Predual Spaces of Banach Completions of Orlicz-Hardy Spaces Associated with Operators

0.3 MB·English

by  Renjin Jiang

J. Fourier Anal. Appl., to appear
Predual Spaces of Banach Completions of
0 Orlicz-Hardy Spaces Associated with Operators
1
0
2
Renjin Jiang and Dachun Yang∗

n
a
J
0 Abstract. Let L be a linear operator in L2(Rn) and generate an analytic semigroup
1 e−tL t≥0 with kernels satisfying an upper bound of Poisson type, whose decay is mea-
{ }
sured by θ(L) (0, ]. Let ω on (0, ) be of upper type 1 and of critical lower type
A] p0(ω) (n/(n∈+θ(L∞)),1] and ρ(t) =∞t−1/ω−1(t−1) for t (0, ). In this paper, the
author∈sfirstintroducetheVMO-typespaceVMO (Rn)a∈ndth∞etentspaceT∞(Rn+1)
C ρ,L ω,v +
h. aend characterizethe space VMOρ,L(Rn) via the space Tω∞,v(Rn++1). Let Tω(Rn++1) be the
t Banach completion of the tent space Tω(Rn++1). The authors then prove that Tω(Rn++1)
a is the dual space of T∞(Rn+1). As an application of this, the autheors finally show
m ω,v +
that the dual space of VMOρ,L∗(Rn) is the space Bω,L(Rn), where L∗ denotees the ad-
[ joint operator of L in L2(Rn) and Bω,L(Rn) the Banach completion of the Orlicz-Hardy
2 spaceHω,L(Rn). Theseresultsgeneralizetheknownrecentresultsbyparticularlytaking
v ω(t)=t for t (0, ).
0 ∈ ∞
8
8
1 1 Introduction
.
6
0 The space VMO(Rn) (the space of functions with vanishing mean oscillation) was first
9 studied by Sarason [26]. Coifman and Weiss [7] introduced the space CMO(Rn) which
0
is defined to be the closure in the BMO norm of the space of continuous functions with
:
v compact support, and moreover, proved that the space CMO(Rn) is the predual of the
i
X Hardy space H1(Rn). When p < 1, Janson [21] introduced the space λ (Rn) which
n(1/p−1)
r is defined to be the closure of the space of Schwartz functions in the norm of the Lipschitz
a
space Λ (Rn), and proved that (λ (Rn))∗ = Bp(Rn), where Bp(Rn) is the
n(1/p−1) n(1/p−1)
Banach completion of the Hardy space Hp(Rn); see also [25, 28] for more properties about
the space λ (Rn).
n(1/p−1)
Inrecentyears,thestudyoffunctionspacesassociatedwithoperatorshasinspiredgreat
interests; see, forexample, [1,2,3,11,12,15,16,17,20,29,30]andtheirreferences. LetL
be a linear operator in L2(Rn) and generate an analytic semigroup e−tL with kernels
t≥0
{ }
satisfying an upper bound of Poisson type, whose decay is measured by θ(L) (0, ].
∈ ∞
Auscher, Duongand McIntosh [1]introducedtheHardyspace H1(Rn)by usingtheLusin-
L
area function and established its molecular characterization. Duong and Yan [16, 18],
2000 Mathematics Subject Classification. Primary 42B35; Secondary 42B30.
Key words and phrases. operator, Orlicz function, Orlicz-Hardy space, VMO, predual space, Banach
completion, tent space, molecule.
DachunYangissupportedbytheNationalNaturalScienceFoundation(GrantNo. 10871025) ofChina.
∗Corresponding author.
1
2 Renjin Jiang and Dachun Yang
and Duong, Xiao and Yan [15] introduced and studied some BMO spaces and Morrey-
Campanato spaces associated with operators. Duongand Yan [17] furtherproved that the
dual space of the Hardy space H1(Rn) is the space BMO (Rn) introduced in [16], where
L L∗
L∗ denotes the adjoint operator of L in L2(Rn). Yan [30] generalized all these results to
the Hardy spaces Hp(Rn) with p (n/(n + θ(L)),1] and their dual spaces. Moreover,
L ∈
recently, Deng, Duong et al in [11] introduced the space VMO (Rn) and proved that
L
(VMO (Rn))∗ = H1(Rn).
L∗ L
Ontheotherhand,theOrlicz-Hardy spacewasstudiedbyJanson[21]andViviani[27].
Let ω on (0, ) be of upper type 1 and of critical lower type p (ω) (n/(n+θ(L)),1] and
0
∞ ∈
ρ(t) t−1/ω−1(t−1) for t (0, ). The Orlicz-Hardy space H (Rn) and its dual space
ω,L
≡ ∈ ∞
BMOρ,L∗(Rn) associated with the aforementioned operator Leand its dual operator L∗ in
L2(Rn) were introduced in [22]. If ω(t) = tp for all t (0, ), then H (Rn) = Hp(Rn)
∈ ∞ ω,L L
and BMO (Rn) becomes BMO (Rn) when p = 1 or the Morrey-Campanato space (see
ρ,L∗ L∗
[18]) when p < 1. The main purpose of this paper is to study the predual space of the
Banach completion of the Orlicz-Hardy space H (Rn).
ω,L
In fact, in this paper, we first introduce the VMO-type space VMO (Rn) and the
ρ,L
tent space T∞(Rn+1) and characterize the space VMO (Rn) via the space T∞ (Rn+1).
ω,v + ρ,L ω,v +
Let T (Rn+1) be the Banach completion of the tent space T (Rn+1). We then prove that
ω + ω +
T (Rn+1) is the dual space of T∞(Rn+1). As an application of this, we finally show that
ω + ω,v +
the deual space of VMO (Rn) is the space B (Rn), where L∗ denotes the adjoint
ρ,L∗ ω,L
oeperator of L in L2(Rn) and B (Rn) the Banach completion of the Orlicz-Hardy space
ω,L
H (Rn). In particular, if p (0,1] and ω(t) = tp for all t (0, ), we obtain the
ω,L
predual space of the Banach com∈pletion of the Hardy space Hp(∈Rn) in∞[30], and if p = 1,
L
we re-obtain that (VMO (Rn))∗ = H1(Rn), which is the main result in [11]. Moreover,
L∗ L
we prove that if L = ∆, p (0,1] and ω(t) = tp for all t (0, ), the space VMO (Rn)
ρ,L
∈ ∈ ∞
coincides with the space λ (Rn) in [21] (see also [25, 28]), where ∆ = n ∂2
is the Laplace operator onnR(1n/.p−1) − i=1 ∂x2i
P
Precisely, this paper is organized as follows. In Section 2, we recall some known def-
initions and notation concerning aforementioned operators, Orlicz functions, the Orlicz-
Hardy spaces and BMO spaces associated with these operators and describe some basic
assumptions on the operator L and the Orlicz function ω considered in this paper. We re-
mark that there exist many operators satisfying these assumptions (see [11, 15, 17, 30, 22]
for examples of such operators). Also, if p (0,1], then ω(t) = tp for all t (0, ) is
∈ ∈ ∞
a typical example of Orlicz functions satisfying our assumptions; see [22] for some other
examples.
In Section 3, we introducethe spaces VMO (Rn) and T∞(Rn+1) and give some basic
ρ,L ω,v +
properties of these spaces. In particular, we characterize the space VMO (Rn) via the
ρ,L
space T∞ (Rn+1); see Theorem 3.2 below. As an application of Theorem 3.2 together
ω,v +
with a characterization of the space λ (Rn) in [28], we obtain that when L = ∆,
n(1/p−1)
p (0,1] and ω(t) = tp for all t (0, ), the space VMO (Rn) coincides with the space
ρ,L
∈ ∈ ∞
λ (Rn); see Corollary 3.1 below.
n(1/p−1)
In Section 4, we introduce the space T (Rn+1), which is defined to be the Banach
ω +
completion ofthetentspaceT (Rn+1)(seeDefinition 4.1below), andprovethatT (Rn+1)
ω + ω +
e
e
Predual Spaces of Banach Completions of Orlicz-Hardy Spaces 3
is the dual space of T∞ (Rn+1); see Theorem 4.2 below. If p (0,1] and ω(t) = tp for all
ω,v + ∈
t (0, ), then the tent space T (Rn+1) = Tp(Rn+1) and its predual space was proved
∈ ∞ ω + 2 +
to be the corresponding space T∞(Rn+1) by Wang in [28], which when p = 1 plays a key
ω,v +
role in [11]. To prove that T (Rn+1) is the dual space of T∞(Rn+1), different from the
ω + ω,v +
approach used in [7] and [28] which strongly depends on an unfamiliar result (Exercise
41 on [13, p.439]) from theefunctional analysis, we only use the basic fact that the dual
space of L2 is itself. Indeed, using this fact, for any ℓ (T∞ (Rn+1))∗, we construct
∈ ω,v +
g T (Rn+1) such that for all f T∞ (Rn+1),
∈ ω + ∈ ω,v +
e dxdt
ℓ(f) = f(x,t)g(x,t) ;
Rn+1 t
Z +
see Theorem 4.2 below. As an application of Theorem 4.2, we further prove that the dual
space of the space VMO (Rn) is the space B (Rn), where the space B (Rn) is the
ρ,L∗ ω,L ω,L
Banach completion of H (Rn); see Definition 4.3 and Theorem 4.4 below. Since all dual
ω,L
spaces are complete, it is necessary here to replace the Orlicz-Hardy space by its Banach
completion, whichisdifferentfrom[11]. In[11], theBanach completion oftheHardyspace
H1(Rn) is just itself. Finally, in Subsection 4.3, we give several examples of operators to
L
which the results of this paper are applicable.
Let us make some conventions. Throughout the paper, we denote by C a positive
constant which is independent of the main parameters, but it may vary from line to line.
The symbol X . Y means that there exists a positive constant C such that X CY;
≤
the symbol α for α R denotes the maximal integer no more than α; B B(z , r )
B B
⌊ ⌋ ∈ ≡
denotes an open ball with center z and radius r and CB(z , r ) B(z , Cr ). Set
B B B B B B
≡
N 1,2, and Z N 0 . For any subsetE of Rn, we denote by E∁ the set Rn E.
+
≡ { ···} ≡ ∪{ } \
2 Preliminaries
In this section, we first describe some basic assumptions on the operators L and Orlicz
functions studied in this paper (see, for example, [14, 23, 21, 27, 15, 16, 17, 11, 22]), and
we then recall some notions about the Orlicz-Hardy space H (Rn) and the BMO-type
ω,L
space BMO (Rn) in [22].
ρ,L
2.1 Two assumptions on the operator L
Let ν (0,π), S z C : arg(z) ν 0 and S0 the interior of S , where
∈ ν ≡ { ∈ | | ≤ }∪{ } ν ν
arg(z) ( π, π] is the argument of z. Assume that L is a linear operator such that
∈ −
σ(L) S , where σ(L) denotes the spectra of L and ν (0,π/2), and that for all γ > ν,
ν
⊂ ∈
there exists a positive constant C such that
γ
(L λI)−1 L2(Rn)→L2(Rn) Cγ λ −1, λ / Sγ,
k − k ≤ | | ∀ ∈
where and in what follows, for any two normed linear spaces X and Y , and any bounded
linear operator T from X to Y , we use T X→Y to denote the operator norm of T from
k k
4 Renjin Jiang and Dachun Yang
X to Y and L(X,Y ) the set of all bounded linear operators from X to Y . Hence L
generates a holomorphic semigroup e−zL, where 0 arg(z) < π ν (see [23]). We make
≤ | | 2 −
the following two assumptions on L (see [17, 15, 11, 30, 22]).
Assumption (a). Assume that for all t > 0, the distribution kernels p of e−tL belong to
t
L∞(Rn Rn) and satisfy the estimate p (x,y) h (x,y) for all x, y Rn, where h is
t t t
× | | ≤ ∈
given by
x y
(2.1) ht(x, y) = t−mng | −1 | ,
(cid:18) tm (cid:19)
in which m is a positive constant and g is a positive, bounded, decreasing function satis-
fying that
(2.2) lim rn+ǫg(r) = 0
r→∞
for some ǫ > 0.
Let H(S0) be the space of all holomorphic functions on S0 and
ν ν
H (S0) b H(S0): b sup b(z) < .
∞ γ ≡ ∈ γ k k∞ ≡ | | ∞
( z∈S0 )
γ
Recall that the operator L is said to have a bounded H -calculus in L2(Rn) (see [23])
∞
provided that for all γ (ν, π), there exists a positive constant C such that for all b
γ
∈ ∈
H∞(Sγ0), b(L) ∈ L(L2(Rn), L2(Rn)) and kb(L)kL2(Rn)→L2(Rn) ≤ Cγkbk∞, where ψ(z) =
z(1+z)−2 for all z S0 and b(L) [ψ(L)]−1(bψ)(L). It was proveed in [23] that b(L) is a
∈ γ ≡
well-defined linear operator in L2(Rn). e
Assumption (b). Assume that the operator L is one-to-one, has dense range in L2(Rn)
and a bounded H -calculus in L2(Rn).
∞
From the assumptions (a) and (b), it is easy to deduce the following useful estimates.
First, if e−tL is a bounded analytic semigroup in L2(Rn) whose kernels p
t≥0 t t≥0
{ } { }
satisfy the estimates (2.1) and (2.2), then for any k N, there exists a positive constant
∈
C such that the time derivatives of p satisfy that
t
∂kp (x,y) C x y
(2.3) tk t g | − |
(cid:12) ∂tk (cid:12)≤ tmn (cid:18) tm1 (cid:19)
(cid:12) (cid:12)
forall t > 0andalmost every(cid:12)wherex, y (cid:12)Rn.Itshouldbepointedoutthatfor anyk N,
(cid:12) (cid:12)
∈ ∈
the function g may depend on k but it always satisfies (2.2); see Theorem 6.17 of [24] and
[8], and also [17, 15, 11, 30, 22].
Secondly, let
Ψ(S0) ψ H(S0) : s,C > 0 such that z S0, ψ(z) C z s(1+ z 2s)−1 .
ν ≡ ∈ ν ∃ ∀ ∈ ν | | ≤ | | | |
(cid:26) (cid:27)
Predual Spaces of Banach Completions of Orlicz-Hardy Spaces 5
ItiswellknownthatLhasaboundedH -calculusinL2(Rn)ifandonlyifforallγ (ν, π]
∞
∈
and any non-zero function ψ Ψ(S0), L satisfies the square function estimate and its
∈ γ
reverse, namely, there exists a positive constant C such that for all f L2(Rn),
∈
∞ dt 1/2
(2.4) C−1kfkL2(Rn) ≤ kψt(L)fk2L2(Rn) t ≤ CkfkL2(Rn),
(cid:18)Z0 (cid:19)
where ψ (ξ) = ψ(tξ) for all t > 0 and ξ Rn. Notice that different choices of γ > ν and
t
∈
ψ Ψ(S0) lead to equivalent quadratic norms of f; see [23] for the details.
∈ γ
As noticed in [23], positive self-adjoint operators satisfy the quadratic estimate (2.4).
So do normal operators with spectra in a sector, and maximal accretive operators. For
definitions of these classes of operators, we refer the reader to [31].
2.2 An acting class of the semigroup e−tL
t≥0
{ }
Duong and Yan [16] introduced the class of functions that the operators e−tL act upon.
Precisely, for any β > 0, let (Rn) be the collection of all functions f L2 (Rn) such
Mβ ∈ loc
that
f(x)2 1/2
kfkMβ(Rn) ≡ Rn 1+| x n|+β dx < ∞.
(cid:18)Z | | (cid:19)
Then Mβ(Rn) is a Banach space under the norm k·kMβ(Rn). For any given operator L,
set
(2.5) θ(L) sup ǫ > 0: (2.2) holds
≡ { }
and define
(Rn) if θ(L)< ;
(Rn) Mθ(L) (Rn) if θ(L)=∞.
M ≡ β
(0<β∪<∞M ∞
Let s Z . For any (x, t) Rn+1 Rn (0, ) and f (Rn), set
∈ + ∈ + ≡ × ∞ ∈M
(2.6) P f(x) f(x) (I e−tL)s+1f(x) and Q f(x) ts+1Ls+1e−tLf(x).
s,t s,t
≡ − − ≡
If s = 0, write
(2.7) P f(x) P f(x)= e−tLf(x) and Q f(x) Q f(x)= tLe−tLf(x).
t 0,t t 0,t
≡ ≡
For any f (Rn), by (2.3), it is easy to show that P f and Q f are well defined.
s,t s,t
∈ M
Moreover, by (2.3) again, we know that the kernels p and q of P of Q satisfy that
s,t s,t s,t s,t
for all t > 0 and x, y Rn,
∈
x y
(2.8) ps,tm(x,y) + qs,tm(x,y) Cst−ng | − | ,
| | | | ≤ t
(cid:18) (cid:19)
wherethefunctiong satisfies thecondition (2.2) andC isapositive constant independent
s
of t, x and y.
It should be pointed out that these operators in (2.6) were introduced by Blunck and
Kunstmann [5].
6 Renjin Jiang and Dachun Yang
2.3 Orlicz functions
Let ω be a positive function defined on R (0, ). The function ω is said to be of
+
≡ ∞
upper type p (resp. lower type p) for some p [0, ), if there exists a positive constant
∈ ∞
C such that for all t 1 (resp. 0< t 1),
≥ ≤
(2.9) ω(st) Ctpω(s).
≤
Obviously, if ω is of lower type p for some p > 0, then lim ω(t) = 0. So for the sake
t→0+
of convenience, if it is necessary, we may assume that ω(0) = 0. If ω is of both upper type
p and lower type p , then ω is said to be of type (p , p ). Let
1 0 0 1
p (ω) inf p > 0 :(2.9) holds for all t (1, ) ,
1
≡ { ∈ ∞ }
and
e
p (ω) sup p > 0 :(2.9) holds for all t (0,1) .
0
≡ { ∈ }
Itiseasytoseethatp (ω) p (ω)forallω.Inwhatfollows, p (ω)andp (ω)arecalledthe
0 1 0 1
e ≤
critical lower type index and the critical upper type index of ω, respectively. Throughout
the whole paper, weealways aessume that ω satisfies the followeing assumeption.
Assumption (c). Suppose that the positive Orlicz function ω on R is continuous,
+
strictly increasing, subadditive, of upper type 1 and p (ω) ( n ,1], where θ(L) is
0 ∈ n+θ(L)
as in (2.5).
e
t ω(s)
Notice that for any ω of type (p ,p ), if we set ω(t) ds for t [0, ), then by
0 1 ≡ 0 s ∈ ∞
[27, Proposition 3.1], ω is equivalent to ω, namely, there exists a positive constant C such
R
that C−1ω(t) ω(t) Cω(t) for all t [0, ), eand moreover, ω is strictly increasing,
≤ ≤ ∈ ∞
subadditive and contineuous function of type (p0, p1). Since all our results in this paper are
invariant on equivealent functions, we may always assume that ω seatisfies the assumption
(c); otherwise, we may replace ω by ω.
We also make the following convention.
e
Convention. From the assumption (c), it follows that n < p (ω) p (ω) 1. In
n+θ(L) 0 ≤ 1 ≤
what follows, if (2.9) holds for p (ω) with t (1, ), then we choose p (ω) p (ω);
1 1 1
∈ ∞ ≡
otherwise p1(ω) < 1 and we choose p1(ω) (p1(ω),1). Similarly, ife(2.9) hoelds for p0(ω)
∈
with t ∈ (0,1), then we choose p0(eω) ≡ p0(ω); otherwise we choose p0(ω)∈ (n+nθ(L),pe0(ω))
such that en( 1 1) = n( 1 1) , wheere m is as in (2.1). e
⌊m p0(ω) − ⌋ ⌊m pe0(ω) − ⌋
e e
Let ω satisfy the assumption (c). A measurable function f on Rn is said to be in the
Lebesgue type space L(ω) if ω(f(x))dx < . Moreover, for any f L(ω), define
Rn | | ∞ ∈
R
f(x)
f inf λ > 0 : ω | | dx 1 .
L(ω)
k k ≡ Rn λ ≤
(cid:26) Z (cid:18) (cid:19) (cid:27)
Predual Spaces of Banach Completions of Orlicz-Hardy Spaces 7
Let ω satisfy the assumption (c). Define the function ρ(t) on R by setting, for all
+
t (0, ),
∈ ∞
t−1
(2.10) ρ(t) ,
≡ ω−1(t−1)
where ω−1 is the inverse function of ω. Then by Proposition 2.1 in [22], ρ is of type
(1/p (ω) 1,1/p (ω) 1), which is denoted by (β (ρ),β (ρ)) in what follows for short.
1 0 0 1
− −
2.4 The Orlicz-Hardy space H (Rn) and its dual space
ω,L
For any function f L1(Rn), the Lusin area function (f) associated with the oper-
L
∈ S
ator L is defined by setting, for all x Rn,
∈
1/2
dydt
SL(f)(x) ≡ |Qtmf(y)|2 tn+1 ,
ZΓ(x) !
where Qtm is as in (2.7). From the assumption (b) together with (2.4), it is easy to deduce
that the Lusin area function is bounded on L2(Rn). Auscher, Duong and McIntosh
L
S
[1] proved that for any p (1, ), there exists a positive constant C such that for all
p
∈ ∞
f Lp(Rn),
∈
(2.11) Cp−1kfkLp(Rn) ≤kSL(f)kLp(Rn) ≤ CpkfkLp(Rn);
see also Duong and McIntosh [14] and Yan [29]. By duality, the operator S also satisfies
L∗
the estimate (2.11), where L∗ is the adjoint operator of L in L2(Rn).
Recall that the Orlicz-Hardy space H (Rn) and the BMO-type space BMO (Rn)
ω,L ρ,L
were introduced in [22].
Definition 2.1. Let L satisfy the assumptions (a) and (b) and ω satisfy the assumption
(c). A function f L2(Rn) is said to be in H (Rn) if (f) L(ω), and moreover,
ω,L L
∈ S ∈
define
e
(f)(x)
L
kfkHω,L(Rn) ≡ kSL(f)kL(ω) = inf λ > 0 : Rnω S λ dx ≤ 1 .
(cid:26) Z (cid:18) (cid:19) (cid:27)
The Orlicz-Hardy space H (Rn) associated with the operator L is defined to be the com-
ω,L
pletion of Hω,L(Rn) in the norm k·kHω,L(Rn).
Definition 2.2. Let L satisfy the assumptions (a) and (b), ω satisfy the assumption (c),
e
ρ be as in (2.10), q [1, ) and s n( 1 1) . A function f (Rn) is said to be
∈ ∞ ≥ ⌊m pe0(ω) − ⌋ ∈ M
in BMOq,s(Rn) if
ρ,L
1/q
1 1
kfkBMOqρ,,sL(Rn) ≡ Bs⊂uRpn ρ(|B|) (cid:20)|B|ZB|f(x)−Ps,(rB)mf(x)|qdx(cid:21) < ∞,
where the supremum is taken over all balls B of Rn.
8 Renjin Jiang and Dachun Yang
Remark2.1. (i)Letp (0,1],q [1, )ands n(1 1) . Ifω(t) = tforallt (0, ),
∈ ∈ ∞ ≥⌊m p− ⌋ ∈ ∞
thenH (Rn)= H1(Rn)andBMOq,s(Rn) = BMO (Rn),whereH1(Rn)andBMO (Rn)
ω,L L ρ,L L L L
were introduced by Duong and Yan [16, 17], respectively. If p (n/(n + θ(L)),1) and
ω(t) = tp forallt (0, ), thenH (Rn) = Hp(Rn)andBMOq,s∈(Rn) = L (1/p 1,q,s),
∈ ∞ ω,L L ρ,L L −
where Hp(Rn) and L (1/p 1,q,s) were introduced by Yan [30] and Duong and Yan [18],
L L −
respectively.
(ii) For q [1, ) and s s n( 1 1) , the spaces BMOq,s(Rn) coincide with
∈ ∞ ≥ 0 ≡ ⌊m pe0(ω) − ⌋ ρ,L
BMO2,s0(Rn); see Corollary 3.1 andRemark 4.4 of [22]. Hence, in whatfollows, we denote
ρ,L
the space BMOq,s(Rn) simply by BMO (Rn).
ρ,L ρ,L
(iii) It was proved in [22] that the dual space of H (Rn) is the space BMO (Rn),
ω,L ρ,L∗
where L∗ denotes the adjoint operator of L in L2(Rn).
3 VMO (Rn)-type spaces
ρ,L
Suppose that the assumptions (a), (b) and (c) hold. In this section, we study the
spaces of functions with vanishing mean oscillation associated with operators and Orlicz
functions. We begin with some notions and notation.
Definition 3.1. Let L satisfy the assumptions (a) and (b), ω satisfy the assumption (c),
ρ be as in (2.10) and s n( 1 1) . A function f BMO (Rn) is said to be in
≥ ⌊m pe0(ω) − ⌋ ∈ ρ,L
VMOs (Rn), if it satisfies the limiting conditions γ (f) = γ (f)= γ (f)= 0, where
ρ,L 1 2 3
1/2
1
γ (f) lim sup f(x) P f(x)2dx ,
1 ≡ c→0ballB:rB≤c(cid:18)|B|[ρ(|B|)]2 ZB| − s,(rB)m | (cid:19)
1/2
1
γ (f) lim sup f(x) P f(x)2dx ,
2 ≡ c→∞ballB:rB≥c(cid:18)|B|[ρ(|B|)]2 ZB| − s,(rB)m | (cid:19)
and
1/2
1
γ (f) lim sup f(x) P f(x)2dx .
3 ≡ c→∞ballB⊂[B(0,c)]∁(cid:18)|B|[ρ(|B|)]2 ZB| − s,(rB)m | (cid:19)
For any function f ∈VMOsρ,L(Rn), we define kfkVMOsρ,L(Rn) ≡ kfkBMOρ,L(Rn).
We next present some properties of the space VMOs (Rn). To this end, we first recall
ρ,L
some notions of tent spaces; see [6, 19, 22].
Let Γ(x) (y,t) Rn+1 : x y < t denote the standard cone (of aperture 1) with
≡ { ∈ + | − | }
vertex x Rn. For any closed set F of Rn, denote by F the union of all cones with
∈ R
vertices in F, namely, F Γ(x); and for any open set O in Rn, denote the tent
x∈F
R ≡ ∪
over O by O, which is defined by O [ (O∁)]∁.
≡ R
For all measurable functions g on Rn+1 and all x Rn, define
+ ∈
b b
1/2
dydt
(g)(x) g(y,t)2
A ≡ | | tn+1
ZΓ(x) !
Predual Spaces of Banach Completions of Orlicz-Hardy Spaces 9
and
1/2
1 1 dydt
(g)(x) sup g(y,t)2 .
ρ
C ≡ ρ(B ) B b| | t
ballB∋x | | (cid:18)| |ZB (cid:19)
For p (0, ), the tent space Tp(Rn+1) is defined to be the set of all measurable
∈ ∞ 2 +
functions g on Rn++1 such that kgkT2p(Rn++1) ≡ kA(g)kLp(Rn) < ∞. The tent space Tω(Rn++1)
associated to the function ω is defined to be the set of all measurable functions g on Rn+1
+
such that (g) L(ω), and its norm is given by
A ∈
(g)(x)
g (g) = inf λ > 0 : ω A dx 1 ;
k kTω(Rn++1) ≡ kA kL(ω) Rn λ ≤
(cid:26) Z (cid:18) (cid:19) (cid:27)
the space T∞(Rn+1) is defined to be the set of all measurable functions g on Rn+1 such
ω + +
that kgkTω∞(Rn++1) ≡ kCρ(g)kL∞(Rn) < ∞.
In what follows, let T∞(Rn+1) be the set of all f T∞(Rn+1) satisfying η (f) =
ω,0 + ∈ ω + 1
η (f)= η (f)= 0, where
2 3
1/2
1 1 dydt
(3.1) η (f) lim sup f(y,t)2 ,
1
≡ c→0ballB:rB≤cρ(|B|) (cid:18)|B|ZBb| | t (cid:19)
1/2
1 1 dydt
η (f) lim sup f(y,t)2 ,
2
≡ c→∞ballB:rB≥c ρ(|B|) (cid:18)|B|ZBb| | t (cid:19)
and
1/2
1 1 dydt
η (f) lim sup f(y,t)2 .
3
≡ c→∞ballB⊂[B(0,c)]∁ ρ(|B|) (cid:18)|B|ZBb| | t (cid:19)
It is easy to see that T∞(Rn+1) is a closed linear subspace of T∞(Rn+1).
ω,0 + ω +
Further, denote by T∞(Rn+1) the space of all f T∞(Rn+1) with η (f) = 0, and
ω,1 + ∈ ω + 1
T2 (Rn+1) the space of all f T2(Rn+1) with compact support. Obviously, we have
2,c + ∈ 2 +
T2 (Rn+1) T∞(Rn+1) T∞(Rn+1). Similarly, let T∞(Rn+1) denote the set of all
2,c + ⊂ ω,0 + ⊂ ω,1 + ω,c +
f T∞(Rn+1) with compact support. It is easy to see that T∞(Rn+1) coincides with
∈ ω + ω,c +
T2 (Rn+1). Define T∞(Rn+1) to be the closure of T∞(Rn+1) in T∞(Rn+1).
2,c + ω,v + ω,c + ω,1 +
Similarly to the proof of Lemma 3.2 in [11], we have the following proposition. We
omit the details.
Proposition 3.1. Let T∞(Rn+1) and T∞(Rn+1) be defined as above. Then T∞(Rn+1)
ω,v + ω,0 + ω,v +
and T∞(Rn+1) coincide with equivalent norms.
ω,0 +
Recall that a measure dµ on Rn+1 is said to be a ρ-Carleson measure if
+
1
sup dµ < ,
B⊂Rn(cid:26)|B|[ρ(|B|)]2 ZBb| |(cid:27) ∞
where the supremum is taken over all balls B of Rn; see [19, 22].
10 Renjin Jiang and Dachun Yang
Let m be the constant in (2.1), s s n( 1 1) , where p (ω) is the critical
≥ 1 ≥ ⌊m pe0(ω) − ⌋ 0
lower type index of ω. Let C be a positive constant such that
m,s,s1
∞ dt e
(3.2) C tm(s+2)e−2tm(1 e−tm)s1+1 = 1.
m,s,s1 − t
Z0
The following Lemma 3.1 and Theorem 3.1 were established in [22].
Lemma 3.1. Let L, ω and ρ be as in Definition 3.1, and s s n( 1 1) . Suppose
≥ 1 ≥ ⌊m pe0(ω)− ⌋
that f ∈ M(Rn)suchthat|Qs,tm(I−Ps1,tm)f(x)|2dxdt/t isaρ-Carlesonmeasure onRn++1
and g H (Rn) L2(Rn). Then
ω,L∗
∈ ∩
dxdt
Rnf(x)g(x)dx = Cm,s,s1 Rn+1Qs,tm(I −Ps1,tm)f(x)Q∗tmg(x) t .
Z Z +
Theorem 3.1. Let L, ω and ρ be as in Definition 3.1, and s s n( 1 1) . Then
≥ 1 ≥ ⌊m pe0(ω)− ⌋
the following conditions are equivalent:
(a) f BMO (Rn);
ρ,L
∈
(b) f ∈ M(Rn) and |Qs,tm(I−Ps1,tm)f(x)|2dxdt/t is a ρ-Carleson measure on Rn++1.
Moreover, kQs,tm(I −Ps1,tm)fkTω∞(Rn++1) is equivalent to kfkBMOρ,L(Rn).
Wenow establishacharacterization ofthespaceVMOs (Rn)viathespaceT∞(Rn+1)
ρ,L ω,v +
by borrowing some ideas from [11]. We remark that if ω(t) = t for all t (0, ), then
∈ ∞
Theorem 3.2 coincides with Proposition 3.3 in [11].
Theorem 3.2. Let L, ω and ρ be as in Definition 3.1, s s n( 1 1) and
1 ≥ 0 ≥ ⌊m pe0(ω) − ⌋
s 2s . Then the following conditions are equivalent:
1
≥
(a) f VMOs0 (Rn);
∈ ρ,L
(b) f ∈ M(Rn) and Qs,tm(I −P2s1,tm)f ∈ Tω∞,v(Rn++1).
Proof. We first notice that by the assumption (c), there exists ǫ (nβ (ρ),θ(L)). Recall
1
∈
that β (ρ) = 1/p (ω) 1 and p (ω) is as in the convention. To see that (a) implies (b),
1 0 0
−
by the fact VMOs0 (Rn) BMO (Rn) together with Theorem 3.1, we only need to
ρ,L ⊂ ρ,L
verify Qs,tm(I −P2s1,tm)f ∈ Tω∞,v(Rn++1). To this end, we need to show that for all balls
B B(x ,r ),
B B
≡
1/2 ∞
1 dxdt
(3.3) ρ(B )B 1/2 b|Qs,tm(I −P2s1,tm)f(x)|2 t . 2−kǫδk(f,B),
| | | | (cid:18)ZB (cid:19) k=1
X
where ǫ (nβ (ρ),θ(L)) and
1
∈
1/2
1
δ (f,B) sup f(x) P f(x)2dx .
k ≡ ρ(B′ )B′ 1/2 | − s0,(rB′)m |
B′⊂2k+1B:rB′∈[rB/2,2rB] | | | | (cid:18)ZB′ (cid:19)
In fact, since f ∈ VMOsρ0,L(Rn)⊂ BMOρ,L(Rn), we have δk(f,B)≤ kfkBMOρ,L(Rn). More-
over, for each k N, we have
∈
lim sup δ (f,B) = lim sup δ (f,B)= lim sup δ (f,B)= 0.
k k k
c→0B:rB≤c c→∞B:rB≥c c→∞B⊂B(0,c)∁