DECOMPOSITION THEOREMS FOR HARDY-ORLICZ SPACES
AND WEAK FACTORIZATION
ALINE BONAMI AND SANDRINE GRELLIER
Abstract. We study the holomorphic Hardy-Orlicz spaces HΦ (Ω), where Ω is
the unit ball or more generally, either a convex domain of finite type or a strictly
pseudoconvex domain in Cn . We prove that these spaces HΦ (Ω) admit FeffermanStein maximal characterizations so that it allows to obtain atomic decomposition.
P
More precisely, each function f ∈ HΦ (Ω) can be written as f = PS ( ∞
j=0 (aj ) =
P∞
and the aj ’s are real variable atoms
j=0 PS (aj ), where PS is the Szegö projection
P∞
on the boundary ∂Ω supported on Bj with j=0 σ(Bj )Φ(kaj k∞ ) <
∼ kf kHΦ(Ω) (the
atomic decomposition of real Hardy-Orlicz spaces has been proven previously by
Viviani in the context of homogeneous spaces). These Hardy-Orlicz spaces appear
naturally since, for instance, the product of a function in BMOA(Ω) with a function
tp
in Hp (Ω), 0 < p ≤ 1 belongs to HΦ (Ω) with Φ(t) = Φp (t) = log(e+t)
p.
Atomic decomposition allows to prove weak factorization theorems that is a
weak converse statement. Each function f ∈ HΦp (Ω) can be written as f =
P
∞
p
j=0 fj gj , where fj ∈ H (Ω), gj ∈ BMOA(Ω). This weak type factorization result
allows to characterize those Hankel operators which are bounded from Hp (Ω) into
H1 (Ω). We also characterize those Hankel operators which are bounded from
HΦ (Ω) into H1 (Ω).
Introduction
Let Bn be the unit ball and Sn be the unit sphere in Cn . We consider the HardyOrlicz space HΦ (Bn ) which generalizes the usual Hardy spaces Hp (Bn ) when Φ(t) =
tp
tp . We are especially interested in the case Φp (t) = log(e+t)
p , 0 < p ≤ 1 since the
Φp
n
space H (B ) arises naturally in the study of pointwise product of functions in
Hp (Bn ) with functions in BMOA(Bn ). Namely, the product of an Hp (Bn )-function
and of a BMOA(Bn )-function belongs to the space HΦp (Bn ). The hard part of this
paper is to prove
a weak
converse
result usually
referenced as a weak factorization
CRW
GL
KL2
BPS
GP
Theorem (see [CRW], [GL],[KL2], [BPS1], [GP]).
For Φ continuous, positive, non-decreasing on R+ and satisfying some technical assumptions precised later, the Hardy-Orlicz space is the space of holomorphic
1991 Mathematics Subject Classification. 32A37 47B35 47B10 46E22.
Key words and phrases. Orlicz spaces, atomic decomposition, finite type domains, convex
domains.
This work was done within the project HARP Network. We thank the European Commission
and the mentioned Network for the support provided.
1
2
A. BONAMI AND S. GRELLIER
functions f so that
sup
0<r<1
Z
Φ(|f (rw)|) dσ(w) < ∞
Sn
where dσ denotes the surface measure on Sn .
The technical assumptions on Φ ensure that H1 (Bn ) ⊂ HΦ (Bn ) ⊂ Hp (Bn ) for
some 0 < p ≤ 1. In particular, any function f in the Orlicz space HΦ (Bn ) admits
aZ unique boundary function still denoted by f which, by Fatou Theorem, satisfies
Φ(|f |)dσ < ∞.
Sn
We also consider the real-variable Hardy-Orlicz space H Φ (Sn ) defined as the space
Φ n
of distributions on Sn which are sums of atoms.
P∞ More precisely, H (S ) is the set
of distributions f which can be written as j=0 aj , where the aj ’s are supported in
P
some ball Bj so that j σ(Bj )Φ(kaj k∞ ) < ∞. The series is assumed to converge in
basic-facts
the sense of distributions (see Section 1 for the precise definition).
We first prove usual maximal characterizations of Hardy-Orlicz spaces. As a
corollary, we obtain that the Hardy-Orlicz space HΦ (Bn ) continuously embeds into
H Φ (Sn ). In other words, every f ∈ HΦ (Bn ) has boundary values that belong to
H Φ (Sn ), so that it admits an atomic decomposition. This follows mainly from the
work of Viviani
where atomic decomposition is proved in the context of homogeneous
Viviani
domains ([V]).
On the other hand, we prove that the Szegö projection PS (the orthogonal projection from L2 (Sn ) into H2 (Bn )) maps continuously H Φ (Sn ) into HΦ (Bn ).
The atomic decomposition allows to prove a (weak) factorization theorem on
Ψ
H (Bn ) which concides with the one for Hp (Bn ) when Ψ(t) = tp . In
particular,
BIJZ
we generalize the factorization theorem proved in the disc for HΦ1 in [BIJZ]. More
precisely, we prove that, given
any f ∈ HΨ (Bn ) there exist fj ∈ HΦ (Bn ), gj ∈
P
∞
BMOA(Bn ) such that f =
Ψ and Φ are linked by the relation
j=0 fj gj where
P∞
t
Ψ(t) = Φ( log(e+t) . Furthermore, one has j=0 kfj kHΦ (Bn ) kgj kBMOA < ∞.
As a consequence, we can characterize the class of symbols for which the Hankel
operators are bounded from HΦ (Bn ) to H1 (Bn ). Those symbols belong to the dual
space of HΨ (Bn ) which can be identified with the BMOA-space with weight ρΨ where
ρΨ (t) = tΨ−11(1/t) . Those spaces are defined by
Z
1
C
BMOA(ρΨ ) := f ∈ H(B );
|f − fB |dx ≤
= CρΨ (|B|) .
|B| B
|B|Ψ−1(1/|B|)
Z
We note kf kBMOA(ρΨ ) := inf C +
|f |. When Ψ = Φ1 , this space is usually referred
n
Bn
as the space LMOA of functions
of logarithmic mean oscillation. This duality result
Janson
n
has been proven in R by [J]. We give the proof in our setting in paragraph??.
Here H(Bn ) denotes the space of holomorphic functions in Bn .
HARDY-ORLICZ SPACES
3
< B to indicate that A ≤ c · B where the constant c does
We use the notation A ∼
not depend on the important parameters on which the functions A and B depend.
(Typically, the constant c will only depend on the geometry of the domain Bn .) We
use the symbols >
∼ and ≈ analogously.
1. Statements of the results
basic-facts
Viviani
1.1. Basic definitions and notations. As in [V], we assume that Φ satisfies some
technical properties given in the following definition.
Definition 1.1. Let 0 < p ≤ 1. A function Φ is said to be a growth function of
order p if it satisfies the following properties:
(1) The function Φ is defined on R+
∗ , real valued, continuous, so that lim0+ Φ = 0
and is strictly increasing.
(2) The function Φ is of lower type p i.e. there exists a constant c > 0 such that,
for any 0 < t ≤ 1,
cond1
Φ(st) ≤ ctp Φ(s).
(1)
Φ(t)
Φ(t)
is non-increasing and the function t 7→ l is quasit
t
increasing for any l ≤ p (there exists a constant c so that, for any s ≤ t,
Φ(t)
Φ(s)
≤ c l .)
l
s
t
(4) The function Φ is subbadditive.
(3) The function t 7→
The last property may be not satisfied in general (our example Φp does not satisfy
Z t
Φ(s)
this property for instance), in that case it suffices to replace Φ(t) by
ds.
s
0
Such a function satisfies the same properties as Φ does but is also subbadditive.
Φ(t)
′
Furthermore, it admits a derivative on R+
∗ so that Φ (t) = t .
Let us begin with some basic remarks on the “norms” on HΦ . If one uses the
so-called Luxembourg norm on HΦ , defined by
Z
|f (rw)|
lux
dσ(w) ≤ 1 ,
|[f ]|HΦ = inf λ > 0 : sup
Φ
λ
0<r<1 Sn
it has homogeneous properties so that it allows to obtain Hölder type inequality.
The disadvantage is that it does not satisfy the triangle inequality. Since we need to
consider infinite sums of atoms in HΦ , it will be helpful to use a quasi-norm which
does satisfy the triangle inequality but which may be non-homogeneous.
We consider
Z
kf kHΦ := sup
Φ(|f (rw)|) dσ(w).
0<r<1
Sn
Both norms give rise to the same space HΦ since Φ is of lower type p. Let us now
give the precise definition of the real-variable Hardy-Orlicz spaces. We first need to
4
A. BONAMI AND S. GRELLIER
introduce the notion of atoms. Let ζ0 ∈ Sn , r0 < 1 and N be a positive integer. We
denote by B(ζ0 , r0 ) the anisotropic ball on Sn defined by
B(ζ0 , r0 ) := {ζ ∈ Sn ; d(ζ, ζ0) := |1 − hζ, ζ0i| ≤ r0 }.
As usual, we denote by Li,j , 1 ≤ i < j ≤ n the complex differential operators in
the complex tangential directions defined by
∂
∂
− zi
Li,j := zj
∂zi
∂zj
SN
and by T the complex tangential differential operator
corresponding to the so-called
P
∂
“missing direction” T := ℑN where N =
j zj ∂zj . For any multi-index µ, we
µ
denote by L any differential operator obtained by composition of |µ| of the Lij ’s.
On C ∞ (B(ζ0 , r0 )) we introduce the norm
X
l+|µ|/2
(2)
kϕkSN (B(ζ0 ,r0 )) =
kT l Lµ ϕkL∞ (B(ζ0 ,r0 )) r0
|µ|+l=N
where the sum runs over all possible choice of differential operators T l Lµ .
atom
momentcond
Hardy-Orlicz
-convergence
Definition 1.2. A measurable function a on Sn is called an atom of order N ∈ N∗
if either it is the constant function on Sn or if, for some ζ0 ∈ Sn and r0 > 0,
(1) Rsupp a ⊆ B(ζ0 , r0 );
(2) Sn a(ζ)dσ(ζ) = 0;
(3) for all ϕ ∈ C ∞ B(ζ0 , r0 ) we have,
Z
≤ kϕkS (B(ζ0 ,r0 )) σ B(ζ0 , r0 ) kak∞
a(ζ)ϕ(ζ)dσ(ζ)
N
n
S
momentcond
Notice that the above condition 3 replaces the classical higher moment condition.
We choose this way to write it since it does not depend on the choice of local
coordinates.
Real-variable Hardy-Orlicz spaces.
Definition 1.3. The real Hardy-Orlicz space H Φ (Sn ) is the space of distributions f
on Sn which can be written as
∞
X
(3)
f=
aj ,
j=0
1
p
P
where the aj ’s are atoms of order N ≥ 2n
− 1 so that
σ(Bj )Φ(kaj k∞ ) < ∞.
The serie is assumed to converge in the sense of distributions.
With
the “norm” on H Φ (Sn ) is defined as kf kH Φ =
P a standard abuse of notation,
P
inf{ j σ(Bj )Φ(kaj k∞ ) : f =
= kf − gkH Φ , H Φ (Sn ) is a
j aj }. With d(f, g) norm-convergence
complete metric space. This
the series in (3) converges in metric. This
Pm2implies thatP
2
is in fact obvious since k j=m1 aj kH Φ . m
j=m1 σ(Bj )Φ(kaj k∞ ) which tends to 0 as
HARDY-ORLICZ SPACES
5
m1 , m2 → ∞. Remark that convergence in H Φ (Sn ) implies convergence in the sense
of distribution.
One of the main tools in the atomic decomposition of Hardy spaces is the use of
their maximal characterizations. For Hardy-Orlicz spaces, we have the corresponding characterizations. Let us give some definitions.
Given ζ ∈ Sn we define the approach region Aα (ζ) as the subset of Bn given by
Aα (ζ) = {z = rw ∈ Bn : d(ζ, w) = |1 − hζ, wi| < α(1 − r)}.
We define the non-tangential maximal function of f by Mα (f )
f-star-gamma
Mα (f )(ζ) = sup |f (z)|,
(4)
z∈Aα (ζ)
and the tangential variant
ouble-star-M
(5)
NM (f )(ζ) =
sup
z=rw∈Bn
1−r
(1 − r) + d(ζ, w)
M
|f (rw)|.
Here d(ζ, w) denotes the pseudodistance on Sn given by d(ζ, w) := |1 − hζ, wi|.
max-charact
sup
Theorem 1.4. Let α > 0 and M large enough. There exists a constant C > 0 so
that, for any f ∈ HΦ ,
sup |f (r·)|)
(6)
1 n ≤ Ckf kHΦ (Bn )
Φ(0<r<1
L (S )
aire
atomic-decom
(7)
kΦ(Mα (f ))kL1 (Sn ) ≤ Ckf kHΦ (Bn )
(8)
kΦ(NM (f ))kL1 (Sn ) ≤ Ckf kHΦ (Bn )
The main part of these maximal characterizations are the characterizations by
the radial maximal function and the non-tangential one. The other one follows by
usual technique.
Theorem 1.5. Let N ∈ N∗ be larger than 2n(p−1 − 1). There exists a constant c
depending only on Bn such that the P
following holds. Given any f ∈ HΦ (Bn ) there
Φ n
exist atoms aj of order N such that ∞
j=0 aj ∈ H (S ) and
X
X
∞
∞
f = PS
aj =
PS (aj ).
j=0
Moreover
∞
X
j=0
σ(Bj )Φ(kaj k∞ ) ≃ kf kHΦ (Bn ) .
j=0
Remark 1.6. As it is well known, the order of ”moment conditions” of the atoms
can be chosen arbitrarily large.
We now introduce the notion of molecules.
6
A. BONAMI AND S. GRELLIER
molecule
Definition 1.7. A holomorphic function A is called a molecule of order N, associated to the ball B := B(z0 , r0 ) ⊂ Sn if it satisfies
Z
1/2
d(z0 , ξ)N
2 dσ(ξ)
|A(ξ)|
kAk2,B,N :=
≤ kak∞ .
σ(B)
r0N
Sn
ule-property
Theorem 1.8.
(1) For any atom a of order L supported on the ball B :=
B(z0 , r0 ) ⊂ Sn , its Poisson extension PS (a) is a molecule associated to B
of any order N < L/2. It satisfies
kAk2,B,N ≤ kak∞ .
(2) Any molecule A belongs to HΦ (Bn ) with
kAkHΦ ≤ Φ(kAk2,B,N )σ(B).
As a consequence we have the following molecular decomposition of functions in
HΦ (Bn ).
Theorem 1.9. For any f ∈ HΦ (Bn ), there exists molecules Aj so that f may be
written as
X
Aj
f=
j
with kf kHΦ (Bn ) ≃
boundedness
P
j
Φ(kAj k2,Bj ,N )σ(Bj ).
Theorem 1.10. The Szegö projection
PS : H Φ (Sn ) → HΦ (Bn )
is bounded.
As a first consequence of the atomic decomposition of the Hardy-Orlicz spaces,
we identify their dual spaces as some BMO-spaces with weight as follows.
duality
Theorem 1.11. Let Φ be a growth function of order p. The dual space of HΦ (Bn )
is the set of function denoted by BMOA(̺) defined by
Z
1
n
BMOA(̺) = f ∈ H(B ); there exists a constant C > 0;
|f − fB |dσ ≤ C̺(σ(B)
σ(B) B
1
where ̺(t) = ̺Φ (t) := −1
. The duality is given as usual by the scalar product
tΦ (1/t)
on L2 (Sn ) which makes sense by taking the limit as r goes to 1 of the integrals over
the spheres of radius r.
The
equality between the dual space of HΦ (Bn ) and BMOA(ρΦ ) has been proven
Janson n
by [J] in R .
Finally, let us remark that the product of a function in HΦ (Bn ) with a function
in BMOA(Bn ) is well defined. If b belongs to BMOA(Bn ) and h to HΦ (Bn ), b × h is
the limit of the products b(r·)h(r·) when r goes to 1. We prove the following.
HARDY-ORLICZ SPACES
7
n
n
Proposition 1.12. Thespace HΦ (B
) × BMOA(B ) is continuously embedded in
t
.
HΨ (Bn ) where Ψ(t) = Φ
log(e + t)
In fact, the product of HΦ (Bn ) with the exponential class is continuously embedded
in HΨ (Bn ).
Proof. One can prove that, for any a, b > 0,
ab
≤ a + eb − 1.
log(e + ab)
It follows that
Ψ(ab) . Φ(a + eb − 1) . Φ(a) + eb − 1.
Applying this inequality on each sphere of radius less than 1, we conclude, by usual
arguments, that the ”Luxembourg”-norm satisfies Hölder inequality so that
kf gklux
HΨ . kf kHΦ kgkexpL ≤ kf kHΦ kgkBMOA
by Jensen Inequality.
For the converse statement, we have the following (weak) factorization theorem.
actorization
Theorem 1.13. There exists a constant c depending only on Bn such that the following holds. Given any f ∈ HΨ (Bn ) there exist fj ∈ HΦ (Bn ), gj ∈ BMOA(Bn ),
j ∈ N such that
∞
X
fj gj .
f=
j=0
This equality holds in the sense of distributions and
P∞
j=0 kfj kHΦ kgj kBMOA
< ∞.
As a corollary, we obtain the following characterization of bounded Hankel operators. Recall that, for b ∈ H2 (Bn ), the Hankel operator hb of symbol b is given, for
smooth functions f , by hb (f ) = PS (bf )
Hankel
Corollary 1.14. Any Hankel operator hb is bounded from HΦ (Bn ) to H1 (Bn ) if and
only if b ∈ (HΨ (Bn ))′ = BMOA(̺Ψ ).
All of these results may be extended to the more general setting of strictly pseudoconvex domains orCasgeneral
of convex domains of finite type in Cn . We give a sketch of
the proofs in section 7.
2. Proof of the maximal characterizations of Hardy-Orlicz spaces
Here we prove the maximal functions FS
characterizations of Hardy-Orlicz spaces.
The proof is analogous to the usual one ([FS] for instance). We first need a lemma
on the boundedness of the Hardy-Littlewood maximal operator. This result is surely
well known but as it is easy to prove, we give it here with a simple proof.
Lemma 2.1. Let Φ be a growth function of order p, then Φ satisfies the following
two properties:
8
A. BONAMI AND S. GRELLIER
(1) For any q > 1/p, there exists a constant C so that, for any s > 0,
Z s
Φ(s)
Φ(t)
dt ≤ C 1/p .
1+1/p
s
0 t
(2) For any q > 1/p, there exists a constant C > 0 so that, for any smooth
function f ,
Z
Z
HL
1/q q
(Φ((M (|f | )) )dσ ≤ C
Φ(|f |)dσ.
Sn
Sn
Proof. Proof of the Lemma.
The first property follows easily from the fact that Φ is of lower type p. We write
for any q > 1/p,
Z s
Z 1
Z
Φ(t)
Φ(s) 1 p−1−1/q
Φ(s)
Φ(st)
−1/q
dt = s
dt ≤ 1/q
t
dt ≤ C 1/q .
1+1/q
1+1/q
s
s
0 t
0 t
0
Let us consider now the maximal Hardy-Littlewood operator.
Denote by Ψ = Ψq the function defined by Ψ(t) := Φ(tq ) hence,
Z sas Φ can be chosen
Ψ(s)
Ψ(t)
Ψ(t)
and
dt ≤ C
.
, the function Ψ satisfies Ψ′ (t) ≃
so that Φ′ (t) = Φ(t)
t
t
t
s
0
We have
Z
Z ∞
HL
(Ψ((M (|g|)))dσ =
Ψ′ (t)σ MHL (|g|) ≥ t dt
Sn
Z0 ∞
Ψ(t)
σ MHL (|g|) ≥ t dt by assumption on Φ
≃
t
Z0 ∞
Z
.
Ψ(t)
|g|dσdt by the weak (1,1) boundedness of MHL
0
=
Z
|g|≥t
|g|
Sn
Z
|g|
0
max-charact
Ψ(t)
dtdσ .
t
Z
Ψ(|g|)dσ.
Sn
Proof. Proof of Theorem 1.4.
Let us prove first the non-tangential maximal characterization of Hardy-Orlicz
spaces. The radial maximal characterization will follow at once. It is as usual clear
that if, for some α > 0, kΦ(Mα (f ))kL1 (Sn ) < ∞ then f belongs to HΦ (Bn ). Let
us prove the converse. Let f be in HΦ (Bn ). As Φ is assumed to be of lower type
p, Φ(t) ≥ Ctp for any t ≥ 1, for some constant C. It follows that f belongs to
˜ Furthermore, these boundary
Hp (Bn ) and hence that f admits boundary values f.
1 n
˜
˜ L1 (Sn ) ≤ kf kHΦ . Let
values satisfy Φ(|f |) ∈ L (S ) by Fatou Theorem with kΦ(|f|)k
q be any real strictly larger than 1/p. As |f |1/q is subharmonic, it admits as least
˜ 1/q , denoted by F .
harmonic majorant the Poisson extension of |f|
As in the standard Hardy spaces theory, we have
˜ 1/q ).
Mα (|f |1/q ) ≤ Mα (F ) ≤ MHL (|f|
HARDY-ORLICZ SPACES
So, we have
Z
Z
Φ(Mα (|f |)dσ ≃
Sn
Φ((Mα (|f |
1/q
q
)) )dσ .
Sn
9
Z
˜ 1/q )q )dσ.
Φ(MHL (|f|
Sn
Hence the Lemma gives
Z
Z
Φ(Mα (|f |)dσ ≤ C
Φ(|f˜|)dσ ≤ kf kHΦ .
Sn
Sn
It finishes the proof for the non-tangential maximal function and also for the
radial maximal function.
The proof of the characterization by its tangential variant
FS
NM is standard([FS] for instance).
As in the usual atomic decomposition, we need to introduce the notion of grand
maximal function. We consider a space of smooth bump functions at ζ:
KαM (ζ) = {ϕ ∈ C ∞ (Sn ) : supp ϕ ⊆ B(ζ0 , t0 ), with ζ0 ∈ Aα (ζ) and kϕkM,ζ0 ,t0 ≤ 1},
where
gamma-M-zeta
(9)
kϕkM,ζ0 ,t0 =
sup
Λ, |µ|+l≤M
|µ|/2+l
t0
σ(B(ζ0 , t0 ))kLµΛ T l ϕkL∞ (B(ζ0 ,t0 )) .
We say that a function ψ is a smooth bump function of order M on B(ζ0 , t0 ) ⊆ Sn
if ψ ∈ C ∞ (B(ζ0 , t0 )) and
|µ|/2+l
|Lµ T l ψ(z)|t0
eta-function
≤ Cψ ,
for all z ∈ B(ζ0 , t0 ) and all choices of l ∈ N and multi-index µ so that |µ| + l ≤ M.
If Cψ = 1, ψ is called a normalized smooth bump function of order M.
The grand maximal function is defined as
Z
(10)
Kα,M (f )(ζ) = sup
f (w)ϕ(w)dσ(w).
ϕ∈KM
α (ζ)
Sn
The following result is proved in many papers and holds independently of the
setting.
one
Lemma 2.2. With the definitions above, there exist c = c(Bn ) and M̃ = M̃ (α, M)
such that
Kα,M f (ζ) <
∼ Mcα (f )(ζ) + NM̃ (f )(ζ).
We now turn to the atomic decomposition. We first prove in the next section that
holomorphic extensions of functions in H Φ (Sn ) are functions of the Hardy-Orlicz
space.
10
A. BONAMI AND S. GRELLIER
boundedness
3. Proof of the extension Theorem 1.10
Let f =
P∞
j=0 aj
∈ H Φ (Sn ). By definition,
PS
X
∞
j=0
aj (z) = h
=
=
∞
X
aj , S(z, ·)i
j=0
∞
X
haj , S(z, ·)i
j=0
∞
X
PS (aj )(z),
j=0
since the serie is assumed to converge in the sense of distributions. It remains to
prove that this last term belongs to HΦ (Bn ), with norm controlled by kf kH Φ (Sn ) .
We claim that there exists C > 0 such that
estim PSa
kPS (a)kHΦ (Bn ) ≤ Cσ(B)Φ(kak∞ )
(11)
for any H Φ -atom a. From this, it then follows that, for any m1 , m2 ∈ N, m1 ≤ m2 ,
k
m2
X
PS (aj )kHΦ ≤ C
j=m1
≤ C
m2
X
j=m1
m2
X
kPS (aj )kHΦ
σ(Bj )Φ(kaj k∞ )
j=m1
so that, by assumption on the P
convergence of this serie and the completeness of
Φ
n
to HΦ (Bn ). Moreover, one has
H (B ), one gets that PS (f ) = ∞
j=0 PS (aj ) belongs
P
P
<
kPS (f )kHΦ ∼
j σ(Bj )Φ(kaj k∞ ) whenever f =
j aj .
Φ
So, one needs to estimate kPS (a)kHΦ for any H -atom a. Let ε > 0. Write
Z
Φ(|PS (a)(ζ)|)dσ(ζ) =
Sn
Z
+
B(ζ0 ,2δ)
= I + II.
Then, as Φ is non-decreasing and t 7→
Φ(t)
t
Z
c B(ζ ,2δ)
0
Φ(|PS (a)(ζ)|)dσ(ζ)
is non-increasing, we have
Φ(|PS (a)|) ≤ Φ(|PS (a)| + kak∞ ) ≤
|PS (a)|
+ 1 Φ(kak∞ ).
kak∞
HARDY-ORLICZ SPACES
11
It allows to obtain
"
#
Z
1/2
1
1/2
|PS (a)(ζ)|2 dσ(ζ)
σ B(ζ0 , 2δ)
+ σ B(ζ0 , 2δ)
I ≤ CΦ(kak∞ )
kak∞
n
S
1/2
kakL2 (Sn )
σ B(ζ0 , 2δ)
+ σ B(ζ0 , 2δ)
≤ CΦ(kak∞ )
kak∞
≤ Cσ(B)Φ(kak∞ )
since PS maps L2 (Sn ) into L2 (Sn ). By assumption on a, we have
Z
S(ζ, w)a(w)dσ(w)
|PS (a)(ζ)| = Sn
≤ kS(ζ, ·)kSN (B(ζ0 ,δ)) · σ B(ζ0 , 2δ) kak∞ ,
Next, set Ek = {ζ ∈ Sn : 2k δ ≤ d(ζ, ζ0) ≤ 2k+1 δ}.
it follows from the size estimates of the derivatives of the Szegö kernel that, for
any ζ ∈ Ek ,
1
< 2−kN/2
kS(ζ, ·)kSN (B(ζ0 ,δ)) ∼
σ(B(ζ0 , δ2k−1 ))
so that, for any ζ ∈ Ek ,
−k(n+N/2)
kak∞ .
|PS (a)(ζ)| <
∼2
Going back to the estimation of II, we have
II =
∞ Z
X
k=1
<
∼
X
k
≤c
−k n+ N
k
2 kak
Φ 2
∞ σ(B(ζ0 , δ2 ))
X
k
Φ(|PS (a)(ζ)|)dσ(ζ)
Ek
Φ(kak∞ )2−pk
n+ N
2
≤ CΦ(kak∞ )σ(B(ζ0 , δ))
× 2kn σ(B(ζ0 , δ))
for N large enough so that n(1 − p) − pN/2 < 0. 2
Once maximal characterizations are proved, the atomic decomposition of Hardy
space on homogeneous domains follows fromViviani
standard techniques. It has been proven
in the context of Hardy-Orlicz spaces by [V]. We give here the main steps of the
proof for completeness.
12
A. BONAMI AND S. GRELLIER
atomic-decom
4. Proof of the atomic decomposition Theorem 1.5
Let f be a fixed function in HΦ . As noticed before, f admits boundary values
defined a.e. on Sn , that we still denote by f .
We fix also N an integer.
We let k0 to be the least integer such that
k-0
kΦ(Kα,M (f ) + Mα (f ))kL1 (Sn ) ≤ 2k0 .
(12)
For a positive integer k, we define
O-k
Ok = {z ∈ Sn : Kα,M f (z) + Mα (f )(z) > 2k0 +k }.
(13)
For each k, we then fix a Whitney covering and a partition of 1lOk , {ϕki } adapted to
this covering {Bik } that is
(i) 0 ≤ ϕki ≤ 1;
(ii) supp ϕki ⊆ Bik ;
(iii) ϕi = 1 on 12 Bik ;
P
k
(iv) ∞
i=0 ϕi = 1lOk ;
(v) for each i, k there exists ζi,k ∈c Ok such that, for any integer M we have
cM ϕki /kϕki kL1 ∈ KαM (ζik ), for some cM = c(M, Bn ).
Let ζ0 be fixed on Sn . We assume chosen a local system of coordinates (w1 , . . . , wn )
around ζ0 so that Re (w1 ) corresponds to the normal direction at ζ0 , (w2 , . . . , wn ) to
the complex tangential space at ζ0 and Im (w1 ) to the so-called missing direction. We
define Vℓ (ζ0 ) to be the space of polynomials of degree ≤ ℓ in Im w1 , w2 , w2 , . . . , wn , wn .
Let ϕ0 be a smooth bump function supported on B(ζ0 , r0 ). We denote by L2ϕ0 (dσ)
the L2 -space with respect to the probability measure (ϕ0 /kϕ0 kL1 )dσ. We let Pϕ0 denote the orthogonal projection of L2ϕ0 (dσ) onto Vℓ (ζ0 ) (which is obviously contained
in L2ϕ0 (dσ)). We now fix an integer N large enough, the order of the atoms that
we are going to construct.
We then take ℓ = N − 1 in the following. The following
GP
lemma is proved in [GP].
P-phi-i-k
Lemma 4.1. With the notation fixed above, there exists c > 0 such that for f ∈
HΦ (Bn )
∞
X
|
Pφki (f )(z)ϕki (z)| ≤ c2k .
i=0
Furthermore,
∞
X
P
φk+1
j
j=0
k
≤ c2k+1 .
f − Pφk+1 (f ) ϕi (z)ϕk+1
(z)
j
j
We write,
∞
∞
∞
X
X
X
f= f−
f − Pφki (f ) ϕki +
f − Pφki (f ) ϕki =: hk +
f − Pφki (f ) ϕki .
i=0
i=0
i=0
HARDY-ORLICZ SPACES
13
P
k
k
As the support of the function ∞
i=0 f − Pφki (f ) ϕi is included in O , it tends to
zero almost everywhere as k goes to infinity, hence hk tends to f so that
∞
X
(hk+1 − hk ).
(14)
f = h0 +
a.e.
k=0
circled-2
Using the equality
∞
X
k
(15)
Pφk+1 f − Pφk+1 (f ) ϕi = Pφk+1 f − Pφk+1 (f ) = 0
j
j
j
j
i=0
we can rewrite hk+1 − hk as
=
=
=
∞
X
∞
k X
f − Pφki (f ) ϕi −
f − Pφk+1 (f ) ϕk+1
j
j
i=0
∞ X
i=0
∞
X
j=0
∞ X
f − Pφki (f ) ϕki −
f − Pφk+1 (f )
j
j=0
ϕki
k
k+1
− Pφk+1 (f − Pφk+1 (f ))ϕi ϕj
j
j
bki ,
i=0
b-i-k
m-equation-1
where
(16)
∞ ∞ X
k X
k
k+1
k
k
bi =
f −Pφki (f ) ϕi −
f −Pφk+1 (f ) ϕi −Pφk+1 (f −Pφk+1 (f ))ϕi ϕj
j
i=0
j
j
j=0
Hence,
(17)
f = h0 +
∞ X
∞
X
bki .
k=0 i=0
It remains to see that the above is the desired atomic decomposition of f
Estimate for h0 . By definition,
h0 = f0 +
∞
X
Pϕ0i (f )ϕ0i .
i=0
We have
∞
X
Pϕ0 (f )ϕ0i ≤ c,
i
i=0
and, by definition,
|f0 | ≤ Mα (f )|c O0 ≤ 2k0 .
Then, kh0 kL∞ ≤ c2Rk0 , so that h0 is a so called
R ”junk atom” bounded and supported
in Sn (in fact h0 − h0 dσ is an atom and h0 dσ is a constant).
14
A. BONAMI AND S. GRELLIER
Size estimates for the bki ’s. We have,
∞ k X
k
k+1 k
k
|bi | = f − Pφki (f ) ϕi −
f − Pφk+1 (f ) ϕi − Pφk+1 (f − Pφk+1 (f ))ϕi ϕj j
≤ f − Pφki (f ) ϕki −
j=0
∞
X
f − Pφk+1 (f )
j
j=0
X
∞ + ϕki ϕk+1
j
Pφk+1 f − Pφk+1 (f )
j
j
j
j
j=0
ϕki
.
ϕk+1
j
P-phi-i-k
The second term on the right hand-side above is bounded by c0 c2k+1 by Lemma 4.1,
while the first one is bounded by
∞
X
k k+1 f − Pφki (f ) − f − Pφk+1 (f ) ϕi ϕj + f − Pφki (f ) ϕki 1lc Ok+1 j
j=0
∞
X
k k+1 Pφk+1 (f ) − Pφki (f ) ϕi ϕj + f − Pφki (f ) ϕki 1lOk \Ok+1 ≤ j
j=0
X
∞
k+1
Pφk+1 (f )ϕj + |Pφki (f )ϕki | + |f 1lOk \Ok+1 | + |Pφki (f )ϕki |
≤ j
j=0
≤ c0 c2k + Mα (f )1lOk \Ok+1
≤ c0 c2k ,
P-phi-i-k
where we have used Lemma 4.1.
b-i-k
Support of the bki ’s. The first term in (16) is supported in Bik . To unsure that a
term in the serie defining bki is not identically 0, the condition Bik ∩ Bjk+1 6= ∅ must
be satisfied for some j.
We claim that if Bik ∩ Bjk+1 6= ∅, then rjk+1 ≤ cαrik . For, let w ∈ Bik ∩ Bjk+1 . Since
Ok+1 ⊆ Ok ,
Crjk+1 ≤ d(Bjk+1, ∂Ok+1 ) ≤ d(Bjk+1, ∂Ok )
≤ d(w, ∂Ok ) ≤ d(Bik , ∂Ok ) + 2d(w, ζik )
≤ cαrik ,
since, by the Whitney property, one has Crik ≤ d(Bik , ∂Ok ) ≤ αrik . Hence, the
support of Bik is included in some dilated ball of Bik .
Moment condition. We wish to estimate
Z
k
n bi (w)ϕ(w)dσ(w) ,
S
for ϕ ∈
SN (Bik ).
HARDY-ORLICZ SPACES
15
On Bik we work in local coordinates and write the Taylor expansion of ϕ around
up to order N −1. We denote by SϕN −1 (ζik ) the corresponding Taylor polynomial.
Notice that SϕN −1 (ζik ) ∈ VN −1 (ζik ).
By definition of SN (Bik ) we have
ζik ,
kϕ − SϕN −1 (ζik )kL∞ (Sn ) ≤ kϕkSN (B(ζ0 ,r0 )) .
By construction, the first term in bki is orthogonal to VN −1 (ζik ) and each nonvanishing term in the series is orthogonal to some VN −1 (ζjk+1), so that Bik ∩Bjk+1 6= ∅
or when rjk+1 ≤ Cαrik and Bjk+1 ⊆ C ′ αBik . In that case it follows that VN −1 (ζjk+1) ⊆
VN −1 (ζik ), since the same coordinate system works, and that
Z
Z
k
bki (w) ϕ(w) − SϕN −1 (ζik ) dσ(w)
bi (w)ϕ(w)dσ(w) =
Sn
Sn
k
k
<
∼ kϕkSN (Bik ) · σ(Bi ) · kbi k∞ .
Summability. It remains to prove that
X
σ(Bik )Φ(kbki k∞ ) < ∞.
i,k
We have
∞ X
∞
X
k=0 i=0
σ(Bik )Φ(kbki k∞ )
≤
∞
X
Φ(2k+k0 )σ(Ok )
k=0
∞
Φ(t)
σ {ζ ∈ Sn : KαM f (ζ) + Mα (f )(ζ) ≥ t} dt
t
Z1 ∞
.c
Φ′ (t)σ {ζ ∈ Sn : KαM f (ζ) + Mα (f )(ζ) ≥ t} dt
≤c
Z
1
≤ kΦ(KαM f )kL1 (Sn ) + kΦ(Mα (f ))kL1 (Sn )
≤ ckf kHΦ .
Convergence in HΦ (Bn )-metric. We proved that the boundaryPvalue of f ∈
Φ
HΦ (Bn ), that we still denote by f , admits a decomposition f =
j aj , for H P
atoms aj such that j σ(Bj )Φ(kaj k∞ ) < ∞.
atomic-decom-equation-1
a.e.
Such equality, that stems from (17), holds a.e., see (14). We now show that it
also valid in the distribution sense.
boundedness
Φ
n
From this fact and Theorem 1.10, it converges
P∞ to f in H (B )-norm.
We now show that the equality f = h0 + k=0 (hk+1 − hk ), that holds a.e., also
holds in the distribution sense.
16
above
A. BONAMI AND S. GRELLIER
P∞ k
Pm
Recall that hk+1 − hk =
i=0 bi , so it suffices to show that
k=0 (hk+1 − hk )
converges in the sense of distributions. We show that
Z X
m Z X
∞
m
X
(18)
bki (z)ψ(z)dσ(z)
(hk+1 − hk )(z)ψ(z)dσ(z) =
Sn k=ℓ
Sn i=0
k=ℓ
m X
∞
X
=
<
∼
<
∼
k=ℓ i=0
m X
∞
X
k=ℓ i=0
m X
∞
X
Z
Sn
bki (z)ψ(z)dσ(z)
2k+k0 σ(Bik )kψkSN (Bik )
2k+k0 σ(Bik )kψkC N (Sn ) (rik )N
k=ℓ i=0
But, by the choice of Φ, we have
Φ(
m X
∞
X
2
k+k0
σ(Bik )kψkC N (Sn ) (rik )N )
≤
k=ℓ i=0
m X
∞
X
Φ(2k+k0 σ(Bik )kψkC N (Sn ) (rik )N )
k=ℓ i=0
∞
m X
X
≤ C
≤ C
Φ(2k+k0 )σ(Bik )
k=ℓ i=0
m
X
Φ(2k+k0 )σ(Ok )
k=ℓ
for N large enough. Hence, the left hand-side is less or equal to a constant times
!
!
∞ Z 2k
m
X
X
Φ(t)
< Φ−1
σ {Mα (f ) + KαM f ≥ t} dt
Φ(2k+k0 )σ(Ok ) ∼
Φ−1
t
k−1
k=ℓ 2
k=ℓ
Z ∞
Z
−1
′
<
dσdt
Φ (t)
∼Φ
M f ≥t}
2ℓ−1
{Mα (f )+Kα
!
Z
< −1
∼Φ
Oℓ−1
Φ(|Mα (f ) + KαM f |)dσ ,
which tends to 0 as ℓ → ∞.
This finishes the proof of the atomic decomposition. 2
factorization
5. Proof of the factorization Theorem 1.13
We first need to estimate the decay of PS (a) whenever a is an atom of order N.
We have the following result.
HARDY-ORLICZ SPACES
estimA
17
Lemma 5.1. Let
Φ be a growth function of order p. Let a be an atom of order
1
N ≥ 2n p − 1 , having support in B = B(ζ0 , r0 ), and let A = PS (a). Then A
satisfies the estimates:
(i) kAkHΦ . σ(B(ζ0 , r0 )) × Φ(kak∞ );
(ii) for C > 1 and d(ζ, ζ0) ≥ Cr0 ,
N/2+n
r0
|A(ζ)| .
kak∞ .
d(ζ, ζ0)
Proof. The estimate in (i) is obtained in the same way as the estimate of the norm
of PS (a) in HΦ . For the pointwise estimate, we use also the same kind of idea.
Denote by SN the Taylor polynomial of w 7→ S(ζ, w) around ζ0 of order N. Then
kS(ζ, w) − SN (w)kL∞ ≤ kS(ζ, ·)kSN (B(ζ0 ,r0 )) .
Hence,
Z
|A(ζ)| = a(w)S(ζ, w)dσ(w)
=
ZS
n
a(w) S(ζ, w) − SN (w) dσ(w)
Sn
≤ kS(ζ, ·)kSN (B(ζ0 ,r0 )) σ B(ζ0 , r0 ) × kak∞ .
Szego-estimates
Using the estimate (26) we have
kS(ζ, ·)kSN (B(ζ0 ,r0 )) =
X
|µ|/2+l
kLµ T l S(ζ, ·)kL∞(B) r0
|µ|+l=N
N/2
1
r0
< sup
·
∼ w∈B d(ζ, w)N/2 σ(B(w, d(ζ, w)) .
Recall that, when d(ζ, ζ0) ≥ Cr0 , for w ∈ B(ζ0 , r0 ) ⊆ B(ζ0 , d(ζ0, ζ)/C), d(ζ0 , ζ) ≈
d(ζ, w). Then,
N/2
−1
r0
<
kS(ζ, ·)kSN (B(ζ0 ,r0 )) ∼
· σ B(ζ0 , d(ζ, ζ0)
d(ζ, ζ0)
so that
<
|A(ζ)| ∼
2
r0
d(ζ, ζ0)
N/2+n
· kak∞ .
Let us prove now the factorization theorem. Let f be in HΨ (Bn ). By the
atomic decomposition, there exist atoms aj of order N larger than 2n( p1 − 1), supP
P
ported in some Bj = B(ζj , rj ) so that f = j PS (aj ) with
Ψ(kaj k∞ )σ(Bj ) <
∞. Let us first factorize each holomorphic atom Aj = PS (aj ). Let Hj (z) =
18
A. BONAMI AND S. GRELLIER
log (1 − hz, (1 − rj )ζj i)−n . We write
Aj =
We first prove that
We write
Z
<
∼
Aj
Hj
Aj
Hj .
Hj
belongs to HΦ and that
X
Aj ≤ C.
Hj Φ
H
j
Aj (ζ) dσ(ζ)
Φ Hj (ζ) Sn
! Z
Z
|Aj (ζ)|
|Aj (ζ)|
Φ
Φ
+
dσ(ζ)
[log(1 − hζ, (1 − rj )ζj i)−n ]
[log(1 − rj−n ]
cB(ζ ,Cr )
B(ζj ,Crj )
j
j
P
since N is larger than 2n( p1 −1). Let us remark now that, since Ψ(kaj k∞ )σ(Bj ) <
∞, necessarily, for j large enough, Ψ(kaj k∞ ) ≤ σ(B1 j ) . Now, by definition of Ψ,
We now prove that Hj belongs uniformly to BMOA(Bn ) or equivalently that (1 −
(1−|z|2 )
|z|2 )|∇Hj |2 ≃ |1−hz,(1−r
2 is a Carleson measure with uniform bound. Let Br =
j )ζj i|
B(x0 , r) be a fixed ball on the boundary of Bn and T (Br ) be the tent over this ball.
We have to prove that
Z
(1 − |z|2 )
(I) =
dV (z) <
2
∼ σ(Br ).
T (Br ) |1 − hz, (1 − rj )ζj i|
If d(x0 , ζj ) ≥ 2r then, for z ∈ T (Br ), d(z, ζj ) ≥ r and we get
Z
−2
< σ(Br ).
(1 − |z|2 )dV (z) ∼
(I) ≤ cr
T (Br )
If d(x0 , ζj ) ≤ 2r we have
Z rZ
1
<
dσ(w)dt
(I) ∼
2
0
d(w,ζj )≤3r (t + rj + d(w, ζj ))
Z r X
∞ Z
1
<
t
dσ(w)dt
2
∼ 0
−j−1 ·3r≤d(w,ζ )≤2−j ·3r (t + rj + d(w, ζj ))
2
j
j=0
Z r
<
tr n−2 dt
∼
0
<
∼ Cσ(Br ).
This finishes the proof of the factorization theorem.
2
HARDY-ORLICZ SPACES
19
6. Continuity of Hankel Operators
Hankel
We are now in position to prove corollary 1.14. Let hb be a Hankel operator of
symbol b. Let us first assume that b belongs to the dual of HΨ hence to BMO(̺)
Then, for any g in BMOA, any f ∈ HΦ (Bn ), we have
|hhb (f ), gi| = |hPS (bf ), gi| = |hb, f gi|
≤ kbkBMO(̺) kf gkHΨ ≤ kbkBMO(̺) kf kHΦ kgkBMOA.
It follows that hb is bounded from HΦ (Bn ) to H1 (Bn ) with |||hb ||| ≤ kbkBMO(̺) . Let
us assume now that hb is bounded from HΦ (Bn ) to H1 (Bn ) and prove that b belongs
Ψ
Ψ
to the
P dual of HP. Let f be in H . By the atomic decomposition, f may be written
as
fj gj with
kfj kHΦ kgj kBMOA < ∞, so we have
X
X
|hb, f i| = |hb,
fj gj | ≤
|hPS (bfj ), gj i|
j
=
X
|hhb(fj ), gj i| ≤ |||hb |||
j
X
kfj kHΦ kgj kBM OA < ∞.
j
It ends the proof.
Casgeneral
7. Extension of the results in a more general setting
Let Ω be a smooth domain in Cn . Define the Hardy-Orlicz spaces as the space of
holomorphic functions f so that
Z
sup
Φ(|f |)(w) dσ(w) < ∞
0<ε<ε0 ; δ(w)=ε
Sn
where Φ is as before of lower type p and δ(w) is the distance from w to ∂Ω (recall
that the usual Hardy space of holomorphic functions Hp (Ω) on Ω corresponds to
the case Φ(t) = tp ).
As in the case of the unit ball, H1 (Ω) ⊂ HΦ (Ω) ⊂ Hp (Ω). In particular, any
function f in the Orlicz space HΦ (Ω) admits a unique boundary function f˜ which,
by Fatou Theorem, satisfies Φ(|f˜|) ∈ L1 (∂Ω).
When Ω is a smoothly bounded convex domain of finite type or a strictly Mc
pseudoconvex domain in Cn , there exists a natural pseudo-distance db on ∂Ω (see [Mc] for
the convex case) which makes ∂Ω into a space of homogenous type. We can consider
the real-variable Hardy-Orlicz space H Φ (∂Ω) defined as the space of distributions
Φ
on ∂Ω which are linear combination of atoms.
P∞ More precisely, H (∂Ω) is the set
of distributions f which can be written as j=0 aj , where the aj ’s are supported in
P
some ball Bj so that j σ(Bj )Φ(kaj k∞ ) < ∞. The serie is assumed to converge in
the sense of distributions.
20
A. BONAMI AND S. GRELLIER
8. Geometry of convex domains of finite type
Let Ω be a smoothly bounded convex domain or a strictly pseudoconvex domain
in Cn . As the geometry of strictly pseudoconvex domain is well known, as well as
the geometry in pseudocconvex domain in C2 , we just explain the geometry when Ω
is a convex domain of finite type in Cn . A point ζ ∈ ∂Ω is said to be of finite
type
BoSt
if the order of contact of complex lines with ∂Ω at this point is finite, see [BS] and
references therein. The type of the point is the least upper bound of the various
orders of contact. We say that Ω is of finite type MΩ if every point on ∂Ω is of finite
type ≤ MΩ .
Let Ω = {z ∈ Cn : ρ(z) < 0}. There exists ε0 > 0 such that for |ε| ≤ ε0 the sets
Ωε = {z ∈ Cn : ρ(z) < ε} are all convex, and the normal projection π : U → ∂Ω is
well defined and smooth, where U = {z ∈ Cn : δ(z) < ε0 }.
The basicMcgeometric facts
about
convex domains
of finite type were first proved
McS1
McS2
DF
by McNeal [Mc], see also [McS1], [McS2] and [DFo]. By recalling the results that are
involved in the present work we take the opportunity to review the main elements
of the construction and set some notation.
For z ∈ U and λ ∈ Cn a unit vector, we denote by τ (z, λ, r) the distance from z
to the surface {z ′ : ̺(z ′ ) = ̺(z) + r} along the complex line determined by λ.
For each z ∈ U and r < ε0 there exists a special set of coordinates {w1z,r , . . . , wnz,r },
that we call r-extremal . The first vector v (1) is given by the direction transversal to
the boundary, in the sense that the shortest distance from z to the set {z ′ : ̺(z ′ ) =
̺(z) + r} is realized in the complex line determined by v (1) .
The vector v (2) is chosen among the vectors orthogonal to v (1) in such a way
that τ (z, v (2) , r) is maximal. We repeat the same process until we determine an
orthonormal basis {v (1) , . . . , v (n) }. We denote by (w1 , . . . , wn ) the coordinates with
respect to this basis. Notice that these coordinates (w1 , . . . , wn ) = (w1z,r , . . . , wnz,r )
depend on z and r. However, the transversal direction w1 does not depend on r.
For k = 1, . . . , n we set
tau-j
(19)
τk (z, r) = τ (z, v (k) , r),
and define the polydisc Q(z, r)
Qzr
(20)
Q(z, r) = {w : |wk | < τk (z, r), k = 1, . . . , n}.
McS2
Basic
relations among these quantities are the following, see [McS2] Prop. 1.1 and
BPS2
also [BPS2] Lemma 2.1.
facts
Proposition 8.1. There exists a constant C > 0 depending only on Ω such that for
any unit vector λ ∈ Cn , 0 < r ≤ ε0 , z ∈ U, and 0 < η < 1 we have:
< τ (z, λ, ηr) < η 1/MΩ τ (z, λ, r);
(i) η 1/2 τ (z, λ, r) ∼
∼
(ii) η 1/2 Q(z, C −1 r) ⊂ Q(z, ηr) ⊂ η 1/MΩ Q(z, Cr);
(iii) if w ∈ Q(z, δ) then τ (z, λ, r) ≈ τ (w, λ, r).
HARDY-ORLICZ SPACES
21
We define the quasi-distance db : U × U by setting
db
db (z, w) = inf {δ : w ∈ Q(z, δ)},
(21)
and the function d
d
(22)
d(z, w) = db (z, w) + δ(z) + δ(w).
Notice that d is initially defined on U × U and we extend it to Cn × Cn by setting
d(z, w) = ψ(̺(z))ψ(̺(w))d(z, w) + (1 − ψ(̺(z)))(1 − ψ(̺(w)))|z − w|,
where ψ is a smooth cut-off function on R such that ψ(t) = 1 for |t| ≤ ε0 /2 and
ψ(t) = 0 for |t| ≥ ε0 .
On the boundary we will use a family of “balls” centered at ζ ∈ ∂Ω of radius δ
defined as
B(ζ, δ) = Q(ζ, δ) ∩ ∂Ω.
For any unit vector λ we introduce the differential operator
L-lambda
(23)
Lλ = (∂λ ̺)∂x1 − (∂x1 ̺)∂λ ,
where w1 = x1 +iy1 is the transversal direction fixed earlier. Here, ∂λ is the standard
vector field defined by λ as ∂λ f = hλ, df i, for a smooth function f , its real differential
df , and where h , i denotes the usual pairing between a one-form and a vector.
Notice that Lλ is always a tangential vector field. If λ ∈ S 2n−1 is itself tangent to
∂Ω, then Lλ is the directional derivative in the direction λ.
For Λ = (λ1 , . . . , λq ) a q-list of vectors in S 2n−1 and µ = (µ1 , . . . , µq ) a q-index we
set |µ| = µ1 + · · · + µn ,
L-Lambda
(24)
µ
LµΛ = Lµλ11 · · · Lλqq ,
and
tau-Lambda
(25)
τ µ (z, Λ, δ) = τ (z, λ1 , δ)µ1 · · · τ (z, λq , δ)µq .
Finally we recall the fundamental estimates forMcS2
the Szegö kernel and its derivatives
[McS2] (called interior estimates of S-type, see [McS2] Def. 4 and Thm. 3.6)
′
−µ
µ µ′
(z, Λ, δ)τ −µ (z ′ , Λ′ , δ)
′ < τ
,
LΛ,z LΛ′ ,z ′ SΩ (z, z ) ∼
(26)
σ B(π(z), δ)
McS2
go-estimates
where δ = d(z, z ′ ), z, z ′ ∈ Ω × Ω \ ∆∂Ω , ∆∂Ω denotes the diagonal on ∂Ω.
SNg
We now define the real-variable Hardy-Orlicz spaces. As in the case of the ball,
we need first to introduce the notion of atoms. Let ζ0 ∈ ∂Ω, r0 < ε0 and N be a
positive integer. On C ∞ (B(ζ0 , r0 )) we introduce the norm
X
(27)
kϕkSN (B(ζ0 ,r0 )) = sup
kLµΛ ϕkL∞ (B(ζ0 ,r0 )) τ µ (ζ0 , Λ, r0).
Λ
atomgen
|µ|=N
Definition 8.2. A measurable function a on ∂Ω is called an atom of order N ∈ N∗
if either it is the constant function on ∂Ω or if there exist ζ0 ∈ ∂Ω, r0 > 0:
(1) supp a ⊆ B(ζ0 , r0 );
nvergencegen
tar-gammagen
le-star-Mgen
x-charactgen
mic-decomgen
22
A. BONAMI AND S. GRELLIER
R
(2) ∂Ω a(ζ)dσ(ζ) = 0;
(3) for all φ ∈ C ∞ B(ζ0 , r0 ) we have,
Z
≤ kφkS (B(ζ0 ,r0 )) σ B(ζ0 , r0 ) kak∞
a(ζ)φ(ζ)dσ(ζ)
N
∂Ω
Real-variable Hardy spaces. The real Hardy-Orlicz space H Φ (∂Ω) is the space
of distributions f on ∂Ω which can be written as
∞
X
(28)
f=
aj ,
j=0
where
σ(Bj )Φ(kaj k∞ ) < ∞, the aj ’s are H Φ -atoms of order N ≥ 1 and the series
is assumed to converge in the sense of distributions.
As before, the key point to prove atomic decomposition is to obtain maximal
characterization of Hardy-Orlicz spaces. Given ζ ∈ ∂Ω, we define the approach
region Aα (ζ) as the subset of Ω given by
P
Aα (ζ) = {z ∈ Ω : d(ζ, π(z)) < αδ(z)}.
We define the non-tangential maximal function of f by Mα (f )
Mα (f )(ζ) = sup |f (z)|,
(29)
z∈Aα (ζ)
and the tangential variant
M
δ(w)
|f (w)|.
(30)
NM (f )(ζ) = sup
w∈Ω δ(w) + d(ζ, π(w))
With the same proof as in the case of the ball, we obtain the following.
Theorem 8.3.
For N large enough,
< kf kHΦ (Ω) .
kMα (f )kL1 (∂Ω) ∼
< kf kHΦ (Ω) .
kNM (f )kL1 (∂Ω) ∼
Once these maximal characterizations obtained, the atomicViviani
decomposition follows
from the structure of homogeneous domains by the work of [V].
Theorem 8.4. Let N ∈ N∗ be larger than n/MΩ (1/p − 1). There exists a constant
Φ
c depending only on Ω such that the following
P∞ holds. ΦGiven any f ∈ H (Ω) there
Φ
exist H -atoms aj of order N such that j=0 aj ∈ H (∂Ω) and
X
X
∞
∞
f = PS
aj =
PS (aj ),
j=0
and moreover
∞
X
j=0
j=0
σ(Bj )Φ(kaj k∞ ) ≃ kf kHΦ (Ω) .
undednessgen
orizationgen
HARDY-ORLICZ SPACES
23
As for the ball, the Poisson extensions of functions in the real Hardy-Orlicz space
are in the Hardy-Orlicz space. The key point is the pointwise estimate
of the Szegö
McS2
kernel and its derivatives (proved in the case of convex domains by [McS2]).
Theorem 8.5. Let Ω be a smoothly bounded convex domain of finite type in Cn .
Let H Φ (∂Ω) be the real-variable Hardy space on ∂Ω. Then, the Szegö projection
PS : H Φ (∂Ω) → HΦ (Ω)
is bounded.
Finally, we have the factorization theorem.
Theorem 8.6. Let Ω be a smoothly bounded domain of finite type. There exists a
constant c depending only on Ω such that the following holds. Given any f ∈ HΨ (Ω)
there exist fj ∈ HΦ (Ω), gj ∈ BMOA(Ω), j = 1, 2, . . . such that
f=
∞
X
fj gj .
j=0
Furthermore, we have
P
j
kfj kHP hi kgj kBMOA < ∞.
As a consequence, we obtain the characterization of bounded Hankel operators.
The proof follows the same lines.
9. Proof of the theorems
The key point to obtain these results is to use the pointwise estimates of the Szegö
kernel. Namely, it follows from the usual estimate of the Szegö kernel that
kSΩ (ζ, ·)kSN (B(ζ0 ,r0 )) =
X
sup kLµΛ,w SΩ (ζ, ·)kL∞(B) τ µ (ζ0 , Λ, r0)
|µ|=N
<
∼
X
|µ|=N
Λ
sup sup
w∈B
Λ
τ µ (ζ0 , Λ, r0)
1
·
.
µ
τ (w, Λ, d(ζ, w)) σ(B(w, d(ζ, w)))
boundednessgen
From this estimate, we get easily Theorem 8.5 by the same method as the one
used in the case of the ball. factorizationgen
For the proof of Theorem 8.6, we prove the following lemma analogous to the one
of the ball.
Lemma 9.1. Let Φ be a growth function of order p. Let a be an atom of order N,
having support in B = B(ζ0 , r0 ), and let A = PS (a). Then A satisfies the estimates:
(i) kAkHΦ ≤ cσ(B(ζ0 , r0 ))Φ(kak∞ );
(ii) for C > 1 and d(ζ, ζ0) ≥ Cr0 ,
1+(N −2+2n)/MΩ
r0
|A(ζ)| ≤ c
kak∞ .
d(ζ, ζ0)
24
A. BONAMI AND S. GRELLIER
Proof. The proof is the same as the one of the ball taking into account the estimate
of kSΩ (ζ, .)kSN (B(ζ0 ,r0 )) . 2
Let us give now the main lines of the proof of the factorization theorem. As
before, it suffices to factorize each holomorphic atom A = PS (a), for a HΦp -atom a
of order N large enough, a having support in some ball B(ζ0 , r0 ).
DF
For this factorization, we use the result of Diederich and Fornæss [DFo] which
gives the existence of support functions on convex domains of finite type. More
precisely, there exists a neighborhood U of ∂Ω and a function H : Ω × U → C such
that H ∈ C ∞ (Ω × U), H(·, w) is holomorphic for each w ∈ U, and
<
d(z, w) <
∼ |H(z, w)| ∼ d(z, w),
BPS2
on Ω × U. This function H was used in [BPS2] to prove the factorization theorem
for H1 .
We set H0 = H(·, ζ̃0), where ζ̃0 = ζ0 − r0 ν(ζ0 ). We write
A=
A
−n
−n log(e + H0 ).
log(e + H0 )
A
log(e+H0−n )
belongs to Hp . The same proof as the one used in
1
2n
+1
−1 .
the case of the ball gives the result as soon as N > MΩ
MΩ
p
2
0|
We now prove that log(e + H0−n ) belongs to BMOA or equivalently that δ |∇H
H02
is a Carleson measure. Let Br = B(x0 , r) be a fixed ball on the boundary of Ω and
T (Br ) be the tent over this ball. We have to prove that
We first prove that
(I) =
Z
δ
T (Br )
|∇H0 |2
< σ(Br ).
dV ∼
H02
Assume first that d(x0 , ζ̃0 ) ≥ 2r then H0 (z) ≃ d(z, ζ̃0 ) ≥ r and, as z 7→ H0 (z) is
smooth on T (B), we get that
(I) ≤ cr
−2
Z
T (Br )
< σ(Br ).
δ(z)dV (z) ∼
HARDY-ORLICZ SPACES
25
DF
< 1 so that
Now, if d(x0 , ζ̃0 ) ≤ 2r, by the estimates given in [DFo], |∇H0 | ∼
Z
δ(z)
<
dV (z)
(I) ∼
2
d(z,ζ̃0 )≤3r (δ(z) + r0 + d(π(z), ζ0 ))
Z 3r Z
t
<
dσ(w)dt
2
∼ 0
d(w,ζ0 )≤3r (t + r0 + d(w, ζ0))
Z 3r X
∞ Z
<
(2−j r)−2 dσ(w)dt
∼ 0 t
−j−1
−j
r≤d(w,ζ0 )≤2 r
j=0 2
Z 3r
X
−2
<C
tr
dt
22j−j((2n−2)/MΩ +1) σ(Br ) ≤ C.
∼
0
j
This finishes the proof of the factorization theorem.
2
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MAPMO-UMR 6628, Département de Mathématiques, Université d’Orleans, 45067
Orléans Cedex 2, France
E-mail address: Aline.Bonami@univ-orleans.fr
MAPMO-UMR 6628, Département de Mathématiques, Université d’Orleans, 45067
Orléans Cedex 2, France
E-mail address: Sandrine.Grellier@univ-orleans.fr