LIPSCHITZ MEASURES AND VECTOR-VALUED HARDY SPACES

© Hindawi Publishing Corp.

LIPSCHITZ MEASURES AND VECTOR-VALUED HARDY SPACES

MAGALI FOLCH-GABAYET, MARTHA GUZMÁN-PARTIDA, and SALVADOR PÉREZ-ESTEVA

(Received 24 January 2000)

Abstract.We define certain spaces of Banach-valued measures called Lipschitz measures.

When the Banach space is a dual spaceX^∗, these spaces can be identified with the duals of the atomic vector-valued Hardy spacesH_X^p(Rⁿ),0< p <1. We also prove that all these measures have Lipschitz densities. This implies that for every real Banach spaceXand 0< p <1,the dualH^p_X(Rⁿ)^∗can be identified with a space of Lipschitz functions with values inX^∗.

2000 Mathematics Subject Classification. Primary 42B30, 46E30.

1. Introduction and notation. An interesting question in the theory of vector- valued Hardy spaces is the representation of their dual spaces. This matter has been considered by several authors in the different versions (not necessarily equivalent) of these spaces, namely theH^p-spaces of holomorphic functions in the disk with values in a Banach spaceXas well as the “maximal” and the atomic Hardy spaces of Banach- valued distributions in the unit circle (cf. [1, 2, 3, 4, 5]). In particular, in [1], it is proved that for every Banach spaceX, the dual of the atomicspaceH¹at(X)in the unit circle is the spaceᏮᏹᏻ(X^∗)of measures in the circle and values inX^∗ of bounded mean oscillation. In that paper, it is proved that this space of measures is in fact the space of functions of bounded mean oscillation if and only ifX^∗has the Radon-Nikodym property and in this caseH¹at(X)^∗=Ꮾᏹᏻ(X^∗).

In this paper, we consider the atomic Hardy spacesHX^p inRⁿ for 0< p <1. The dual space of each of theseH_X^pwill be a space of measures with values inX^∗modulo polynomials of degreeN≤[n(1/p−1)](measures with polynomial densities). These measures will be finitely additive measures defined in the ring of all bounded Borel sets inRⁿand have a control imposed on cubes analogous to that one shared by the Lipschitz functions:f∈L¹_loc(Rⁿ)is a Lipschitz function with exponentα >0 if and only if there exists a positive constantC >0 such that

f−P_Q^[α](f )

L²(Q)≤C|Q|^(α/n)+1/2, (1.1) for every cubeQ, whereP_Q^[α](f )is the unique polynomial of degree less than or equal to[α]having the same moments of order less than or equal to[α]asf.

The exponentp∈(0,1)makes possible to adapt the arguments of the scalar theory to approximate these measures by measures with smooth density. The final result is that every measure belonging to our space of measures denoted byᏹ_X^αhas a Radon- Nikodým derivative that is a Lipschitz function with exponentα=n(1/p−1). Then

the dual ofHX^p is the space of Lipschitz functions inRⁿwith values in X^∗ modulo polynomials of degree less than or equal to [α].This holds for every real Banach spaceX, contrasting with the casep=1 mentioned above.

The following notation will be used throughout this paper,᏿(Rⁿ,X)will denote the space of rapidly decreasing functions onRⁿwith values in a real Banach spaceX. We will denote byᏰ(Rⁿ,X)the space ofX-valued distributions inRⁿ,that is, the space of all continuous linear operators fromCc^∞(Rⁿ)intoXand᏿(Rⁿ,X)will be the space of all the linear and continuous operators from᏿(Rⁿ)with values inX. ᏿_s(Rⁿ,X) and ᏿_b(Rⁿ,X)will denote, respectively, ᏿(Rⁿ,X)with the topologies of pointwise convergence and uniform convergence in bounded subsets of᏿(Rⁿ).

For a cubeQ⊂Rⁿand 1≤p <∞,L^p_X(Q)will denote the space of all theX-valued Bochner measurable functions onQsuch thatfbelongs toL^p(Q).

For 1≤q <∞, V_X^q(Q)will be the space of all countably additive measuresµon the Borel sets on the cubeQwith values inXand with finiteq-variation

µ_VX^q_(Q)=sup

A∈π

µ(A)^q m(A)^q−1

1/q

, (1.2)

where the supremum is taken over all finite partitions (measurable)π ofQandm is the Lebesgue measure. Forp >1,the dual space ofL^p_X(Q)can be identified with V_X^q∗(Q), where 1/p+1/q=1. For complete expositions of vector measures and vector integration see [6, 7].

All the cubes considered here will be compact and will have sides parallel to the axes.

0will be the ring of bounded Borel sets inRⁿ. Forf∈᏿(Rⁿ,X)andϕ∈᏿(Rⁿ), we define

M_ϕ^∗(f )(x)= sup

|y−x|<t

ϕt∗f (y)

X. (1.3)

∆hg(x)denote the differenceg(x+h)−g(x), wheregis anX-valued function and

∆²_hg=∆h∆hg. For α >0, the spaces Λ^α_X will be the vector-valued versions of the spacesΛ^αof Lipschitz functions, namely, ifα=[α]+α, 0< α<1, theng∈Λ^αXif g∈C_X^[α](Rⁿ)and

gΛ^α_X= sup

|β|=[α] sup

h∈Rⁿ\0|h|^−α∆hD^βg_L∞

X<∞. (1.4)

Forα∈Z, g∈Λ^α_Xifg∈C_X^α−1(Rⁿ)and gΛ^α_X= sup

|β|=α−1 sup

h∈Rⁿ\0|h|⁻¹∆²_hD^βg_L∞

X. (1.5)

Notice that, as in the case of scalar functions,g_Λ^α_X=0 implies thatD^βgis constant for all|β| =[α]ifα∉ Z, andD^βgis an affine linear function for every|β| =α−1 when α∈Z. Using this fact and Taylor’s theorem for Banach-valued functions, we conclude thatgis a polynomial of degree less than or equal to[α].

We defineΛ^αXto be the normed quotient space ofΛ^αXmodulo polynomials of degree less than or equal to[α].

As usual, the letterCdenote a constant that could be different at each occurrence.

2. Interpolation polynomials and Lipschitz measures. The aim of this section is to define an appropriate space of vector-valued measures, which turn out to be the dual of a vector-valued Hardy space.

LetQbe a cube onRⁿwith centerx0and letf∈L¹(Q). The linear independence of the family of functions{x^α}|α|≤N implies the existence of a unique polynomialP_Q^N of degree less than or equal toNsuch that, for every multi-indexαwith|α| ≤N,

Q

x−x0 αf (x )dx

|Q|=

Q

x−x0 αP_Q^N(x)dx

|Q|. (2.1)

This polynomialP_Q^N can be constructed in the standard way using the dual basis of the set{(x−x0)^α:|α| ≤N}inL²(Q,dx/|Q|), that is, the set of polynomials{ψ^Qα}|α|≤N

of degree less than or equal toN, such that

Q

x−x0 αψ^Q_β(x)dx

|Q|=δαβ, (2.2)

whereδαβis the Kronecker delta.

Then the interpolating polynomial is given by P_Q^Nf (x )=

|α|≤N

aαψ^Qα(x), (2.3)

where

aα=

Qf (x )

x−x0 αdx

|Q|. (2.4)

LetQ1be the cube centered at zero and side length 1 and take any cubeQwith center x0and side lengthδ. If we letψα=ψ^Qα¹for every|α| ≤N, then

ψ^Qα(x)=δ^−|α|ψα

x−x0

δ

. (2.5)

Then we conclude that there exists a constantC >0 independent ofQandα, such

that ψ^Qα(x)≤C|Q|^−|α|/n, x∈Q. (2.6)

Theorem2.1. Given a cubeQcentered atx0,N∈N∪ {0}, q≥1andf∈L^q_X(Q), there exists a unique polynomialP_Q^Nf:Rⁿ→Xof degree less than or equal toNsuch

that

QP_Q^Nf (x )

x−x0 αdx

|Q|=

Qf (x )

x−x0 αdx

|Q| (2.7)

for every multi-indexαwith|α| ≤N. This polynomialP_Q^Nfsatisfies PQ^Nf (x )

X≤ C

|Q|^1/qf_L^q_X(Q), x∈Q. (2.8)

Thus P_Q^Nf

L^q_X(Q)≤Cf_L^q_X(Q). (2.9)

Proof. Letaα=

Qf (x)(x−x0)^αdx/|Q|andPQ^Nf (x )=

|α|≤Naαψ^Qα(x). Then if βis a multi-index with|β| ≤N, then

QP_Q^Nf (x )

x−x0 βdx

|Q|=aβ=

Qf (x )

x−x0 βdx

|Q|. (2.10)

To prove the uniqueness, suppose that

QP(x)(x−x0)^αdx/|Q| =0 for everyαwith

|α| ≤N, whereP is a polynomial of degree less than or equal toN. Then, for every ξ^∗∈X^∗,

Qξ^∗◦P(x)

x−x0 αdx

|Q|=0 (2.11)

and sinceξ^∗◦Pis a polynomial with scalar coefficients whose degree is less than or equal toN, it follows thatξ^∗◦P =0 for everyξ^∗∈X, thereforeP=0. Finally, given x∈Q,

P_Q^Nf (x )

X≤

|α|≤N

aα

X|Q|^−|α|/n. (2.12)

From Hölder’s inequality, we have

aα_X≤C|Q|(|α|/n)−1/q

Q

f (x )^q_Xdx _1/q

(2.13)

and this implies the desired estimates.

LetQbe any cube centered atx0andµ∈V_X^q(Q). As in Theorem 2.1, we can con- struct

P_Q^Nµ(x)=

|α|≤N

aαψα^Q(x), (2.14)

whereaα=1/|Q|

Q(x−x0)^αdµ.

As in Theorem 2.1 we can prove the following corollary.

Corollary2.2. Letµ∈V_X^q(Q). ThenP_Q^Nµ is the unique polynomial of degree less than or equal toNsatisfying

QP_Q^Nf (x )

x−x0 αdx

|Q|= 1

|Q|

Q

x−x0 αdµ, (2.15)

for every|α| ≤N. The polynomialP_Q^Nµverifies P_Q^Nµ(x)_X≤ C

|Q|^1/qµ_VX^q_(Q), x∈Q, P_Q^Nµ

L^q_X(Q)≤Cµ_VX^q_(Q). (2.16) To abbreviate notation, we often writePQf orPQµ if the context does not cause confusion. We introduce the following space of vector-valued measures.

Definition2.3. Letµ:

0→Xbe a vector measure,mthe Lebesgue measure on

0, 0≤α <∞. We say thatµ∈ᏹ^α_X, if the following conditions hold:

(i) µ∈V_X²(Q), for every cubeQ.

(ii) There exists a constantCsuch that, for every cubeQ, µ−P_Q^[α](µ)m

V_X²(Q)≤C|Q|^(α/n)+1/2. (2.17) Forµ∈ᏹ_X^α, we define

µᏹ_X^α=inf

C: (ii) holds

. (2.18)

· ᏹ_X^α is a seminorm onᏹ^α_X andµᏹ^α_X=0 if and only ifµis a measure with poly- nomial densityPsuch that degP≤[α]. If we form the quotient space ofᏹ^α_Xmodulo polynomials of degree less than or equal to[α](measures with polynomial density) we obtain a normed spaceᏹ^α_X.

As in the scalar case, we have

µ_ᏹX^α∼sup

Q |Q|^−α/n inf

degP≤[α]

1

|Q|^1/2µ−Pm

V_X²(Q). (2.19) From now on, we refer to this space as the space of Lipschitz measuresᏹ_X^α.

Every function g∈ Λ^αX defines a measure in ᏹ_X^α, in fact, as in [9, Lemma 5.18, Chapter III], givenx0∈Rⁿ andr >0, there exists a polynomial P(x)of degree[α]

such that

g(x)−P(x)_X≤Cr^αgΛ^α_X (2.20) for everyxin the cube centered inx0and side lengthr, whereConly depends onα andn. This implies that

g_ᏹ^α_X≤CgΛ^α_X, (2.21)

whereg_ᏹ^α_Xdenotes the norm of the measure with densityg.In Section 3, we prove that everyµ∈ᏹ^α_Xhas a densityg∈Λ^α_X. Now, we can prove a weak version of this fact.

Lemma2.4. Letα >0andg∈C_X^m(Rⁿ), wherem=[α]ifαis not an integer and m=α−1otherwise. If the measure with densitygbelongs toᏹ^α_X, theng∈Λ^αX and g_Λ^α_X≤Cg_ᏹX^α.

Proof. For every ξ^∗ ∈ X^∗ and every cube Q, we have that P_Q^[α](ξ^∗ ◦g) = ξ^∗◦P_Q^[α](g). Then it is easy to see thatξ^∗◦g∈ᏹ^α_Randξ^∗◦g_ᏹ^α_R≤ ξ^∗X^∗g_ᏹ^α_X. By the scalar theory we have thatξ^∗◦g∈Λ^αand

ξ^∗◦g

Λ^α≤Cξ^∗◦g

ᏹ^α_R≤Cξ^∗

X^∗gᏹ^α_X. (2.22) The lemma follows from this inequality.

3. Vector-valued Hardy spaces. We start with the classical definition of a vector- valued atom.

Definition3.1. Let 0< p≤1.A functiona∈L¹_X(Rⁿ)is called an(X,p)-atom, or simply an atom inX, if the following conditions hold:

(1) suppa⊂Q, whereQis a cube onRⁿ.

(2) 1

|Q|

Q

a(x)²_Xdx _1/2

≤ |Q|^−1/p. (3.1) (3) For every multi-indexα, with|α| ≤n(1/p−1),

Rⁿx^αa(x)dx=0. (3.2)

Definition3.2. A(p,∞)-atom inX,is a functionasatisfying(1),(3)above, and aL^∞_X ≤ |Q|^−1/p. (3.3) Definition3.3. Given 0< p≤1 as above,H_X^pis the space of vector-valued dis- tributionsf∈᏿_s(Rⁿ,X)such thatf=_∞

i=1λiai, whereaiis an atom inXfor every i∈Nand_∞

i=1|λi|^p<∞.

We define

f_HX^p=inf _∞

i=1

λi^p_1/p

. (3.4)

As usual

d

f ,g =f−g^p

H^p_X (3.5)

defines an invariant metricinH_X^p.

Remark3.4. (1) Letϕ∈᏿(Rⁿ). Ifais an atom inX, then

Rⁿ

M_ϕ^∗(a)(x) ^pdx≤Cϕ, (3.6)

whereCdoes not depend ona.

(2) There exists a continuous seminormρin᏿(Rⁿ)such that

Rⁿaϕ(x)dx

≤ρ(ϕ) (3.7)

for every atomainXandϕ∈᏿(Rⁿ).

(3) Ifaiis an(X,p)-atom and_∞

i=1|λi|^p<∞, then_∞

i=1λiaiconverges in᏿_b(Rⁿ,X), hence, the series always defines an element ofH_X^pand the convergence is in this space.

(4)H_X^pis a complete metric space.

The proof of (1) forϕ≥0 radial and decreasing is the same as in [9, Theorem 4.3, page 275]. The general case follows from the fact that all the gaugesM_ϕ^∗(f )pare equivalent forϕ∈᏿(Rⁿ), with

Rⁿϕ(x)dx=0.This fact is proved as in the scalar case (cf. [8]).

To prove(2), letϕ∈᏿(Rⁿ)and an atomainX. Then

Rⁿa(x)ϕ(x)dx

=a∗ϕ(0)ˇ ≤M_ϕ^∗_ˇ(a)(y) (3.8) for every|y| ≤1, where ˇϕ(x)=ϕ(−x). Raising the inequality above to thepth power and integrating over the unit ball, we obtain

Rⁿa(x)ϕ(x)dx

≤Cϕ. (3.9)

The Banach-Steinhaus theorem implies that the set of atoms defines an equicontin- uous family in᏿(Rⁿ,X), which implies(2).

Statement(3)follows directly from(2).

ThatHX^pis complete follows from(3)taking a subsequence of a Cauchy sequence {fn}^∞_n=1inH_X^psatisfying

fn_k+1−fn_k^p

H_X^p< 1

2^k. (3.10)

If we denote byHmax^p (X)the space of elementsf∈᏿(Rⁿ,X)such thatM_ϕ^∗(f )∈ L^p(Rⁿ),0< p≤1, then(1)in Remark 3.4 implies that we have a continuous inclusion H_X^p

 //

Rⁿϕ(x)dx=0. Moreover, we can use the grand maximal function to define it. Latter’s proof of the atomic decomposition of the scalar Hardy spaces can be adapted to prove that, every atomainXcan be decom- posed as a seriesa=_∞

i=1λiai, whereai is a(p,∞)-atom inX,with_∞

i=1|λi|^p< C, whereCis independent ofa.

4. Duality. In this section, we state the duality between the vector-valued Hardy spaces and the space of Lipschitz vector measures with values inX^∗. Then we prove that all these measures have densities that are Lipschitz continuous functions, hence we have the representation of the space(H_X^p)^∗ that holds in the scalar case without any restriction on the Banach spaceX.Henceforth, we denote

α=n1 p−1

, N= n1

p−1

. (4.1)

The first step is to prove that the elements of the dual(H_X^p)^∗can be represented by measuresµ∈ᏹ^α_X∗ acting on atoms inXby

Rⁿa(x)dµ. (4.2)

Proposition4.1. Let0< p≤1. For everyΦ∈(H_X^p)^∗, there exists a measureµ∈ ᏹ^α_X∗ unique modulo polynomials of degree less than or equal toN (measures with polynomial density), such that (4.2) holds for every atom inX.

Proof. Denote byL²N(Q,X) the space of functionsf ∈L²X(Rⁿ), supported inQ and with vanishing moments up to orderN,and let

ΘN(X)=

Q

L²N(Q,X). (4.3)

Iff∈L²_N(Q,X), the functiona(x)= |Q|^1/2−1/pf⁻¹_L2

Xf (x )is an atom inX, then Φ(f )≤ Φf_L²

X|Q|^1/p−1/2. (4.4)

We can extend this functional to the whole space L²_X(Q)without increasing the norm, and obtain a measureµQ∈V_X²∗(Q), withµQ_V²

X∗(Q)≤ Φ|Q|^1/p−1/2such that Φ(f )=

Qf dµQ (4.5)

for everyf∈L²_N(Q,X).

We show thatµQis uniquely determined up to addition of a measure with polyno- mial density with values inX^∗.

Indeed, sincef∈L²_N(Q,X)has vanishing moments up to orderN, then for every polynomialP of degree less than or equal toN,µQ+P·malso satisfies (4.5). Con- versely, suppose thatµ1andµ2belong toV_X²∗(Q)and satisfy (4.5). Letf∈L²_X(Q), ν= µ1−µ2and consider the interpolation polynomialsPQf andPQνof degreeN. Then f−PQf∈L²N(Q,X)and

0=

Q

f−PQf dν=

Q

f−PQf dν−PQν(x)dx =

Qf

dν−PQν(x)dx (4.6) for everyf∈L²_X(Q), thereforedν=PQν(x)dx.

We conclude in particular that there exists a unique measureνQ∈V_X²∗(Q)satisfying (4.5) with vanishing moments of order less than or equal toN. This measure is

νQ=µQ−

PQµQ m. (4.7)

By Corollary 2.2, we have the estimate νQ

V_X²∗(Q)≤CΦ|Q|^1/p−1/2. (4.8) Now, we decomposeRⁿ= ∪^∞_j=1Qj, where(Qj)is an increasing sequence of cubes.

We can adjust the measuresµQj adding measures with polynomial density of degree less than or equal toNto obtain a single measureµwith values onX^∗ and defined

on

0, such that

Φ(f )=

Rⁿf dµ (4.9)

for everyf∈ΘN(X). Since the restriction to every cubeQofµ−(PQµ)misνQ, then by (4.8) we conclude thatµ∈ᏹ^α_X∗ and

µᏹ^α_X∗ ≤CΦ. (4.10)

SinceΘN(X)is dense inHX^p, we have shown that

HX^p ∗

 //

Now, let 0< p <1 andµ∈ᏹ^α_X∗. We prove that the natural action ofµ inΘN(X) given by

Φ(f )=

Rⁿf dµ (4.12)

can be extended toH_X^pas a continuous functional.

Since

Qa(x)dµ(x) =

Qa(x)d

µ−PQ(µ)m

≤ a_L²_Xµ−PQ(µ)m_V2

X∗(Q)≤ µᏹ^α_X∗,

(4.13)

the proof of the continuity ofΦwould be complete if we could prove that Φ(f )=

λjΦ

aj , (4.14)

for everyf =

λjaj∈ΘN(X)with atomicrepresentationf =

λjaj. The identity (4.14) holds ifµhas a smooth density, as the following lemma shows.

Lemma 4.2. Letψ ∈᏿(Rⁿ,X^∗)and f ∈ΘN(X)with atomic representation f =

λjaj. Then

Rⁿ

f ,ψ

dx= λj

Rⁿ

aj,ψ

dx. (4.15)

(here,denotes the duality inX).

Proof. Every atomainXdefines a continuous functional on᏿(Rⁿ,X^∗) Ta(ψ)=

Rⁿ

a,ψ

dx. (4.16)

Letρbe the continuous seminorm in᏿(Rⁿ)of Remark 3.4(2). Then for anyψ= ϕi⊗ ξ^∗_i ∈᏿(Rⁿ)⊗X^∗, we have

Ta(ψ)≤ ρ

ϕi ξ^∗_i

X^∗. (4.17)

Hence,

Ta(ψ)≤ρ(ψ),˜ (4.18)

where ˜ρis the continuous seminormρ⊗·in the projective tensor product᏿(Rⁿ)⊗π

X^∗. It follows that the family {Ta : a is an atom inX} is equicontinuous in (᏿(Rⁿ)⊗πX^∗)=᏿(Rⁿ,X^∗)(see [12, Chapter 51]). We conclude that the series

λjTa_j

converges in the strong topology of the dual space᏿(Rⁿ,X^∗)to a continuous func- tional. Since (4.15) holds forψin the dense subset᏿(Rⁿ)⊗X^∗of᏿(Rⁿ,X^∗), the proof of the lemma is now complete.

Now we are in position to prove the continuous inclusionᏹ_X^α⊂Λ^α_X.

Proposition 4.3. Every µ ∈ ᏹ^α_X has a density g∈ Λ^αX(Rⁿ) such that gΛ^α_X ≤ Cµ_ᏹ^α_X.

Proof. Let ϕ∈C_c^∞ radial, ϕ≥0, with supp ϕ in the unit ball and such that

Rⁿϕ = 1. Consider the family of measures (µt)t>0 with density ϕt∗µ(x) = µ(ϕt(x−·)), whereϕt(x)=(1/tⁿ)ϕ(x/t). Notice thatµtis aC^∞function with values inX. In fact, for everyξ^∗∈X^∗, the measureξ^∗◦µhas a Radon-Nikodym derivative inL¹_loc. Thenξ^∗◦µt=(ξ^∗◦µ)t is a smooth function, and this implies thatµtis also smooth (see [11]).

By Lemma 2.4 we have

ξ^∗◦µt_Λα=ξ^∗◦µ _tΛα≤Cξ^∗◦µ _tᏹα

R, (4.19)

and [9, proof of Theorem 5.22, Chapter III]

ξ^∗◦µ _tᏹα

R≤Cξ^∗◦µ_ᏹα

R. (4.20)

But we have

ξ^∗◦µ_ᏹα

R≤ξ^∗_X∗µ_ᏹα

X (4.21)

and thus, we conclude that

µt_Λα

X≤Cµᏹ^α_X. (4.22)

We know thatµtconverges toµinᏰ(Rⁿ,X)ast→0. Then the proof of the propo- sition will be complete once we prove thatD^βµtconverges uniformly on compact sets ofRⁿ for|β| ≤[α]. To this end, recall that ifkis a nonnegative integer andβis a multi-index such thatk+|β|> α. Then

∂^k

∂t^kD^βx

ϕt(x) =Ct^{α−k−|β|}at(x), x∈Rⁿ, t >0, (4.23) whereat(x)is a scalar(p,∞)-atom for everyt, andCis a constant depending onϕ, α,n, and the order of differentiation, but not ont(see [9, Lemma 5.20, Chapter III]).

Then

∂^k

∂t^kD^βxµt(x,t) _X=

Rⁿ

∂^k

∂t^kD^βy

ϕt(x−y) dµ(y) _X

=Ct^{α−k−|β|}

Rⁿat(x−y)dµ(y) X.

(4.24)

Proceeding as in (4.13), we see that

Rⁿat(x−y)dµ(y)

_X≤ µᏹ^α_X. (4.25)

Therefore, for any integerk≥0 and any multi-indexβsuch thatk+|β|> α, ∂^k

∂t^kD^βxµt(x)

_X≤Ct^{α−k−|β|}µᏹ^α_X, (4.26) whereCdoes not depend ontandµ.

With this inequality and the fundamental theorem of calculus it is easy to prove thatD^βxµt satisfies a uniform Cauchy estimate on compact sets onRⁿasttends to zero. The proof is complete.

Now, we are ready to prove that

Rⁿf dµ= λj

ajdµ (4.27)

for everyf∈ΘN(X)with atomicrepresentationf=

λjajand everyµ∈ᏹ^α_X∗. Letg∈Λ^α_X∗ be the density ofµ. Lethm(x)=g(x)ψ(x/m), where ψ is a radial andC^∞scalar function supported in the unit ball such thatψ(x)=1 for|x| ≤1/2 and 0≤ψ(x)≤1 for all x. The family{hm}m∈N is a sequence of functions inΛ^α_X∗

with norm bounded byCgΛ^α_X∗. This cut-off process combined with the regulariza- tion described in the proof of Proposition 4.3 implies that there exists a sequence of functions(gm)^∞_m=1inC_X^∞∗ with compact support such that

(1) gmΛ^α_X∗ ≤CgΛ^α_X∗ for everym∈N, whereCis an absolute constant, (2)

f ,gmdx→

f ,gdxasm→ ∞for everyf∈ΘN(X).

Lemma 4.2 and the bounded convergence theorem imply (4.14). Thus, we have proved the following theorem.

Theorem4.4. LetXbe a Banach space overR. F or0< p <1andα=n(1/p−1),

HX^p ∗=Λ^α_X∗ (4.28)

with equivalent norms.

Acknowledgement. This work was partially supported by PAPIIT-UNAM IN1027 99 and CONACyT 32408-E.

References

[1] O. Blasco,Hardy spaces of vector-valued functions: duality, Trans. Amer. Math. Soc.308 (1988), no. 2, 495–507. MR 90a:46082. Zbl 658.46021.

[2] , On the dual space of H^1,∞_B , Colloq. Math. 55 (1988), no. 2, 253–259.

MR 89m:46068. Zbl 663.46028.

[3] ,Vector valued measures of bounded mean oscillation.Conference on Mathematical Analysis (El Escorial, 1989), Publ. Mat.35(1991), no. 1, 119–130. MR 92i:46051.

Zbl 746.46040.

[4] J. Bourgain,Vector-valued singular integrals and theH¹-BMO duality, Probability The- ory and HarmonicAnalysis (Cleveland, Ohio, 1983), Monographs Textbooks Pure Appl. Math., vol. 98, Marcel Dekker, New York, 1986, pp. 1–19. MR 87j:42049b.

Zbl 602.42015.

[5] A. V. Buhvalov,The conjugates of spaces of analytic vector-valued functions and the du- ality of functors generated by these spaces.Investigations on linear operators and the theory of functions, IX, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov.

(LOMI)92(1979), 30–50, 318 (Russian). MR 81b:46052. Zbl 431.46050.

[6] J. Diestel and J. J. Uhl, Jr., Vector Measures, Mathematical Surveys, no. 15, Ameri- can Mathematical Society, Providence, R.I., 1977, with a foreword by B. J. Pettis.

MR 56#12216. Zbl 369.46039.

[7] N. Dinculeanu,Vector Measures, International Series of Monographs in Pure and Ap- plied Mathematics, vol. 95, Pergamon Press, Oxford; VEB Deutscher Verlag der Wissenschaften, Berlin, 1967. MR 34#6011b.

[8] C. Fefferman and E. M. Stein,H^pspaces of several variables, Acta Math.129(1972), no. 3- 4, 137–193. MR 56#6263. Zbl 257.46078.

[9] J. García-Cuerva and J. L. Rubio de Francia,Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Ams- terdam, 1985, Notas de Matematica [Mathematical Notes], vol. 104. MR 87d:42023.

Zbl 578.46046.

[10] R. H. Latter,A characterization ofH^p(Rⁿ)in terms of atoms, Studia Math. 62(1978), no. 1, 93–101. MR 58#2198. Zbl 398.42017.

[11] L. Schwartz,Espaces de fonctions différentiables à valeurs vectorielles, J. Analyse Math.4 (1954/55), 88–148 (French). MR 18,220a. Zbl 066.09601.

[12] F. Trèves,Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, 1967. MR 37#726. Zbl 171.10402.

Magali Folch-Gabayet: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, D.F.,04510, Mexico

E-mail address:[email protected]

Martha Guzmán-Partida: Universidad de Sonora, Departamento de Matemáticas, Blvd. Luis Encinas y Rosales, Hermosillo, Sonora,83000, Mexico

E-mail address:[email protected]

Salvador Pérez-Esteva: Instituto de Matemáticas, Universidad Nacional Autónoma de México, Unidad Cuernavaca Apartado Postal273-3, Administración de Correos #3, Cuernavaca, Morelos,62251, Mexico

E-mail address:[email protected]