BMO (see[B1]). In this paper we shall study such space in little more detail and we shall consider the H-BMO duality for vector-valued functions in the more general setting of spaces of homogeneous type (see [CW]). Throughout the paper X will stand for a Banach space, Ω will be a space of homogeneous type (see definition in the preliminary section) and we write Lp(Ω, X) for the space of measurable functions on Ω with values in X such that ‖f(x)‖ belongs to Lp(Ω). As usual C will denote a constant not necessarily the same at each occurrence.


Introduction
The duality between Hl and BMO, the space of functions of bounded mean oscillation (see [JN]), was first proved by C. Fefferman (see [F], [FS]) and then other proofs of it were obtained.Using the atomic decomposition approach ( [C], [L]) the author studied the problem of characterizing the dual space of Hl of vector-valued functions .In [B2] the author showed, for the case SZ = {Iz1 = 1}, that the expected duality result Hl-BMO holds in the vector valued setting if and only if X* has the Radon-Nikodym property.If we want to get a duality result valid for all Banach spaces we may consider vector valued measures (see [BT], where the vector valued L7, case is treated, for an explanation) and therefore to deal with the general case it was necessary to consider a new space of vector valued measures closely related to BMO (see [B1]) .
In this paper we shall study such space in little more detail and we shall consider the H'-BMO duality for vector-valued functions in the more general setting of spaces of homogeneous type (see [CW]) .
Throughout the paper X will stand for a Banach space, 9 will be a space of homogeneous type (see definition in the preliminary section) and we write Lp (S2, X) for the space of measurable functions on 9 with values in X such that lif(x)jj belongs to L,(Q) .As usual C will denote a constant not necessarily the same at each occurrence.

. Preliminaries
A space of homogeneous type 52 is a topological space endowed with a Borel measure m and a quasi-distance d, that is d : X x X -+ ué -} with a) From (1 .0)we can assume that 0 < m(B) < oo for every ball B (otherwise m would be identically 0 or oo) and therefore m is a o-finite measure on 9. Denote by Eo the ring of bounded measurable sets.The o-finiteness condition implies that the o-algebra generated by Eo coincides with the Borel o-algebra that we shall denote by E .
Let us now recall the notion of atom with values in X.Given In the case m(9) < oo the constant function *.~(s~) b, where b E X with Ilbll = 1, is also considered as a (X,p)-atom.Note that the atoms are in the unit ball of L1 (St, X) .
Following [CW] we define Hp(9, X) as the space of functions f in L, (Q, X ) admitting an atomic decomposition 00 where the aj's are (X,p)-atoms and Ei °o lAjl < oo .(The convergence of (1 .1) is taken in L1 (2, X) ).
We get a Banach space if we consider the norm 00 IlfllH; =infElAj 1 j-o where the infimum is taken over all representations f = J :i°o Ajaj .
Let us denote by li911m = sup{ (rra(B) 1B 119(x) -9B11 q dm(x))1/q : B ball} When m(S2) = oo then jIgliBMO, = ¡¡gil, q gives a norm on the set of equivalence classes of functions which differ by a constant in X.
Let us recall now a few definitions about vector-valued measures we shall use later on.Let (52, E, m) be any o-finite measure space, A a measurable set and 1 < p < oo.Given a vector valued measure G, we denote by IGI the variation of G, that is n (1 .4)IGI(A) = sup{j: JIG(E;)jj : (E ;) partition of A} i-1 and by IGI p (A) the p-variation on A, that is n JIG(Ei)11 P 1/p where the supremum is taken over all finite partitions (E;) of disjoint measurables sets contained in A with m(E;) > 0 .
For the case p = oo we shall denote by V'(Q, X) the space of X-valued measures G satisfying (1 .6) JIG(E)jj < C -(E) for all measurable set E Defining the norm by the infimum of the constants satisfying (1 .6)we get a Banach space.
Remark 1.1 .It is not hard to see that in fact JIG(E;)jj can be replaced by IGI(E ;) in the definition of p-variation .(See Lemma 1 in [H3]) Remark 1.2 .If G is a vector valued measure defined on E o which is absolutely continuous with respect to m, that is lim G(E) = 0, then it can be m(E)-0 extended to a measure on E, being still absolutely continuous with respect to m .(See [D], [DU]) We refer the reader to ( [DU], [D]) and to ( [J], [GC-RF]) for general theory and the properties we shall use about vector valued measures and Hardy spaces respectively.Definition 2.1.Let 1 <_ q < oo.Given a countably additive measure G defined on E and with values in X, it is said that G belongs to MBMO q (S2, X ) if where the supremun is taken over all balls B and over all finite partitions of B in pairwise disjoint measurable sets E; with m(Ej) > 0 .
When m(S2) = oo then IIGII MBMO, = IGI .,qgives a norm on the set of equivalence classes of measures : GI -G2 if there is b in X such that G, (E) - G2 (E) = bm(E) for all measurable set E .
It is obvious that if 1 < ql < q2 < oo then Remark 2.1.Let us assume G belong to MBMO q (S2,X) .Given a ball B and a measurable set E C B, it is quite immediate to find a constant CB depending on B satisfying IIG(E)II < CB max(m(E), m(E) 1-11 q) Suposse we consider B,, = {y E 9 : d(x o , y) < n} and denote by GB the measure G concentrated on Bn, that is GB (E) = G(EnBn ) .A glance at (2.3) allows us to say that for any 1 < q < oo if G belongs to MBMO q (SZ,X) then GB are necessarily absolutely continuous with respect to m and this clearly implies that also G is absolutely continuous with respect to m. (Recall that for vector-measures on a-algebras it suffices to check that they vanish on m-null sets) .
Proposition 2 .1.Le¡ 1 <_ q < oo, g be locally in L q (S2, X) and G be an Xvalued measure such that G(E) = fE g(x) dm(x) for all measurable bounded set.

E .
Then g belongs to BMO q (S2, X) if and only if G belongs lo MBMO q (S2, X) .
Moreover IIGIIMBMO, = II9IIBMO,-Proof: Given any ball B, consider GB coincides with the q-variation of GB on 9 divided by m(B) 1 /q and GB is a measure represented by the function (g -9B)XB, that is E Therefore the proposition follows from the equality between the q-variation and the norm in Lq of the function which represents the measure (see [D]) .
Remark 2.2.In general it is not true that any measure in MBMO q(S2,X) is representable by a function, this depends on the Radon-Nikodym property.We refer the reader to [B1] for the case Sl = {Izl = 1}, but a similar result and proof can be established also in this general setting.
Proposition 2 .2. Leí 1 _< q < oo .G belongs to MBMO,(Q,X) if and only if there exists a family of vectors in X, say {as : B ball}, such that n {(j where the supremum is taken oven all balls B and over all finite partitions of B in pairwise disjoini mensurable sets E; with m(E;) > 0 Proof: The direct implication is obvious by taking aB = m~B~.To show the converse let us assume that we have {as : B ball} with the above property, and notice that 11-Bm( B) II C for all B (simply take the partition of B given only by B).
Therefore for any B and any partition As in the case of functions we can define an equivalent norm in MBMO q(2, X ) .Let us take (2.5) IGI., q = sup { inf m(B)1/q I G -a ml q(B)}.

balIB -Ex
Note that essentially the same argument as in Proposition 2 .2 .shows the following (2.6)I GI*,q :5 IGI*,q :5 C IGI ., q Proposition 2.3 .Let 1 < q < oo .If G belongs to MBMO q f,X) then there exists a non negative function 0 in BMOq (Q) such that Moreover II0IIBMO, < C IIGII MEMO, Take a B Then = m B~, and observe that Proof.. Since G is countably additive and m-continuous then the same is true for the variation of G, IGI .Therefore using the Radon-Nikodym theorem there exists a non negative measurable function 0 which represents the measure IGI.
To show that 0 belongs to BMO,(Q), we shall use Propositions 2 .2 and 2 .1 .We simply have to find a family of real numbers {aB} such that The last inequality follows from Remark 1 .1 .

. The theorem and its proof
In the sequel 1 < p, q < oo, with r -f-9 = 1 .In this section we shall achieve the duality result between HP (S2, X) and MBMOg(S2, X*) .We shall need several lemmas before we prove the result .The next result was done in [B1] for the circle and for q = 2, and here we present a different approach which is valid for general spaces of homogeneous type.The author would like to point out that a similar and independent proof of the following lemma has been obtained by T. Wolniewicz (personal communication) .
Lemma 3.1 .Let G be a measure in MBMO q (S2, X) .Then for each integer n E N we can find a measure Gn in V'(Q,X) and a constant Cn satisfying IGni , ,q < Cn and such that (3 .1)IGI=,9 :5 lim Cn < KIGI .,9 for all measurable bounded set E.
n-oo Proof. .Using Proposition 2.3 we first get a function ~in BMO 9 (SZ) .
Denote by Pn = {x E X : «x) > n } and 0,, (x) = min(1, n/O(x» .Let us define now This, using Remark 1.2., allows to extend G n to E and shows that G n belongs to V,(9, X).
On the other hand Therefore if E is contained in some ball B Since OXB is in Ll (Q) then taking limit as n ---> oo shows (3.2) .From (2.6) we have finally to estimate m(B) -iwIG .-a mi 9 (B) fór all balls B .Using (3.4) we have that for any E C B If ¡¡al¡ < n then Therefore we have (3.5)I Gn -GI q(B) : IG -a m19(B fl s2n) 0 .BLASCO Though IGIq is not a measure for q > 1 the q-variation es subadditive and therefore we get that for all ¡la¡¡ < n (3 .6)m(B) -1/gIG namlq(B)<2m(B)-1 /qIGaml q (B) Denoting now by Dn = sup inf {m(B)-1/qIG-aMIq(B)} 6aNB Ijalj<n we get (3.1) for Cn = 2 C Dn where C is the constant appearing in (2.6) .
Notice that V'(Q, X*) can be obviously identified with the dual of L 1 (9, X) .Indeed any measure Gin V°°(52, X*) defines a functional TG acting on X-valued simple functions (which are dense in L1 (9, X) ) by the formula n n where <, > means duality between X and X* .
For such an atom we can write Therefore For a general atom a supported in B in Hp (S2, X) we can use approximation by simple functions in Lp (Q, X), and find a sequence of simple functions dk supported in B converging to a in Lp (f?, X), and take the sequence sk = (dk -fB dk(x) dm(x))XB which clearly also converges toa in Lp (S2,X) .Hence IIskilp _< 2IIajip for k large enough, and therefore sk/2 are "simple atoms" .
Using now that TG is continuous as operator on Ll (S2, X), and that sk converges to a in L, (S?, X), then (3.9) ITG(a)I = lim IT(sk)I = 2 lim IT(sk/2)I < 2IGI=,q For a general function f, take any representation of f in HI (S2, X), say f = ~j °_o Ajaj, where the al are (X,p)-atom and j:', IAjj < oo and notice that (3.8) follows from (3.9) and the fact that the series f = ~.i °o Ajaj is absolutely convergent in Ll (S2, X) what implies that TG(f) Ai TG(ai) .
Theorem 3 .1.Le¡ 1 < p, q < oo and ñ From the definition of Hp (S2, X) we can easily see that simple functions with support in balls are dense in the space, therefore it is enough to see that For the converse we shall deal first with the case m(S2) < oo .Let us take now a functional T in (Hp (S2, X)) * .Since constant functions are also considered as X-atoms in the case of finite measure we have that aXE E Hp (52, X), what allows us to define the following X* valued measure .
show that if E l IIbiliX = 1 then a is a (X,p)-atom.Therefore we obtain This shows that T defines a bounded functional on Lp (B, X) and hence from the Hahn-Banach extension theorem, we get an element in the dual of Lp (B, X) .The characterization of the dual space (Lp (B, X » * in terms of X*-valued measures of bounded q-variation allows us to find a measure GB with values in X* verifying (3 .13)T(f) = f dGB f E LP(B,X) B (Note that this measure is uniquely determined up to a measure F(E) = f m(Ef1B) for some 1 E X*) .Now ifwe take an increasing sequence of balls converging to 52, say B, and we determine GB by the assumption GB (B1 ) = 0, then we can construct a vector-valued measure on Eo , given by G(E) = GB (E) for E C B. .It is clear that GB are absolutely continuous and hence the same is true for G .Now from remark 1 .2we get an extension to E. For each f E Lp (B, X), consider a = 2,n(á)1/9 (f -fB)XB and therefore ,y) < K(d(x,z) + d(z,y)).*Partially suported by the grant C .A .I .C .Y .T .PB85-033 8 C .BLASCO and we assume that the halls B,(x) = {y E 2 : d(x, y) < r} form a basis of open neighborhoods of the point x and there exists a constant A satisfying (1 .0)m(B,(x)) < Am(Bg2(x)) 11) we first invoke Lemma 3.1, to find a sequence of measures Gn in V°°(S2, X *), that according to (3.2) verifies lim a-,>~T Gn (s) = TG(s) for all simple function supported in a ball.Secondly we use Lemma 3.2, together with (3.1) to get ITG(s)I < lim ITG.(s)I < C lim IGni*,gJISIIHI < n-oo n-oo n C lira Cn II81IHI < C IGI*,g1Is1IH-n~oo n * a <_ 2 ¡¡TI¡ .Since T and TG coincide over simple atoms, we have T = TG .On the othér hand IIG(Q)II :5 sup{ IT(bX9)I : IIbII _< 1 } < m(2) IITII and this finishes the proof for the finite measure case.It is elementary to Let us deal now with the case of m(9) = oo .Take a functional T in (HP (12, X» * and a ball B in S2.Let us consider the following space Lp(B,X) = {f E Lp (2,X) : supp f C B and f(x)dm(x) = 0} IBThe following function is an (X, p) his completes the proof.B .M .O .VECTOR VALUED MEASURES 129 jifijHp <_ m(B)1 /v jiflip IlTf11 C ¡¡TI¡ m(B) 1/9 lifllp Remark 3.1 .For 1 < p, r < oa, MBMO q (9, X) = MBA10,(S2, X) with equivalent norms