Blasco, O.

Revistes Catalanes amb Accés Obert

Publicacions Matemátiques, Vol 35 (1991), 119-130 .
VECTOR VALUED MEASURES OF
BOUNDED MEAN OSCILLATION
OSCAR BLASCO *
0. 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 .
1 . 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)
	
d(x,y) = d(y,x),
b) d(x, y) = 0 if and only if x = y,
c) d(x,y) < K(d(x,z) + d(z,y)).
*Partially suported by the grant C.A .I .C.Y.T . PB85-0338
120
	
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))
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 1 < p -< oo, a
function a in LP(Q,X) is called (X, p)-atom if
a) the support is contained in a ball B = B,(xo)
b) (m(B)
Álla(x)JIPdm(x»1/P<
m(B)
(p < 00)
b') Ila(x)II _< 1m(B)
m-a.e .
c) f
s
a(x) dm(x) = 0
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
(1 .1) f =EAjaj
j=0
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 .
The same arguments as in [CW] show that, in fact, for 1 < p, r < oo
(1 .2)
	
HP (S2, X) = H,1. (S2, X) (with equivalent norms) .
Let us also recall the definition of vector-valued BMO . Let 1 <_ q < oo, an X-
valued function which is locally in Lq(9, X) is said to belong to BMOq(S2, X)
provided that
(1 .3)
t
tá(m(B)Á119(x)-9Bligdm(x))1/q<C
where gB = m1B .fB g(x) dm(x) .
Let us denote by
B .M.O . VECTOR VALUED MEASURES 121
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 equiva-
lence classes of functions which differ by a constant in X.
For m(2) < oo we consider the norm 1191IBMO 9 = ¡¡9il*,q -f- 11 .f,, 9(x) dm(x)jj .
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 IGIp(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 measur-
ables 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 Eo which is abso-
lutely 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.
122
	
O. BLASCO
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 MBMOq (S2, X)
if
(2.3)
2 . Vector valued measures of bounded mean oscillation
G(Ej) - G(B) m(Ej) 1lq
IGI " ,a = supi~~ II m(Ei) m(B) IIq m(B) ) } <
o0
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 IIGIIMBMO, = IGI .,q gives 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 .
For m(S2) < oo we consider the norm IIGIIMBMO, = IGI .,q + IIG(Q)II .
It is obvious that if 1 < ql < q2 < oo then
(2.2) V-(9,X) C MBM0,(S2,X) C MBMO 91 (S2,X)
Remark 2.1 . Let us assume G belong to MBMOq (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 MBMOq (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 Lq(S2, X) and G be an X -
valued measure such that G(E) = fE g(x) dm(x) for all measurable bounded set.
E .
Then g belongs to BMOq (S2, X) if and only if G belongs lo MBMOq(S2, X) .
Moreover IIGIIMBMO, = II9IIBMO,-
Proof: Given any ball B, consider GB(E) = G(E (1 B) - m(B m(E (1 B) .
Observe that
n
G(1i) - G(B) m(Ei) 1/q , artition of Bsup{ ~~ II m(Ei) m(B)
II q m(B) ) (E¡ ) p }
B.M .O . VECTOR VALUED MEASURES
	
123
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
GB(E) = f (g(x) - gB)XB dra(x) .
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 MBMOq(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
:
Ilm
G(
(
E
E
i
q
)
(2.4) sup )
_
aBll q
m(Ez) 1/q
) } < 00m(B)
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-B -
m(B)
II C
for all B (simply take the partition of B given only by B) .
Therefore for any B and any partition
(
II
G(E=)
- G(B) II9 m(E' ) )
1/q
<1=: m(E;) m(B) m(B) -
G(Es) _ g m(Et) 1/q G(B) q m(Ej ) 1/q
(~ II,,~(Ej) aBll
,m(B)
) + ~~ (I aB -
m(B)
II
m(
B)) < C
As in the case of functions we can define an equivalent norm in
MBMOq(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) IGI*,q :5 IGI*,q :5 C IGI ., q
124
	
O. BLASCO
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 IIGIIMEMO,
Take aB
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
sup{(~~ IGI(Ei) -
Ig m(E,) )1/q}
m(Ei) aB m(B)
< o0
IGI(E) -
IIG(B)jj m(E) I < IG -
m(B)
ml(E)
{(~I
IGI(Ei)
-
IIG(B)II Ig m(E;) )1/qsup } <
i-1 m(E1) m(B) m(B)
sup{m(B)1/q (~(IG m(B) mi_
(Ei))gm(Ei)1-q)1/q} <
IGI*,q
t=1
The last inequality follows from Remark 1 .1 .
3 . 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) .
B .M .O . VECTOR VALUED MEASURES
	
125
Lemma 3.1 . Let G be a measure in MBMOq(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
(3 .2) lim Gn(E) = G(E) for all measurable bounded set E.n-oo
Proof. Using Proposition 2.3 we first get a function ~ in BMO9(SZ) .
Denote by Pn = {x E X : «x) > n } and 0,, (x) = min(1, n/O(x» . Let us
define now
(3.3) Gn(E) = On(x) dG(x) (E E ro )
E
Notice that
IIGn(E)II :5 IGnI(E) :5 f On(x) di GI(x) <- Á Wn(x) O(x) dm(x) <n m(E)
E
This, using Remark 1.2 ., allows to extend Gn to E and shows that Gn belongs
to V,(9,X) .
On the other hand
(3.4) G(E) - Gn(E) = ~ (1 - ~n(x))dG(x)
Enf2
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 mi9(B) fór all balls
B . Using (3.4) we have that for any E C B
If ¡¡al¡ < n then
IIG(E) - Gn (E)II < 2 ~ «x)dm(x)
EnS2
IIG(E) - G.(E)II :5
J ( 1 - n10(x)) dI
GI(x)
EnS2
IIG(E) - Gn(E)II <
JEnS2
(O(x) - n) dm (x) < fEnS2 .,
(O(x) - ¡¡al¡) dm(x)
Therefore we have
(3.5) I Gn - GIq(B) : IG - a m19(B fl s2n)
126
	
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/gIGn - amlq(B)<2m(B)-1 /qIG- aml q (B)
Denoting now by
Dn = sup inf {m(B)-1/qIG- aMIq(B)}
6aNB Ijalj<n
we get (3.1) for Cn = 2 CDn 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
(3.7) TG(~aiXEj) _ < G(Ej),a¡ >
i=1 i=1
where <, > means duality between X and X* .
Lemma 3.2 . Le¡ 1 < p, q < co, 1 -}- 1 = 1 and G belong to V'(Q,X*).
Then
(3 .8) ITG(f)I :5 CIIGIIMBMo, lifllxy for all f in HP(9,X) .
Proof. Let us first take a "simple atom" in HP'(9, X), that is
s = E1 biXEi, Ei C B for some ball B,
E
1 bi m(Ej) = 0
and E l IIb¡IIX -(E¡) < m(B)1-P .
For such an atom we can write
Therefore
biXEi) _ ~, < G(Ei)>bi >= ~<G(E;) - m~B)
m(Ej),bi >
i=1 i=1 i=1
ITG(s)I IIG(Ej) - ,G(B) m(Ei)Ilx.m(E¡)-1/nm(Ei)1/vllbillx <
i=1
C (~IIm(Ei) m(B)IIX
"m(E;))1_ l9¡~IIbi IIxm(Ei))1/n
<
i=1 i=1
n
G(Ei) _ G(B) q m(Ei) 1/q< ~r II
m(Ej) m(B)
IIX " m(B) ) C IGI*~q
B.M .O . VECTOR VALUED MEASURES
	
127
For a general atom a supported in B in Hp(S2, X) we can use approxima-
tion by simple functions in Lp (Q, X), and find a sequence of simple func-
tions 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 con-
verges 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 + 1 = 1. Then
(3.10) (Hp(S2,X))* = MBMOq (S2;X*)-.(equivalent norms)
Proof. Let us take G in MBMO g(S2,X*), and define as above
n
biXE,) _ < G(E?),bi >
i=1 4=1
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
(3.11) ITG(EbiXE;)I s CIGI*, q II(EbiXEJIIHI
i-1 i=1
To see (3.11) 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-,>~ TGn (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
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
128
	
O. BLASCO
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 .
(3.12) < G(E), a >= T(aXE) ( a E X )
Given a ball B and a partition of B, say {Ei}, of pairwise disjoint sets, using
the duality (1P(X»* = 11(X*), we have
i-1 m(Ej) m(B) m(B)
(~1I(G(Ei)
_ G(B) )( m(~ )1/9llX
"
)1/9_
m(Ej) m(B) m(B)4=1
sup{~ < (m(E¡) -
G(B)B»
(m(B)
)1/e b i > I ; IIb¡IIX = 1} .
i=1 i-1
On the other hand we have
G(E;) - G(B) m(E;) 1/q
I < (m(E,) m(B)) ( m(B)) , bi
> I =
s=1
1 1 I n < G(E=) b i > _ < G(B) n 7rt(Ei)1/9 b=) > I1: 1: =m(B) /9 i-1 m(Ei ) /P	m B)
n
1= IT(Em(Ej)-1/PbiXEj-T(bXB)I
where b = m(1B) (E 1 m(E;)1/qbi) .
Denote by a = 2,n(B)lI4 (E 1
m(Ei)-1
/PbiXEi - bXB) .
show that if E l IIbiliX = 1 then a is a (X,p)-atom .
Therefore we obtain
(~ II
G(E4)
_
G(B) 11
X
.
m(E$)
)1/° < 2IT(a)I <_ 2 ¡¡TI¡
i=1 m(Ej) m(B) m(B)
_
This shows IGI* 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}
IB
The following function is an (X, p)-atom
a(x) ( )(x)~~f~~p
for f E LP(B,X) .
m B lq
hence
and therefore
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 mea-
sures 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 con-
verging 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 .2 we get an extension to E .
n
G(E;) - G(B) v _m(E;) 1/9 - 1 G(B)
m (E¡ ) m(B) ~~x " m(B) ) IMlpp1 I m(B)1/9 fB f d ( G m(B) m)
For each f E Lp(B, X), consider a = 2,n(á)1/9 (f - fB)XB and therefore
G(E=)
- G(B)
11X
"
m(Eí) 1/9 -
t-1 m(E¡)
m(B) m(B) )
- SáP 2 IT(a)I C 2 IIT~~
This 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,
MBMOq(9, X) = MBA10,(S2, X) with equivalent norms
130
	
O . BLASCO
For dual spaces follows from the theorem and (1.1), and the general case is
consequence of the embedding X C X**
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Departamento de Matemáticas
Universidad de Zaragoza
Zaragoza 50009
SPAIN


Vector valued measures of bounded mean oscillation

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