Journal of Inequalities in Pure and
Applied Mathematics
http://jipam.vu.edu.au/
Volume 6, Issue 1, Article 13, 2005
CARLESON MEASURES FOR ANALYTIC BESOV SPACES: THE UPPER
TRIANGLE CASE
NICOLA ARCOZZI
U NIVERSITÀ DI B OLOGNA
D IPARTIMENTO DI M ATEMATICA
P IAZZA DI P ORTA S AN D ONATO , 5
40126 B OLOGNA , ITALY.
arcozzi@dm.unibo.it
Received May, 2004; accepted 14 January, 2005
Communicated by S.S. Dragomir
A BSTRACT. For a large family of weights ρ in the unit disc and for fixed 1 < q < p < ∞, we
give a characterization of those measures µ such that, for all functions f holomorphic in the unit
disc,
! p1
Z
m(dz)
2 0
p
p
kf kLq (µ) ≤ C(µ)
|(1 − |z| )f (z)| ρ(z)
+ |f (0)|
.
2
(1 − |z| )2
D
Key words and phrases: Analytic Besov Spaces, Carleson measures, Discrete model.
2000 Mathematics Subject Classification. Primary: 30H05; Secondary: 46E15 46E35.
1. I NTRODUCTION
Given indices p, q, 1 < p, q < ∞, and given a positive weight ρ on the unit disc, D, a positive
measure µ on is a Carleson measure for (Bp (ρ), q) if the following Sobolev-type inequality
holds whenever f is a function which is holomorphic in D,
Z
p1
m(dz)
2
(1.1)
kf kLq (µ) ≤ C(µ)
|(1 − |z| )f 0 (z)|p ρ(z)
+ |f (0)|p
.
(1 − |z|2 )2
D
Throughout the paper, m denotes the Lebesgue measure. If ρ is positive throughout D and, say,
continuous, the right-hand side of (1.1) defines a norm for a Banach space of analytic functions,
the analytic Besov space Bp (ρ).
ISSN (electronic): 1443-5756
c 2005 Victoria University. All rights reserved.
Most of this work was done while the author was visiting the Institute Mittag-Leffler, with a grant of the Royal Swedish Academy of
Sciences.
It is a pleasure to thank D. R. Adams for pointing out the reference [5].
Work partly supported by the COFIN project "Analisi Armonica", funded by the Italian Minister of Research.
124-04
2
N ICOLA A RCOZZI
2
m(dz)
0
The measure (1−|z|
2 2 and the differential operator |(1 − |z| )f (z)| should be read, respec)
tively, as the volume element and the gradient’s modulus with respect to the hyperbolic metric
in D,
|dz|2
ds2 =
.
(1 − |z|2 )2
In [2], a characterization of the Carleson measures for (Bp (ρ), q) was given, when p ≤ q and
ρ is a p-admissible weight, to be defined below. Loosely speaking, a weight ρ is p-admissible
0
if one can naturally identify the dual space of Bp (ρ) with Bp0 (ρ1−p ). The weights of the form
(1 − |z|2 )s , s ∈ R, are p-admissible if and only if −1 < s < p − 1. (Here and throughout
p−1 + p0−1 = q −1 + q 0−1 = 1).
Theorem 1.1 ([2]). Suppose that 1 < p ≤ q < ∞ and that ρ is a p-admissible weight. A
positive Borel measure µ on D is Carleson for (Bp (ρ), q) if, and only if, there is a C1 (µ) > 0,
so that for all a ∈ D
Z
q00
p
0
0
p
≤ C1 (µ)µ (S(a)) .
(1.2)
ρ(z)−p /p (µ (S(z) ∩ S(a))) mh (dz)
S(a)
For a ∈ D,
arg(az̄) ≤ 1 − |a|
2π is the Carleson box with center a. The proof was based on a discretization procedure and on the
solution of a two-weight inequality for the “Hardy operator on trees”. Actually, when q > p,
Theorem 1.1 holds with a “single box” condition which is simpler than (1.2). For different
characterizations of the Carleson measures for analytic Besov spaces in different generality, see
[13], [8], [14], [17], [18]. A short survey of results and problems is contained in [1].
In this note, we consider the Carleson measures for (Bp (ρ), q) in the case 1 < q < p < ∞.
The new tool is a method allowing work in this “upper triangle” case, developed by C. Cascante,
J.M. Ortega and I.E. Verbitsky in [5]. Before we state the main theorem, we introduce some
notation.
For a ∈ D, let P (a) = [0, a] ∈ D, the segment with endpoints 0 and a. Let 1 < p < ∞ and
let ρ be a positive weight on D. Given a positive, Borel measure µ on D, we define its boundary
Wolff potential, Wco (µ) = Wco (ρ, p; µ) to be
Z
|dw|
0
0
Wco (µ)(a) =
ρ(w)p −1 µ(S(w))p −1
.
1 − |w|2
P (a)
S(a) = z ∈ D : 1 − |z| ≤ 2(1 − |a|),
The main result of this note is the theorem below. Its statement certainly does not come as a
surprise to the experts.
Theorem 1.2. Let 1 < q < p < ∞ and let ρ be a p-admissible weight. A positive Borel measure
µ on D is a Carleson measure for (Bp (ρ), q) if and only if
Z
q(p−1)
(1.3)
(Wco (µ)(z)) p−q µ(dz) < ∞.
D
We say that a weight ρ is p-admissible if the following two conditions are satisfied:
(i) ρ is regular, i.e., there exist > 0, C > 0 such that ρ(z1 ) ≤ Cρ(z2 ) whenever z1
and z2 are within hyperbolic distance . Equivalently, there are δ < 1, C 0 > 0 so that
ρ(z1 ) ≤ C 0 ρ(z2 ) whenever
z1 − z2 1 − z1 z2 ≤ δ < 1.
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C ARLESON M EASURES FOR A NALYTIC B ESOV S PACES
3
(ii) the weight ρp (z) = (1 − |z|2 )p−2 ρ(z) satisfies the Bekollé-Bonami Bp condition ([4],
[3]): There is a C(ρ, p) so that for all a ∈ D
 01


p −1
Z
Z
0



(1.4)
ρp (z)m(dz) 
ρp (z)1−p m(dz)
≤ C(ρ, p)m(S(a))p .

S(a)
S(a)
Inequalities like (1.1) have been extensively studied in the setting of Sobolev spaces. For
instance, given 1 < p, q < ∞, consider the problem of characterizing the Maz’ya measures for
(p, q); that is, the class of the positive Borel measures µ on R such that the Poincarè inequality
1q
Z
Z
p1
q
p
|u| dµ
(1.5)
≤ C(µ)
|∇u| dm
Rn
Rn
holds for all functions u in C0∞ (Rn ), with a constant independent of u. Here, we only consider
the case when 1 < q < p < ∞, and refer the reader to [16] for a comprehensive survey of
these “trace inequalities”. Maz’ya [11], and then Maz’ya and Netrusov [12], gave a characterization of such measures that involves suitable capacities. Later, Verbitsky [15], gave a first
noncapacitary characterization.
The following noncapacitary characterization of the Maz’ya measures for q < p is in [5]. For
0 < α < n, let Iα (x) = c(n, α)|x|α−n be the Riesz kernel in Rn . Recall that, for 1 < p < ∞,
(1.5) is equivalent, for α = 1, to the inequality
Z
1q
Z
p1
q
p
(1.6)
|Iα ? v| dµ
≤ C(µ)
|v| dm
Rn
Rn
with a constant C(µ), independent of v ∈ Lp (Rn ).
Now, let B(x, r) denote the ball in Rn , having its center at x and radius r. The Hedberg-Wolff
potential Wα,p of µ is
p0 −1
Z ∞
µ(B(x, r))
dr
Wα,p (µ)(x) =
.
n−αp
r
r
0
Theorem 1.3 ([5]). If 1 < q < p < ∞ and 0 < α < n, µ satisfies (1.6) if and only if
Z
q(p−1)
(1.7)
(Wα,p (µ)) p−q dµ < ∞.
Rn
Comparing the different characterizations for the analytic-Besov and the Sobolev case, we
see at work the heuristic principle according to which the relevant objects for the analysis in
Sobolev spaces (e.g., Euclidean balls, or the potential Wα,p ) have as holomorphic counterparts
similar objects, who live near the boundary (e.g., Carleson boxes, or the potential Wco ). This is
expected, since a holomorphic function cannot behave badly inside its domain. Another simple,
but important, heuristic principle is that holomorphic functions are essentially discrete. By this,
we mean that, for many purposes, we can consider a holomorphic function in the unit disc as
if it were constant on discs having radius comparable to their distance to the boundary. (For
positive harmonic functions, this is just Harnack’s inequality). Based on these considerations,
one might think that the problem of characterizing the Carleson measures for (Bp (ρ), q) might
be reduced to some discrete problem. This is in fact true, and it is the main tool in the proof of
Theorem 1.2.
The idea, already exploited in [2], is to consider (1 − |z|2 )f 0 (z) constants on sets that form a
Whitney decomposition of D. The Whitney decomposition has a natural tree structure, hence,
the Carleson measure problem leads to a weighted inequality on trees.
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N ICOLA A RCOZZI
The discrete result is the following. Let T be a tree, i.e., a connected, loopless graph, that
we do not assume to be locally finite; see Section 3 for complete definitions and notation. Let
o ∈ T be a fixed vertex, the root of T . There is a partial order on T defined by: x ≤ y, x, y ∈ T ,
if x ∈ [o, y], the geodesic joining o and y. Let ϕ : T → C. We define Iϕ, the Hardy operator
on T , with respect to o, applied to ϕ, by
Iϕ(x) =
x
X
ϕ(y) =
o
X
ϕ(y).
y∈[o,x]
A weight ρ on T is a positive function on T .
For x ∈ T , let S(x) = {y ∈ T : y ≥ x}. S(x) is the Carleson box with vertex x or the
successors’ set of x. Also, let P (x) = {z ∈ T : o ≤ z ≤ x} be the predecessors’ set of x.
Given a positive weight ρ and a nonnegative function µ on T , and given 1 < p < ∞, define
W (µ) = W (ρ, p; µ), the discrete Wolff potential of µ,
X
0
0
(1.8)
W (µ)(x) =
ρ(y)1−p µ(S(y))p −1 .
y∈P (x)
Theorem 1.4. Let 1 < q < p < ∞ and let ρ be a weight on T . For a nonnegative function µ
on T , the following are equivalent :
(1) For some constant C(µ) > 0 and all functions ϕ
! p1
! 1q
X
X
≤ C(µ)
|ϕ(x)|p ρ(x)
.
(1.9)
|Iϕ(x)|q µ(x)
x∈T
x∈T
(2) We have the inequality
(1.10)
X
µ(x) (W (µ)(x))
q(p−1)
p−q
< ∞.
x∈T
A different characterization of the measures µ for which (1.9) holds is given in [6] Theorem
3.3, in the more general context of thick trees (i.e., trees in which the edges are copies of
intervals of the real line). The necessary and sufficient condition given in [6], however, seems
more difficult to verify than (1.10), at least in our simple context.
The paper is structured as follows. In Section 2, we show that the problem of characterizing the measures µ for which (1.1) holds is completely equivalent, if ρ is p-admissible, to the
corresponding problem for the Hardy operator on trees, I. The idea, already present in [2], is
to replace the unit disc by one of its Whitney decompositions, endowed with its natural tree
structure, and the integral along segments by the sum along tree-geodesics. In Section 3, following [5], Theorem 1.4 is proved, and the problem on trees is solved. Section 2 and Section
3 together, show that (1.1) is equivalent to a condition which is the discrete analogue of (1.3).
Unfortunately, this discrete condition depends on the chosen Whitney decomposition. In Section 4, we show that the discrete condition is in fact equivalent to (1.3). In the course of the
proof, we will see that the Carleson measure problem in the unit disc is equivalent to a number
of its different discrete “metaphors” on a suitable graph.
Actually, this route to the proof of Theorem 1.2 is not the shortest possible. In fact, we could
have carried the discretization directly over a graph, skipping the repetition of some arguments.
We chose to do otherwise for two reasons. First, the tree situation is slightly easier to handle,
and it leads, already in Section 2, to a characterization of the Carleson measures for (Bp (ρ), q).
Second, it can be more easily compared with the proof of the characterization theorem for the
case q ≥ p, which was obtained in [2] working on trees.
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C ARLESON M EASURES FOR A NALYTIC B ESOV S PACES
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It should be mentioned that [5] has some results for spaces of holomorphic functions, which
are different from those considered in this article.
2. D ISCRETIZATION
In this section, Theorem 2.5, we show that, if ρ is p-admissible, then the problem of characterizing the Carleson measures for Bp (ρ) is equivalent to a two-weight inequality on trees. This
fact is already implicit in [2], but, here, our formulation stresses more clearly the interplay between the discrete and the continuous situation. In the context of the weighted Bergman spaces,
a similar approach to the Carleson measures problem was employed by Luecking [9], who also
obtained a characterization theorem in the upper triangle case [10].
First, we recall some facts on Bergman and analytic Besov spaces.
Let 1 < p < ∞ be fixed and let ρ be a weight on D. The Bergman space Ap (ρ) is the space
of those functions f that are holomorphic in D and such that
Z
p
kf kAp (ρ) =
|f (z)|p ρ(z)m(dz)
D
is finite. Define, for f, g ∈ A2 ≡ A2 (1),
Z
hf, giA2 =
f (z)g(z)m(dz).
D
0
Let Ap (ρ)∗ be the dual space of Ap (ρ). We identify g ∈ Ap0 (ρ1−p ) with the functional on Ap (ρ)
Λg : f 7→ hf, giA2 .
(2.1)
0
By Hölder’s inequality we have that A∗p (ρ) ⊆ Ap0 (ρ1−p ). Condition (1.4) shows that the reverse
inclusion holds.
Theorem 2.1 (Bekollé-Bonami [4], [3]). If the weight ρ satisfies (1.4) then g 7→ Λg , where Λg
0
defined in (2.1) is an isomorphism of Ap0 (ρ1−p ) onto A∗p (ρ).
We need some consequences of Theorem 2.1, whose proof can be found in [2], §2 and §4.
Let F, G be holomorphic functions in D,
F (z) =
∞
X
an z n ,
G(z) =
0
∞
X
bn zn .
0
Define
hF, GiD∗ =
∞
X
Z
F 0 (z)G0 (z)m(dz)
nan bn =
D
1
and
hF, GiD = a0 b0 +
∞
X
nan bn = F (0)G(0) + hF, GiD∗ .
1
0
Lemma 2.2. Let ρ be a weight satisfying (1.4). Then Bp0 (ρ1−p ) is the dual of Bp (ρ) under the
pairing h·, ·iD . i.e., each functional Λ on Bp (ρ) can be represented as
Λf = hf, giD ,
f ∈ Bp (ρ)
0
for a unique g ∈ Bp0 (ρ1−p ).
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N ICOLA A RCOZZI
The reproducing kernel of D with respect to the product h·, ·iD is
1
φz (w) = 1 + log
1 − wz̄
i.e., if f ∈ D, then
0
Z
1
0
m(dw) + f (0).
f (z) = hf, φz iD =
f (w) 1 + log
1 − z̄w
D
Lemma 2.3. Let ρ be an admissible weight, 1 < p < ∞. Then, φz is a reproducing kernel for
0
0
Bp0 (ρ1−p ). i.e., if G ∈ Bp0 (ρ1−p ), then
G(z) = hG, φz iD .
(2.2)
0
In particular, point evaluation is bounded on Bp0 (ρ1−p ).
0
Observe that (1.4) is symmetric in (ρ, p) and (ρ1−p , p0 ) and hence the same conclusion holds
for Bp (ρ).
Now, let µ be a positive bounded measure on D and define
Z
hF, Giµ = hF, GiL2 (µ) =
F (z)G(z)µ(dz).
D
µ is Carleson for (Bp (ρ), p, q) if and only if
Id : Bp (ρ) → Lq (µ)
is bounded. In turn, this is equivalent to the boundedness, with the same norm, of its adjoint
Θ = Id∗ ,
0
0
Θ : Lq (µ) → (Bp (ρ))∗ ≡ Bp0 (ρ1−p ),
where we have used the duality pairings h·, ·iD and h·, ·iµ , and Lemma 2.2.
By Lemma 2.3,
ΘG(z) = hΘG, φz iD = hG, φz iL2 (µ)
Z 1
=
1 + log
G(w)µ(dw).
1 − z w̄
D
For future reference, we state this as
Lemma 2.4. If ρ is a p-admissible weight, the adjoint of Id : Bp (ρ) → Lq (µ) is the operator
0
0
Θ : Lq (µ) → (Bp (ρ))∗ ≡ Bp0 (ρ1−p )
defined by
Z (2.3)
ΘG(z) =
D
1
1 + log
1 − z w̄
G(w)µ(dw).
Consider, now, a dyadic Whitney decomposition of D. Namely, for integer n ≥ 0, 1 ≤ m ≤
2n , let
m −n−1
−n arg(z)
−(n+1)
∆n,m = z ∈ D : 2
≤ 1 − |z| ≤ 2 , − n ≤ 2
.
2π
2
These boxes are best seen in polar coordinates. It is natural to consider the Whitney squares as
indexed by the vertices of a dyadic tree, T2 . Thus the vertices of T2 are
(2.4)
{α|α = (n, m), n ≥ 0 and 1 ≤ m ≤ 2n , m, n ∈ N}
and we say that there is an edge between (n, m), (n0 , m0 ) if ∆(n,m) and ∆(n0 ,m0 ) share an arc of
a circle. The root of T2 is, by definition, (0, 1). Here and throughout we will abuse notation and,
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C ARLESON M EASURES FOR A NALYTIC B ESOV S PACES
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when convenient, identify the vertices of such a tree with the sets for which they are indices.
Here we identify α and ∆α . Thus, there are four edges having (0, 1) as an endpoint, each other
box being the endpoint of exactly three edges.
Given a positive, regular weight ρ on D, we define a weight on T2 , still denoted by ρ. If
α ∈ T2 and if zα be, say, the center of the box α ⊂ D, then, ρ(α) = ρ(zα ). By the regularity
assumption, the choice of zα does not matter in the estimates that follow.
Theorem 2.5. Let 1 < q, p < ∞ and let ρ be a p-admissible weight. A positive Borel measure
µ on D is a Carleson measure for (Bp (ρ), q) if and only if the following inequality holds, with
a constant C which is independent of ϕ : T2 → R.
! p1
! 1q
X
(2.5)
|Iϕ(x)|q µ(x)
≤C
X
|ϕ(y)|p ρ(y)
.
y∈T2
x∈T2
Proof. It is proved in [2] (§4, Theorem 12, proof of the sufficiency condition) that (2.5) is
sufficient for µ to be a Carleson measure.
We come, now, to necessity. Without loss of generality, assume that supp(µ) ⊆ {z : |z| ≤
1/2}. By the remarks preceding the proof, Lemma 2.2 and Lemma 2.3, IF µ is Carleson, then
0
0
0
Θ is bounded from Lq (µ) to Bp0 (ρ1−p ). Consider, now, functions g ∈ Lq (µ), having the form
g(w) =
|w|
h(w),
w
where h ≥ 0 and h is constant on each box α ∈ T2 , h|α = h(α). The boundedness of Θ implies
! 10
q
C
X
q0
|h(α)| µ(α)
Z
=
10
q
|g| dµ
q0
D
α∈T2
≥ kΘgkBp0 (ρ1−p0 )
≥
! 10
p0
Z Z
p
1 − |z|2
m(dz)
0
ρ(z)1−p
|w|h(w)µ(dw)
.
1 − zw
(1 − |z|2 )2
D
D
For z ∈ D, let α(z) ∈ T2 be the Whitney box containing z. By elementary estimates,
Re
(2.6)
|w|(1 − |z|2 )
1 − zw
≥0
if w ∈ D, and
Re
|w|(1 − |z|2 )
1 − zw
≥ c > 0, if w ∈ S(α(z))
for some universal constant c. If α(z) = o is the root of T2 , the latter estimate holds, say, only
on one half of the box o, and this suffices for the calculations below.
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N ICOLA A RCOZZI
Using this, and the fact that all our Whitney boxes have comparable hyperbolic measure, we
can continue the chain of inequalities
! 10
p0
Z Z
p
2
1 − |z|
m(dz)
1−p0
≥
|w|h(w)µ(dw) ρ
(z)
(1 − |z|2 )2
D
S(α(z)) 1 − zw
 10
 
p0
p
Z
X
m(dz)
0


h(β)µ(β) ρ1−p (z)
≥ c 

2
2
(1 − |z| )
D
β∈S(α(z))

 10
p0

p
0
X  X
≥ c
h(β)µ(β) ρ(α)1−p  .
α
β∈S(α)
Let I ∗ , defined on functions ϕ : T2 → R, be the operator
X
I ∗ ϕ(α) =
ϕ(β)µ(β).
β∈S(α)
It is readily verified that I ∗ is the adjoint of I, in the sense that
X
X
I ∗ ψ(α)ϕ(α) =
ψ(α)Iϕ(α)µ(α).
T2
T2
Then, the chain of inequalities above shows that
0
0
0
I ∗ : Lq (µ) → Lp (ρ1−p )
is a bounded operator. In turn, this is equivalent to the boundedness of
I : Lp (ρ) → Lq (µ).
3. A T WO - WEIGHT H ARDY I NEQUALITY ON T REES
In this section we prove Theorem 1.4.
Let T be a tree. We use the same name T for the tree and for its set of vertices. We do not
assume that T is locally finite; a vertex of T can be the endpoint of infinitely many edges. If
x, y ∈ T , the geodesic between x and y, [x, y], is the set {x0 , . . . , xn }, where x0 = x, xn =
y, xj−1 is adjacent to xj (i.e., xj−1 and xj are endpoints of an edge), and the vertices in [x, y]
are all distinct. We let [x, x] = {x}. If x, y are as above, we let d(x, y) = n. Let o ∈ T be a
fixed root. We say that x ≤ y, x, y ∈ T , if x ∈ [o, y]. ≤ is a partial order on T . For x ∈ T , the
Carleson box of vertex x (or the set of successors of x) is S(x) = {y ∈ T : y ≥ x}. We will
sometimes write [o, x] = P (x), the set of predecessors of x.
Theorem 3.1. Let 1 < q < p < ∞ and let ρ be a weight on T . For a nonnegative function µ
on T , the following are equivalent:
(1) For some constant C(µ) > 0 and all functions ϕ
! 1q
! p1
X
X
q
p
(3.1)
|Iϕ(x)| µ(x)
≤ C(µ)
|ϕ(x)| ρ(x)
.
x∈T
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C ARLESON M EASURES FOR A NALYTIC B ESOV S PACES
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(2) We have the inequality
X
(3.2)
µ(x) (W (µ)(x))
q(p−1)
p−q
< ∞.
x∈T
As a consequence of Theorems 3.1 and 2.5, we obtain a characterization of the Carleson
measures for (Bp (ρ), q).
Corollary 3.2. A measure µ in the unit disc is Carleson for (Bp (ρ), q) if, and only if,
X
q(p−1)
(3.3)
µ(α) (W (µ)(α)) p−q < ∞.
α∈T2
Proof. First, we show that (3.2) implies (3.1). By duality, it suffices to show that, if (3.2) holds,
then I ∗ ,
X
I ∗ ϕ(x) =
ϕ(y)µ(y)
y∈S(x)
q0
p0
is a bounded map from L (µ) to L (ρ
positive functions. Let g ≥ 0. Then
1−p0
). Without loss of generality, we can test I ∗ on
p0

0
kI ∗ gkpLp0 (ρ1−p0 ) =
X
by definition of W, =
X
0
ρ(x)1−p 
x∈T
X
g(y)µ(y)
y∈S(x)
g(y)µ(y)W (gµ)(y).
y∈T
Define, now, the maximal function
P
t∈S(z)
Mµ g(y) = max
(3.4)
g(t)µ(t)
µ(S(z))
z∈P (y)
.
The following lemma will be proved at the end of the proof of Theorem 3.1. It can be considered
as a discrete, boundary version of the weighted maximal theorem of R. Fefferman [7].
Lemma 3.3. If 1 < s < ∞ and µ is a bounded measure on T , then Mµ is bounded on Ls (µ).
Since g ≥ 0,
W (gµ)(y) ≤
X
0
0
0
ρ(x)1−p µ(S(x))p −1 (Mµ g(y))p −1
P (y)
0
= W (µ)(y) (Mµ g(y))p −1 .
Thus,
0
kI ∗ gkpLp0 (ρ1−p0 ) ≤
X
0
g(y)µ(y)W (µ)(y) (Mµ g(y))p −1
y∈T
! r1
≤
X
(p0 −1)r
Mµ g(y)
µ(y)
! 10
r
X
y∈T
by Hölder’s inequality, with r =
≤ CkgkLq0 (µ)
r0
g(y) (W (µ)(y)) µ(y)
y∈T
q0
p0 −1
> 1,
!
p0 −1
r0
X
0
g(y)λr µ(y)
y∈T
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1
λr 0
!
X
1
λ0 r 0
0 0
(W (µ)(y))λ r µ(y)
y∈T
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10
N ICOLA A RCOZZI
by Lemma 3.3 and Hölder’s inequality, with λ = q 0 − p0 + 1 > 1,
p−q
! (p−1)q
X
(p−1)q
0
= CkgkpLq0 (µ)
(W (µ)(y)) p−q µ(y)
.
y∈T
This proves one implication.
We show that, conversely, (3.1) implies (3.2). By hypothesis and duality, we have, for g ≥ 0,
X
0
0
0
kgkpLq0 (µ) ≥ C
I ∗ g(y)p ρ(y)1−p
y∈T
=
X
1−p0
ρ(y)
P
x∈S(y) g(x)µ(x)
p0
µ(S(y))
!p0
µ(S(y))
y∈T
.
1
Replace g = (Mµ h) p0 , with h ≥ 0. By Lemma 3.3, since q 0 > p0 ,
khk
q0
L p0 (µ)
≥ CkMµ hk
q0
L p0 (µ)
0
0
= Ck (Mµ h)1/p kpLq0 (µ)
P
p0
1
0
p
X
x∈S(y) (Mµ h(x)) µ(x)
 ,
≥C
e(y) 
µ(S(y))
y∈T
0
0
where e(y) = ρ(y)1−p µ(S(y))p ,

≥C
X
P
x∈S(y)

e(y) 
y∈T

≥C
X
y∈T
=C
X
e(y) 
µ(S(y))
µ(t)h(t)
t∈T
p0
.
10
p
µ(x) 
t∈S(y) h(t)µ(t) µ(S(y))

µ(S(y))
P

X
h(t)µ(t)
t∈S(y)
X
y∈T
e(y)
χS(y)
µ(S(y))
!
(t).
By duality, then, we have that
X
y∈T
q(p−1)
e(y)
0
0 0
χS(y) ∈ L(q /p ) (µ) = L p−q (µ).
µ(S(y))
Hence,
∞>
X X
x∈T
=
X
y∈T
! q(p−1)
p−q
e(y)
χS(y) (x)
µ(x)
µ(S(y))
µ(x) (W (µ)(x))
q(p−1)
p−q
,
x∈T
which is the desired conclusion.
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C ARLESON M EASURES FOR A NALYTIC B ESOV S PACES
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As a final remark, let us observe that the potential W admits the following, suggestive formulation:
p0 −1
∗
1−p0
(I Id)
,
W (µ) = I ρ
where Id is the identity operator.
Condition (1.10) might, then, be reformulated as
q(p−1)
0
0
I ρ1−p (I ∗ Id)p −1 ∈ L p−q (µ).
Proof of Lemma 3.3. The argument is a caricature of the classical one. By intepolation, it suffices to show that Mµ is of weak type (1, 1). For f ≥ 0 on T and t > 0, let E(t) = {Mµ > f >
t}. If x ∈ E(t), there exists z = z(x) ∈ P (x), such that
X
tµ(S(z)) <
f (y)µ(y).
y∈S(z)
Let I be the set of such z’s. By the tree structure, there exists a subset J of I, which is maximal,
in the sense that, for each z in I, there is a w in J so that w ≥ z. Hence,
E(t) ⊆ ∪z∈I S(z) = ∪w∈J S(w)
the latter union being disjoint. Thus,
µ(E(t)) ≤
1
µ(S(w)) ≤ kf kL1 (µ) ,
t
w∈J
X
which is the desired inequality.
4. E QUIVALENCE OF T WO C ONDITIONS
The last step in the proof of Theorem 1.2 consists of showing that condition (3.3) is equivalent
to (1.3). In order to do so, we introduce in T2 , the set of the Whitney boxes in which D was
partitioned, a graph structure, which is richer than the tree structure we have considered so far.
Let T2 be the set defined in Section 2. We make T2 into a graph G structure as follows. For
α, β ∈ T2 , to say that “there is an edge of G between α and β” is to say that the closures α and
β share an arc or a straight line. For α, β ∈ G, the distance between α and β, dG (α, β), is the
minimum number of edges in a path between α and β. The ball of center α and radius k ∈ N in
G will be denoted by B(α, k) = {β ∈ G : dG (α, β) ≤ k}. In G, we maintain the partial order
given by the original tree structure. In particular, we still have the tree geodesics [0, α].
Let k ≥ 0 be an integer. For α ∈ G, define
Pk (α) = {β ∈ G : dG (β, [o, α]) ≤ k}
and, dually,
Sk (α) = {β ∈ G : α ∈ Pk (β)}.
Observe that β ∈ Sk (α) if and only if [0, β] ∩ B(α, k) is nonempty. Clearly, P0 = P and
S0 = S are the sets defined in the tree case. The corresponding operators Ik and Ik∗ are defined
as follows. For f : G → R,
X
(4.1)
Ik f (α) =
f (β)
β∈Pk (α)
and
(4.2)
Ik∗ f (α) =
X
f (β)µ(β).
β∈Sk (α)
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N ICOLA A RCOZZI
As before, Ik and Ik∗ are dual to each other. That is, if L2 (G) is the L2 space on G, with respect
to the counting measure,
hIk ϕ, ψiL2 (µ) = hϕ, Ik∗ ψiL2 (G) .
For each k, we have a discrete potential
X
0
0
ρ(y)1−p µ(Sk (y))p −1
Wk (µ)(x) =
y∈Pk (x)
and a [COV]-condition
X
(4.3)
µ(x) (Wk (µ)(x))
q(p−1)
p−q
< ∞.
x∈T
If f ≥ 0 on G, then Ik f ≥ If , pointwise on G. The estimates for all these operators, however,
behave in the same way.
Proposition 4.1. Let µ be a measure and ρ a p-admissible weight on D, 1 < q < p < ∞. Also,
let µ and ρ denote the corresponding weights on G.
Then, the following conditions are equivalent
(i) There exists C > 0 such that (1.1) holds, that is
Z
p1
m(dz)
2
kf kLq (µ) ≤ C(µ)
|(1 − |z| )f 0 (z)|p ρ(z)
+ |f (0)|p
(1 − |z|2 )2
D
whenever f is holomorphic on D.
(ii) For k ≥ 2, there exists Ck > 0 such that
! 1q
! p1
X
X
q
p
(4.4)
|Ik ϕ(x)| µ(x)
≤ Ck (µ)
|ϕ(x)| ρ(x)
.
x∈T
x∈T
(iii) The following inequality holds,
X
q(p−1)
(4.5)
µ(x) (Wk (µ)(x)) p−q < ∞.
x∈T
(iv) (1.3) holds,
Z
(Wco (µ)(z))
(4.6)
q(p−1)
p−q
µ(dz) < ∞.
D
(v) (1.10) holds,
X
(4.7)
µ(x) (W (µ)(x))
q(p−1)
p−q
< ∞.
x∈T
(vi) (1.9) holds for some C > 0,
! 1q
(4.8)
X
|Iϕ(x)|q µ(x)
x∈T
! p1
≤ C(µ)
X
|ϕ(x)|p ρ(x)
.
x∈T
Proof. We prove that (i) =⇒ (ii) =⇒ (iii) =⇒ (iv) =⇒ (v) =⇒ (vi) =⇒ (i).
The implications (v) =⇒ (vi) =⇒ (i) were proved in Theorems 1.4 and 1.2, respectively.
(i) =⇒ (ii) can be proved by the same argument used in the proof of Theorem 2.5, with minor
changes only. The key is the estimate (2.6).
The proof that (iii) =⇒ (iv) =⇒ (v) is easy. Observe that Wk (µ) increases with k, hence that
(iii) with k = n implies (iii) with k = n − 1. In particular, it implies (v), which corresponds to
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C ARLESON M EASURES FOR A NALYTIC B ESOV S PACES
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k = 0. Let z ∈ D, and let α(z) ∈ G be the box containing z. Then, it is easily checked that, if
k ≥ 2,
S(α(z)) ⊂ S(z) ⊂ Sk (α(z)).
To show the implication (iii) =⇒ (iv), observe that
Z
|dw|
0
0
Wco (µ)(z) =
ρ(w)p −1 µ(S(w))p −1
1 − |w|2
P (z)
≤C
α(z)
X
0
0
ρ(β)p −1 µ(S(β))p −1
β=o
≤C
X
0
0
ρ(β)p −1 µ(Sk (β))p −1
β∈Pk (α(z))
= CWk (µ)(α(z)),
hence
Z
(Wco (µ)(z))
q(p−1)
p−q
µ(dz) ≤ C
D
X
α∈G
≤C
X
sup (Wco (µ)(z))
q(p−1)
p−q
µ(α)
z∈α
(Wk (µ)(α))
q(p−1)
p−q
µ(α)
α∈G
as wished.
For γ ∈ G, let γ − be the predecessor of γ : γ − ∈ [o, γ] and dG (γ, γ − ) = 1. For the
implication (iv) =⇒ (v), we have
Z
|dw|
0
0
Wco (µ)(z) =
ρ(w)p −1 µ(S(w))p −1
1 − |w|2
P (z)
α(z)−
≥C
X
0
0
ρ(β)p −1 µ(S(β))p −1
β=o
α(z)
≥C
X
0
0
ρ(β)p −1 µ(S(β))p −1
β=o
= CW (µ)(α(z)).
In the second last inequality, we used the fact that S(α) ⊂ S(α− ). Then,
Z
X
q(p−1)
q(p−1)
(Wco (µ)(z)) p−q µ(dz) ≥ C
inf (Wco (µ)(z)) p−q µ(α)
D
α∈G
≥C
X
z∈α
(W (µ)(α))
q(p−1)
p−q
µ(α)
α∈G
and this shows that (iv) =⇒ (v).
We are left with the implication (ii) =⇒ (iii). The proof follows, line by line, that of Theorem
1.4 in Section 3. One only has to modify the definition of the maximal function
P
t∈Sk (z) g(t)µ(t)
(4.9)
Mk,µ g(y) = max
.
z∈Pk (y)
µ(Sk (z))
We just have to verify that Mk,µ is bounded on Ls (µ), if 1 < s < ∞. It suffices to show that
Mk,µ is of type weak (1, 1) and this, in turn, boils down to the covering lemma that follows.
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N ICOLA A RCOZZI
Lemma 4.2. There exists a constant L > 0 with the following property. Let F be any set in G,
F ⊆ ∪z∈I Sk (z)
where I ⊆ G is an index set. Then, there exists J ⊆ I such that
F ⊆ ∪z∈J Sk (z)
and, for all x ∈ G,
]{w ∈ J : x ∈ Sk (w)} ≤ L
where ]A is the number of elements in the set A.
Proof of the lemma. For simplicity, we prove the lemma when k = 2. Incidentally, this suffices
to finish the proof of Theorem 1.2.
It suffices to show that, if zj are points in G, j = 1, 2, 3, and ∩3j=1 S2 (zj ) is nonempty, then
one of the S2 (zj )’s, say S2 (z1 ), is contained in the union of the other two. In fact, this gives
L = 2 in the lemma.
Let z ∈ G, dG (z, w) ≥ 3. Let z −2 be the point w in [o, z] such that dG (o, w) = 2 and let z ∗
be the only point w in G such that
w ∈ S2 (z), dG (o, w) = dG (o, z) − 1 and w ∈
/ S0 (z −2 )
where S0 (z) = S(z) is the same Carleson box introduced in Section 3. Then, one can easily
see that
S2 (z) = S0 (z −2 ) ∪ S0 (z ∗ )
the union being disjoint.
Let now z1 , z2 , z3 be as above, with dG (o, z1 ) ≥ dG (o, z2 ) ≥ dG (o, z3 ). Then, dG (o, z1−2 ) ≥
dG (o, z2−2 ) ≥ dG (o, z3−2 ) and z2 is a point within dG distance 1 from S2 (z3 ). If S2 (z2 ) ⊆ S2 (z3 ),
there is nothing to prove. Otherwise,
S2 (z2 ) ∪ S2 (z3 ) = S0 (z3−2 ) ∪ S0 (z3∗ ) ∪ S0 (w),
where w = z2−2 or w = z2∗ , respectively, the union being disjoint, and
S2 (z2 ) ∩ S2 (z3 ) = S0 (ξ),
where ξ = z2∗ or ξ = z2−2 , respectively. In the first case, since dG (o, z1 ) ≥ dG (o, z2 ), if S2 (z1 )
intersects S0 (w), then S2 (z1 ) must be contained in the union of S2 (z2 ) and S2 (z3 ). The same
holds in the second case, unless dG (o, z1 ) = dG (o, z2 ). In this last case, one of the following
three holds: (i) S2 (z1 ) ⊂ S2 (z3 ), (ii) S2 (z1 ) = S2 (z2 ), (iii) S2 (z2 ) ⊂ S2 (z3 ) ∪ S2 (z1 ). In all
three cases, the claim holds, hence the lemma.
An extension of the results in this paper to higher complex dimensions is in N. ARCOZZI,
R.ROCHBERG, E. SAWYER, “Carleson Measures and Interpolating Sequences for Besov
Spaces on Complex Balls", to appear in Memoirs of the A.M.S.
The covering lemma might also be proved taking into account the interpretation of the graph
elements as Whitney boxes, then using elementary geometry.
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