Duality for large Bergman-Orlicz spaces and Boundedness of Hankel Operators

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by  Benoit F. Sehba

DUALITY FOR LARGE BERGMAN-ORLICZ SPACES AND
BOUNDEDNESS OF HANKEL OPERATORS
BENOˆITF. SEHBA ANDEDGAR TCHOUNDJA
Abstract. For Bn the unit ball of Cn, we consider Bergman-Orlicz
5
spacesofholomorphicfunctionsinLΦ,whicharegeneralizationsofclas-
1 α
sicalBergmanspaces. WecharacterizethedualspaceoflargeBergman-
0
2 Orlicz space, and bounded Hankel operators between some Bergman-
OrliczspacesAΦα1(Bn)andAΦα2(Bn)whereΦ1 andΦ2 areeitherconvex
n
or concave growth functions.
a
J

4
1
1. Introduction

]
A Let Bn be the unit ball of Cn. We denote by dν the Lebesgue measure
C on Bn and dσ the normalized measure on Sn = ∂Bn (the boundary of Bn).
. The space (Bn) is the set of holomorphic functions on Bn.
th For z = (Hz1, ,zn) and w = (w1, ,wn) in Cn, we let
a ··· ···
m z,w =z w + +z w
1 1 n n
h i ···
[ so that z 2 = z,z = z1 2+ + zn 2.
| | h i | | ··· | |
1 We say that a function Φ is a growth function if it is a continuous and
v
non-decreasing function from [0, ) onto itself.
6 ∞
Forα > 1,wedenotebydν thenormalizedLebesguemeasuredν (z) =
1 α α
4 cα(1 z 2)−αdν(z), with cα such that να(Bn) = 1. For Φ a growth function,
3 the w−ei|gh|ted Bergman-Orlicz space Φ(Bn) is the space of holomorphic
0 Aα
functions f such that
.
1
0 f := Φ(f(z))dν (z) < .
α,Φ α
5 || || Bn | | ∞
1 Z
: More generally, we will also consider the Orlicz space LΦ(Bn), that is, the
v α
space of functions f such that
i
X
r f α,Φ := Φ(f(z))dνα(z) < .
a || || Bn | | ∞
Z
Hence Φ(Bn) is the subspace of LΦ(Bn) consisting of holomorphic func-
Aα α
tions.
We define on Φ(Bn) the following (quasi)-norm
Aα
f(z)
(1) f lux := inf λ > 0 : Φ | | dν (z) 1
|| ||α,Φ { ZBn (cid:18) λ (cid:19) α ≤ }
which is finite for f Φ(Bn) (see [10]).
∈ Aα
Date: January 15, 2015.
2000 Mathematics Subject Classification. Primary 47B35, Secondary 32A35, 32A37.
Key words and phrases. Hankel operator, Bergman-Orlicz spaces, Atomic decomposi-
tion, Weak factorization.
1
2 BENOˆITF.SEHBAANDEDGARTCHOUNDJA
When Φ(t) = tp, we recover the classical weighted Bergman spaces de-
noted by p(Bn) and defined by
α
A
f p = f p := f(z)pdν (z) < .
k kp,α || ||Apα Bn| | α ∞
Z
We say that a growth function Φ is of upper type q 1 if there exists C > 0
≥
such that, for s > 0 and t 1,
≥
(2) Φ(st) CtqΦ(s).
≤
We denote by Uq the set of growth functions Φ of upper type q, (for some
Φ(t)
q 1), such that the function t is non-decreasing.
≥ 7→ t
We say that Φ is of lower type p > 0 if there exists C > 0 such that, for
s > 0 and 0 < t 1,
≤
(3) Φ(st) CtpΦ(s).
≤
We denote by L the set of growth functions Φ of lower type p, (for some
p
Φ(t)
p 1), such that the function t is non-increasing.
≤ 7→ t
We say that Φ satisfies the ∆ -condition if there exists a constant K > 1
2
such that, for any t 0,
≥
(4) Φ(2t) KΦ(t).
≤
Recall that two growth functions Φ and Φ are said equivalent if there
1 2
exists some constant c such that
cΦ (ct) Φ (t) c−1Φ (c−1t).
1 2 1
≤ ≤
SuchequivalentgrowthfunctionsdefinethesameOrliczspace. Notethatwe
may always suppose that any Φ L (resp. U ), is concave (resp. convex)
p q
∈
andthatΦis aC1 functionwithderivative Φ′(t)⋍ Φ(t) (see [3]for thelower
t
type functions).
Letusobservethat ifΦis ofuppertype(resp. lower type)p , thenitisof
1
upper type (resp. lower type) p for any > p > p (resp. p < p < ).
2 2 1 2 1
Hence, when we say Φ Uq (resp. Φ ∞L ), we suppose that q (resp.∞p)
p
∈ ∈
is the smallest (resp. biggest) number q (resp. p ) such that Φ is of upper
1 1
type q (resp. lower type p ).
1 1
We denoted by Lp(Bn), 0 < p < , the Lebesgue space with respect to
α
∞
the measure dν .
α
The orthogonal projection of L2(Bn) onto 2(Bn) is called the Bergman
α Aα
projection and denoted P . It is given by
α
(5) P (f)(z) = K (z,ξ)f(ξ)dν (ξ),
α α α
Bn
Z
where
1
K (z,ξ) =
α (1 z,ξ )n+1+α
−h i
istheweightedBergmankernelonBn. WedenoteaswellbyP itsextension
α
to L1(Bn).
α
It is well known that the Bergman projection P is bounded on Lp(Bn)
α α
for all p (1, ). One of the consequences of this result is the fact that the
∈ ∞
DUALITY AND HANKEL OPERATORS ON BERGMAN-ORLICZ 3
topological dual space of the Bergman space Ap(Bn) identifies with Aq(Bn),
α α
with 1 + 1 = 1, under the integral pairing
p q
(6) f,g := f(z)g(z)dν (z),
α α
h i Bn
Z
f Lp(Bn), g Lq(Bn).
α α
∈ ∈
The boundedness of the Bergman projection has been extended to the
setting of Orlicz spaces for the class of Young functions in [4], and this
provides as a consequence that the dual space of the Bergman-Orlicz space
AΦ(Bn) can be identifies with another Bergman-Orlicz space that we will
α
specify in the next section.
Our first interest in this paper is the characterization of the dual space
of the Bergman-Orlicz spaces defined from the class L . We note that this
p
class generalizes the class of power functions φ(t) = tp, 0 < p < 1, and we
have that for Φ L , the following inclusions hold
p
∈
(7) 1(Bn) Φ(Bn) p(Bn).
Aα ⊂ Aα ⊂ Aα
We recall that given an analytic function f on Bn, the radial derivative Rf
of f is defined by
n
∂f
Rf(z) = z (z).
j
∂z
j
j=1
X
For β 0, we denote by Γ (Bn) the space of holomorphic functions f for
β
≥
which there exists and integer k > β and a positive constant C such that
Rkf(z) C(1 z 2)β−k.
| | ≤ −| |
Remarkthatforβ = 0, theclass Γ (Bn)coincides withtheusualBloch class
β
. The Bloch class is the space of holomorphic functions in Bn such that
B
sup Rf(z)(1 z 2) < .
z∈Bn| | −| | ∞
For β >0, it coincides with the class of Lipschitz functions of order β.
It is known that, for 0 < p 1, the dual of the Bergman space p(Bn),
α
coincides with Γ (Bn) with β =≤(n+1+α)(1 1) under the integralApairing
β p−
(8) lim f(rz)g(z)dν (z)
α
r→1 Bn
Z
(see [16]).
To Φ a growth, we associate the function
1
ρ(t) = .
tΦ−1(1/t)
The function ρ is quite relevant in the study of Orlicz space of analytic
functions (see [3, 5, 11, 12] and the references therein). Note in particular
that in the case of Apα(Bn), Φ(t) = tp and ρ(t) = tp1−1, hence f Γβ(Bn)
∈
can be written as
Rkf(z) C(1 z 2)−kρ (1 z 2)n+1+α .
| |≤ −| | −| |
(cid:0) (cid:1)
4 BENOˆITF.SEHBAANDEDGARTCHOUNDJA
From this observation, we will make the following generalization. Let ρ be
a positive continuous increasing function from [0, ) onto itself. Let γ > 0.
∞
We say that ρ is of upper type γ on [0,1] if there exists a constant C such
that
(9) ρ(st) Csγρ(t),
≤
fors> 1andst 1. Wewillcallaweight,afunctionρwhichisacontinuous
≤
increasingfunctionfrom[0, )ontoitself, whichisofuppertypeγ,forsome
∞
γ > 0.
Now for α> 1 and a weight ρ (of upper type γ), we define the weighted
Lipschitz space−Γ (Bn) as the space of holomorphic functions f in Bn such
α,ρ
that, for some integer k > γ(n+1+α) and a positive constant C > 0, we
have
Rkf(z) C(1 z 2)−kρ (1 z 2)n+1+α .
| |≤ −| | −| |
We will show that, as in the classical Lipschitz spaces, these spaces are
(cid:0) (cid:1)
independent of k. This allows us to see Γ (Bn) as a Banach space under
α,ρ
the following norm
Rkf(z)(1 z 2)k
||f||Γα,ρ(Bn) = |f(0)|+zs∈uBpn |ρ((1 |z 2)−n+|1+| α).
−| |
The following is our first main result which extends the duality result for
classicalBergmanspaceswithsmallexponenttoBergman-Orliczspaceswith
concave exponent.
THEOREM 1.1. Let α > 1, Φ L and ρ(t) = 1 . Then the
− ∈ p tΦ−1(1/t)
topological dual space Φ(Bn) ∗ of Φ(Bn) identifies with Γ (Bn) under
Aα Aα α,ρ
the duality pairing
(cid:0) (cid:1)
(10) f,g := lim f(rz)g(z)dν (z),
α α
h i r→1 Bn
Z
where f Φ(Bn) and g Γ (Bn).
∈ Aα ∈ α,ρ
The proof of the above result required two main steps. First one has to
insure that the above definition of the space Γ (Bn) does not depend on
α,ρ
the choice of the number of derivatives. Second, one needs a nice example
of functions in the Bergman-Orlicz space Φ(Bn) and a generalization to
Aα
large Bergman-Orlicz spaces of the following inequality known for Bergman
spaces with exponent 0 < p 1 (see [15]),
≤
(11) Bn|f(z)|(1−|z|2)(p1−1)(n+1+α)dνα(z) ≤ Ckfkpα,p.
Z
For b 2(Bn), the small Hankel operator with symbol b is defined for
∈ Aα
f a bounded holomorphic function by h (f):= P (bf).
b α
Boundedness of the small Hankel operator between classical weighted
Bergman spaces has been considered in [2] where using duality and test
functions, the authors obtained a full characterization of bounded Han-
kel operators between Bergman spaces except for estimations with loss i.e.
h : p(Bn) q(Bn) with 1 q < p < . The estimations with loss
b α α
A 7→ A ≤ ∞
DUALITY AND HANKEL OPERATORS ON BERGMAN-ORLICZ 5
have been recently handled by J. Pau and R. Zhao in [7], closing the subject
for the classical weighted Bergman spaces.
Our second interest in this paper is for the boundedness of the small
Hankel operators h from Φ1(Bn) to Φ2(Bn). We do not use a specific
b Aα Aα
methodbutcombineseveraltechniquessomeofthemappearingin[2]orused
inthecaseofHardy-Orliczspacesin[11,12]. Inparticular,whenconsidering
boundedness of h on Φ(Bn) with Φ Uq, we use a weak factorization
result of the Bergbman Aspαace 1(Bn) in∈term of Bergman-Orlicz functions,
Aα
extending the usual weak factorization for this space. Nevertheless, we do
not generalize this method for the whole situation, as when Φ and Φ are
1 2
growth functions with Φ Uq, we are dealing only with the uppertriangle
2
case, i.e Φ−1(t)Ψ−1(t) U∈ q, Ψ being the complementary function of Φ
1 2 ∈ 2 2
to be defined later.
The ranges of the symbols of bounded Hankel operators obtained here
are some weighted Lipschitz spaces related to the dual spaces of Bergman
Orlicz spaces with concave growth functions as given above. This will allow
us to study the boundedness of the Hankel operators between Bergman-
Orlicz spaces in the same range of growth functions as in [3, 11]. However,
when Φ is a concave growth function, we will suppose that Φ satisfies a
2 2
Dini condition to be defined later. This will cause additional restrictions
on the growth functions Φ and Φ for which we are able to extend the
1 2
Hankel operators, h , into bounded operators from Φ1(Bn) into Φ2(Bn).
b Aα Aα
Inparticular,weareableheretohandlethecaseswherethegrowthfunctions
p s
are given by Φ (t) = t , p 1 and Φ (t) = t , s < 1.
1 log(e+t) ≤ 2 log(e+t)
Moreover, our results g(cid:16)eneraliz(cid:17)e the results obtained in(cid:16)[2]. (cid:17)
Thepaperisorganizedasfollows, insection2,wecollectsomeresultsthat
areneededtocharacterizethedualspacesoflargeBergman-Orliczspaceand
tostudytheboundednesspropoertiesofHankeloperatorsbetweenBergman-
Orlicz spaces. In section 3, we deal with the duality question for Bergman-
Orlicz spaces with concave exponent. In the last section, each subsection
is devoted to the study, in each case, of the boundedness of the Hankel
operator h from Φ1(Bn) into Φ2(Bn), when Φ L or Uq, i= 1,2.
b i p
A A ∈
Finally, all over the text, C will be a constant not necessary the same
at each occurrence. We will also use the notation C(k) to express the fact
that the constant depends on the underlined parameter. Given two positive
quantities A and B, the notation A . B means that A CB for some
≤
positive constant C. When A . B and B . A, we write A ⋍B.
2. Preliminaries
Inthis section, werecall someknown resultsthat areneeded inour study,
we also extend to Orlicz setting many some classical results known for the
Bergman spaces.
2.1. Some propertiesof growth functions. Wecollectinthissubsection
some properties of growth functions we shall used later. For Φ a convex
growth function, we recall that the complementary function, Ψ : R R ,
+ +
→
6 BENOˆITF.SEHBAANDEDGARTCHOUNDJA
is defined by
(12) Ψ(s)= sup ts Φ(t) .
t∈R+{ − }
One easily checks that if Φ Uq, then Ψ is also a growth function of lower
∈
Ψ(t)
type such that t is non-decreasing but which may not satisfy the
7→ t
∆ conditon. WesaythatthegrowthfunctionΦsatisfiesthe condition
2 2
− ▽ −
whenever both Φ and its complementary satisfy the ∆ conditon.
2
−
For Φ a 1 growth function, the lower and the upper indices of Φ are
C
respectively defined by
tΦ′(t) tΦ′(t)
a := inf and b := sup .
Φ Φ
t>0 Φ(t) t>0 Φ(t)
WerecallthatwhenΦisconvex, then1 a b < and,ifΦisconcave,
Φ Φ
≤ ≤ ∞
then 0 < a b 1. We have the following useful fact.
Φ Φ
≤ ≤
LEMMA 2.1. Let Φ be a 1 growth function. Denote by p and q its lower
C
Φ(t) Φ−1(t)
and its upper indices respectively. Then the functions and are
tp t1/q
increasing.
Φ(t)
Proof. We only prove that is increasing. The proof is the same for the
tp
tΦ′(t)
second function. Recall that by definition of p, we have p for any
≤ Φ(t)
t > 0. It easily follows that
Φ(t) ′ Φ′(t) Φ(t) Φ′(t) tΦ′(t) Φ(t)
= p = 0.
tp tp − tp+1 ≥ tp − Φ(t) × tp+1
(cid:18) (cid:19)
The proof is complete. (cid:3)
REMARK 2.2. One useful way to use this lemma is to observe that it
allow us to say that, if Φ L , then the growth function Φ , defined by
p p
∈
Φ (t) = Φ(t1/p), is in Uq for some q 1. So we may assume that Φ is
p p
≥
convex.
We also observe that a (resp. b )coincides with the biggest(resp. small-
Φ Φ
est) number p such that Φ is of lower (resp. upper) type p.
We say that Φ Uq satisfies the Dini condition if there exists a constant
∈
C > 0 such that, for t > 0,
t Φ(s) Φ(t)
(13) ds C .
s2 ≤ t
Z0
We observe that if Φ satisfies (13), then Φ satisfies the condition.
2
▽ −
We will also make use of the following properties of growth functions
established in section 2 of [11]. We recall them here for quick references.
PROPOSITION 2.3. The following assertion holds:
Φ L if and only if Φ−1 U1/p.
p
∈ ∈
LEMMA 2.4. Let Φ L and Φ Uq, and Ψ the complementary
1 p 2 2
∈ ∈
function of Φ . Let Φ be such that
2
Φ−1(t):= Φ−1(t)Ψ−1(t).
1 2
Then Φ L for some r p.
r
∈ ≤
DUALITY AND HANKEL OPERATORS ON BERGMAN-ORLICZ 7
LEMMA 2.5. Let Φ be a growth function and Φ Uq, ρ (t) = 1
1 2 ∈ i tΦ−1(1/t)
i
and Ψ the complementary of Φ . Then, if
2 2
ρ
1
ρ := ,
Φ
ρ
2
we also have
(14) Φ−1(t) Φ−1(t)Ψ−1(t)
≃ 1 2
and vice-versa.
LEMMA 2.6. Let Φ and Φ be in Uq, and Ψ the complementary function
1 2 2
of Φ . Let Φ be such that Φ−1(t) = Φ−1(t)Ψ−1(t). We suppose that Φ
2 1 2 2
satisfies the Dini condition (13) and that
Φ−1 Φ (t)
2 ◦ 1 is non-increasing.
t
Then Φ L for some p > 0.
p
∈
The following can be adapted from [14].
PROPOSITION 2.7. For Φ and Φ two growth functions of lower type,
1 2
α > 1, the bilinear map (f,g) fg sends LΦ1 LΦ2 onto LΦ, with the
− 7→ α × α α
inverse mappings of Φ ,Φ and Φ related by
1 2
(15) Φ−1 = Φ−1 Φ−1.
1 × 2
Moreover, there exists some constant c such that
fg lux c f lux g lux .
|| ||LΦα ≤ || ||LΦα1|| ||LΦα2
2.2. Boundedness of the Bergman projection. We start by recalling
the following result in [1].
PROPOSITION 2.8. Let α > 1, there exists a constant C > 0 such that
for f 1(Bn) −
∈ Aα
f
ν ( z Bn : P f(z) > λ ) Ck k1,α.
α α
{ ∈ | | } ≤ λ
The next result follows from interpolation with Orlicz functions (see [4]).
PROPOSITION 2.9. Let α > 1 and Φ Uq. Suppose that Φ satisfies
− ∈
the condition. Then the Bergman projection P extends into a bounded
2 α
oper▽ato−r on LΦ(Bn).
α
From the duality result in [8], since P is bounded in LΦ for Φ Uq
α α ∈
that satisfies the condition, we obtain the following duality result in
2
▽ −
this case.
PROPOSITION 2.10. Let Φ Uq and α > 1. Suppose that Φ satisfies
∈ −
the condition and denote by Ψ its complementary function. Then the
2
dual▽sp−ace Φ(Bn) ∗ of Φ(Bn) identifies with Ψ(Bn) under the integral
Aα Aα Aα
pairing
(cid:0) (cid:1)
f,g = f(z)g(z)dν (z), f Φ(Bn), g Ψ(Bn).
h iα Bn α ∈Aα ∈ Aα
Z
8 BENOˆITF.SEHBAANDEDGARTCHOUNDJA
2.3. Lipschitz-type spaces. Weconsiderinthissubsectionsomeweighted
Lipschitz spaces and their logarithmic counterparts. We recall that given
an analytic function f on Bn, the radial derivative Rf of f is defined by
n
∂f
Rf(z) = z (z).
j
∂z
j
j=1
X
For α> 1 and a weight ρ (of upper type γ), we recall that the weighted
Lipschitz sp−ace Γ (Bn) has been defined as the space of holomorphic func-
α,ρ
tions f in Bn such that, for some integer k > γ(n+1+α) and a positive
constant C > 0, we have
Rkf(z) C(1 z 2)−kρ (1 z 2)n+1+α .
| |≤ −| | −| |
We will also need a logarithmic version of(cid:0)the above spac(cid:1)e, LΓα,ρ(Bn), de-
fined as the space of holomorphic functions f in Bn such that, for some
k > γ(n+1+α) and a positive constant C > 0, we have
−1
1
Rkf(z) C(1 z 2)−kρ (1 z 2)n+1+α log .
| | ≤ −| | −| | 1 z 2
(cid:18) −| | (cid:19)
(cid:0) (cid:1)
One can show that, as in the classical Lipschitz spaces, these spaces are
independent of k. We prove this in the following proposition.
PROPOSITION 2.11. Let ρ be a weight satisfying (9). The weighted (resp.
logarithmic weighted) Lipschitz spaces Γ (Bn) (resp. LΓ (Bn)) are inde-
α,ρ α,ρ
pendent of various values of k.
Proof. Let us provide a proof for Γ (Bn), the proof for LΓ (Bn) requires
α,ρ α,ρ
only few harmless modifications.
Let f be an holomorphic function in Bn. Let us first suppose that there
is a constant C > 0 such that for some nonnegative integer k > (n+1+α)γ
with γ as in (9), and any z Bn,
∈
(16) Rkf(z) C(1 z 2)−kρ (1 z 2)n+1+α .
| |≤ −| | −| |
We want to show that (16) holds for k+1(cid:0). (cid:1)
From (16), it follows in particular that as ρ is increasing on (0,1], the
function Rkf(z)(1 z 2)k is bounded on Bn. Hence, for β sufficiently
| | − | |
large (in fact β > k α 1 will do ), we have the representation
− −
Rkf(w)(1 w 2)β
Rkf(z) = C −| | dν (w).
αβ Bn (1 z,w )n+1+β+α α
Z −h i
Thus
(n+1+β +α) z,w Rkf(w)(1 w 2)β
Rk+1f(z) = C h i −| | dν (w)
| | (cid:12) αβ Bn (1 z,w )n+2+β+α α (cid:12)
(cid:12) Z −h i (cid:12)
(cid:12)(cid:12) Rkf(w)(1 w 2)β (cid:12)(cid:12)
. (cid:12)Bn |1 z,w| n−+2|+β|+α dνα(w) (cid:12)
Z | −h i|
ρ (1 w 2)n+1+α (1 w 2)β−k
. −| | −| | dν (w)
ZBn (cid:0) |1−hz,wi|n(cid:1)+2+β+α α
= I +I ,
1 2
DUALITY AND HANKEL OPERATORS ON BERGMAN-ORLICZ 9
where
ρ (1 w 2)n+1+α (1 w 2)β−k
I = −| | −| | dν (w)
1 1 z,w n+2+β+α α
Z1−|w|2≤1−|z|2 (cid:0) | −h i| (cid:1)
and
ρ (1 w 2)n+1+α (1 w 2)β−k
I = −| | −| | dν (w).
2 1 z,w n+2+β+α α
Z1−|w|2>1−|z|2 (cid:0) | −h i|(cid:1)
Let us start by estimating the integral I . Using the monotonicity of the
1
weight ρ and [9, Propostion 1.4.10], we obtain
ρ (1 w 2)n+1+α (1 w 2)β−k
I = −| | −| | dν (w)
1 1 z,w n+2+β+α α
Z1−|w|2≤1−|z|2 (cid:0) | −h i|(cid:1)
(1 w 2)β−k
. ρ (1 z 2)n+1+α −| | dν (w)
−| | 1 z,w n+2+β+α α
Z1−|w|2≤1−|z|2 | −h i|
(cid:0) (cid:1)
. (1 z 2)−k−1ρ (1 z 2)n+1+α .
−| | −| |
To handle the integral I , w(cid:0)e use (9) and [9(cid:1), Proposition 1.4.10] once more,
2
and we obtain
ρ (1 w 2)n+1+α (1 w 2)β−k
I = −| | −| | dν (w)
2 1 z,w n+2+β+α α
Z1−|w|2>1−|z|2 (cid:0) | −h i| (cid:1)
ρ (1 z 2)n+1+α (1 w 2)β−k+(n+1+α)γ
C −| | −| | dν (w)
≤ (1(cid:0)−|z|2)(n+1+α)γ(cid:1)ZBn |1−hz,wi|n+2+β+α α
ρ (1 z 2)n+1+α
C −| | (1 z 2)−k−1+(n+1+α)γ
≤ (1 z 2)(n+1+α)γ × −| |
(cid:0) (cid:1)
−| |
= C(1 z 2)−k−1ρ (1 z 2)n+1+α .
−| | −| |
We conclude that there is also (cid:0)a constant C > 0(cid:1) such that for any z Bn,
∈
Rk+1f(z) C(1 z 2)−k−1ρ (1 z 2)n+1+α .
| |≤ −| | −| |
Now, let us suppose that there is a const(cid:0)ant C > 0 such(cid:1)that for some
integer k > (n+1+α)γ with γ being the upper-type of ρ, and any z Bn,
∈
(17) Rk+1f(z) C(1 z 2)−k−1ρ (1 z 2)n+1+α .
| |≤ −| | −| |
We will show that this implies that we can(cid:0)find a constant(cid:1)C˜ > 0 such that
for any z Bn,
∈
Rkf(z) C˜(1 z 2)−kρ (1 z 2)n+1+α .
| |≤ −| | −| |
We start by the following lemma. (cid:0) (cid:1)
LEMMA 2.12. Let ρ be a weight. Then for any integer k > (n+1+α)γ,
there is a constant C˜ depending on n,α,C,γ where C and γ are as in (9),
such that for any 0 < t < 1,
1 ρ(sn+1+α) ρ(tn+1+α)
(18) ds C˜ .
sk+1 ≤ tk
Zt
10 BENOˆITF.SEHBAANDEDGARTCHOUNDJA
Proof. Let N be the smallest integer such that t2N+1 1. Then we have
≥
using the monotonicity of the function ρ and (9) that
1 ρ(sn+1+α) N t2l+1 ρ(sn+1+α)
ds ds
sk+1 ≤ sk+1
Zt l=0Zt2l
X
N ρ(tn+1+α2(l+1)(n+1+α))
2lt
≤ tk+12l(k+1)
l=0
X
N ρ(tn+1+α2(l+1)(n+1+α))
=
tk2lk
l=0
X
N ρ(tn+1+α)
C2(n+1+α)γ 2−l(k−(n+1+α)γ)
≤ tk
l=0
X
ρ(tn+1+α) N
= C2(n+1+α)γ 2−l(k−(n+1+α)γ)
tk
l=0
X
ρ(tn+1+α)
C˜ .
≤ tk
The proof of the lemma is complete. (cid:3)
Coming back to our proof, we first consider the case of z > 1/2, z = rξ,
ξ Sn. We recall the following identity for any holomorph|ic| function g:
∈
1 Rg(tz)
(19) g(z) g(rξ/2) = dt.
− t
Z1/2
Applying (19) with g = Rkf, we obtain using (18) that for z >1/2,
| |
1 Rk+1g(tz)
Rkf(z) Rkf(rξ/2) dt
| − | ≤ t
Z1/2 (cid:12) (cid:12)
(cid:12) (cid:12)
1
C (1 t z )−k−1ρ (1 t z )n+1+α dt
≤ − | | − | |
Z1/2
1 ρ(sn+1+α) (cid:0) (cid:1)
C ds
≤ sk+1
Z1−|z|
C (1 z 2)−kρ (1 z 2)n+1+α .
1
≤ −| | −| |
(cid:0) (cid:1)
Hence for z > 1/2,
| |
Rkf(z) S(1/2)+C (1 z 2)−kρ (1 z 2)n+1+α
1
| | ≤ −| | −| |
(cid:0) (cid:1)
with
S(1/2) = max Rkf(z).
|z|≤1/2| |