Journal of Inequalities in Pure and
Applied Mathematics
CARLESON MEASURES FOR ANALYTIC BESOV SPACES:
THE UPPER TRIANGLE CASE
volume 6, issue 1, article 13,
2005.
NICOLA ARCOZZI
Università di Bologna
Dipartimento di Matematica
Piazza di Porta San Donato, 5
40126 Bologna, ITALY.
Received 30 May, 2004;
accepted 14 January, 2005.
Communicated by: S.S. Dragomir
EMail: arcozzi@dm.unibo.it
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
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,
Z
kfkLq (µ) ≤ C(µ)
m(dz)
|(1 − |z| )f (z)| ρ(z)
+ |f(0)|p
(1 − |z|2 )2
D
2
0
p
1
p
.
2000 Mathematics Subject Classification: Primary: 30H05; Secondary: 46E15
46E35.
Key words: Analytic Besov Spaces, Carleson measures, Discrete model
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.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3
A Two-weight Hardy Inequality on Trees . . . . . . . . . . . . . . . . 19
4
Equivalence of Two Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 25
References
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1.
Introduction
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
(1.1) kf kLq (µ) ≤ C(µ)
m(dz)
p
|(1 − |z| )f (z)| ρ(z)
2 2 + |f (0)|
(1
−
|z|
)
D
2
0
p
p1
.
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).
2
m(dz)
0
The measure (1−|z|
2 2 and the differential operator |(1 − |z| )f (z)| should
)
be read, respectively, as the volume element and the gradient’s modulus with
respect to the hyperbolic metric in D,
|dz|2
ds =
.
(1 − |z|2 )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 if one can naturally identify the dual space
0
of Bp (ρ) with Bp0 (ρ1−p ). The weights of the form (1 − |z|2 )s , s ∈ R, are padmissible if and only if −1 < s < p − 1. (Here and throughout p−1 + p0−1 =
q −1 + q 0−1 = 1).
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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
−p0 /p
ρ(z)
(1.2)
q00
p
(µ (S(z) ∩ S(a))) mh (dz)
≤ C1 (µ)µ (S(a)) .
p0
S(a)
For a ∈ D,
S(a) =
arg(az̄) z ∈ D : 1 − |z| ≤ 2(1 − |a|), ≤ 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)
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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.
(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


Z
(1.4)


S(a)

Z

ρp (z)m(dz) 
1
p0 −1
0

ρp (z)1−p m(dz)
Carleson Measures for Analytic
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S(a)
≤ C(ρ, p)m(S(a))p .
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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
Z
p1
Z
1q
q
p
|∇u| dm
(1.5)
|u| dµ
≤ C(µ)
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
(1.6)
Rn
1q
Z
|Iα ? v| dµ
≤ C(µ)
q
p
p1
|v| dm
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) =
.
rn−αp
r
0
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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.
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
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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)
1. For some constant C(µ) > 0 and all functions ϕ
! p1
! 1q
X
X
(1.9)
|Iϕ(x)|q µ(x)
≤ C(µ)
|ϕ(x)|p ρ(x)
.
x∈T
2. We have the inequality
X
q(p−1)
(1.10)
µ(x) (W (µ)(x)) p−q < ∞.
x∈T
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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 :
x∈T
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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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It should be mentioned that [5] has some results for spaces of holomorphic
functions, which are different from those considered in this article.
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2.
Discretization
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)
Carleson Measures for Analytic
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D
Contents
is finite. Define, for f, g ∈ A2 ≡ A2 (1),
Z
hf, giA2 =
f (z)g(z)m(dz).
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D
0
Let Ap (ρ)∗ be the dual space of Ap (ρ). We identify g ∈ Ap0 (ρ1−p ) with the
functional on Ap (ρ)
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(2.1)
Λg : f 7→ hf, giA2 .
By Hölder’s inequality we have that
that the reverse inclusion holds.
A∗p (ρ)
⊆ Ap0 (ρ
Page 11 of 33
1−p0
). Condition (1.4) shows
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Theorem 2.1 (Bekollé-Bonami [4], [3]). If the weight ρ satisfies (1.4) then
0
g 7→ Λg , where Λg 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) =
∞
X
0
Define
hF, GiD∗ =
bn zn .
0
∞
X
Z
hF, GiD = a0 b0 +
F 0 (z)G0 (z)m(dz)
nan bn =
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D
1
and
Carleson Measures for Analytic
Besov Spaces: The Upper
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∞
X
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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)
for a unique g ∈ Bp0 (ρ
1−p0
).
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The reproducing kernel of D with respect to the product h·, ·iD is
φz (w) = 1 + log
Contents
1
1 − wz̄
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i.e., if f ∈ D, then
f (w) 1 + log
Z
0
f (z) = hf, φz iD =
D
1
1 − z̄w
0
m(dw) + f (0).
Lemma 2.3. Let ρ be an admissible weight, 1 < p < ∞. Then, φz is a repro0
0
ducing kernel for Bp0 (ρ1−p ). i.e., if G ∈ Bp0 (ρ1−p ), then
G(z) = hG, φz iD .
(2.2)
In particular, point evaluation is bounded on Bp0 (ρ
1−p0
).
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).
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µ is Carleson for (Bp (ρ), p, q) if and only if
Id : Bp (ρ) → Lq (µ)
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is bounded. In turn, this is equivalent to the boundedness, with the same norm,
of its adjoint Θ = Id∗ ,
0
Carleson Measures for Analytic
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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.
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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
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Z (2.3)
Carleson Measures for Analytic
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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−1
−n arg(z)
∆n,m = z ∈ D : 2
≤ 1 − |z| ≤ 2 , − n ≤ 2
.
2π
2
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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
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{α|α = (n, m), n ≥ 0 and 1 ≤ m ≤ 2n , m, n ∈ N}
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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, 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.
! 1q
! p1
X
X
(2.5)
|Iϕ(x)|q µ(x)
≤C
|ϕ(y)|p ρ(y)
.
x∈T2
y∈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
0
0
2.3, IF µ is Carleson, then Θ is bounded from Lq (µ) to Bp0 (ρ1−p ). Consider,
0
now, functions g ∈ Lq (µ), having the form
g(w) =
|w|
h(w),
w
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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)
α∈T2
Z
=
10
q
|g| dµ
q0
D
≥ 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,
|w|(1 − |z|2 )
≥0
(2.6)
Re
1 − zw
if w ∈ D, and
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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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Using this, and the fact that all our Whitney boxes have comparable hyperbolic measure, we can continue the chain of inequalities
! 10
p0
p
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
2
Z Z
β∈S(α(z))

 10
p0

p
0
X  X
≥ c
h(β)µ(β) ρ(α)1−p  .
α
Carleson Measures for Analytic
Besov Spaces: The Upper
Triangle Case
Nicola Arcozzi
β∈S(α)
Title Page
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
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T2
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Then, the chain of inequalities above shows that
0
Contents
0
0
I ∗ : Lq (µ) → Lp (ρ1−p )
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is a bounded operator. In turn, this is equivalent to the boundedness of
I : Lp (ρ) → Lq (µ).
Carleson Measures for Analytic
Besov Spaces: The Upper
Triangle Case
Nicola Arcozzi
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3.
A Two-weight Hardy Inequality on Trees
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:
! 1q
(3.1)
|Iϕ(x)|q µ(x)
! p1
≤ C(µ)
x∈T
X
|ϕ(x)|p ρ(x)
x∈T
X
x∈T
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2. We have the inequality
(3.2)
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1. For some constant C(µ) > 0 and all functions ϕ
X
Carleson Measures for Analytic
Besov Spaces: The Upper
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µ(x) (W (µ)(x))
q(p−1)
p−q
< ∞.
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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)
Carleson Measures for Analytic
Besov Spaces: The Upper
Triangle Case
Nicola Arcozzi
y∈S(x)
0
0
0
is a bounded map from Lq (µ) to Lp (ρ1−p ). Without loss of generality, we can
test I ∗ on positive functions. Let g ≥ 0. Then
p0

p0
kI ∗ gkLp0 (ρ1−p0 ) =
X
by definition of W, =
X
0
ρ(x)1−p 
x∈T
X
g(y)µ(y)
y∈S(x)
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g(y)µ(y)W (gµ)(y).
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Define, now, the maximal function
Page 20 of 33
P
(3.4)
Mµ g(y) = max
z∈P (y)
t∈S(z)
g(t)µ(t)
µ(S(z))
.
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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
Carleson Measures for Analytic
Besov Spaces: The Upper
Triangle Case
P (y)
0
= W (µ)(y) (Mµ g(y))p −1 .
Nicola Arcozzi
Thus,
0
kI ∗ gkpLp0 (ρ1−p0 ) ≤
X
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0
g(y)µ(y)W (µ)(y) (Mµ g(y))p −1
Contents
y∈T
! r1
≤
X
(p0 −1)r
Mµ g(y)
! 10
r
X
µ(y)
y∈T
≤ CkgkLq0 (µ)
y∈T
g(y) (W (µ)(y)) µ(y)
λr0
g(y)
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q0
p0 −1
> 1,
!
X
r0
y∈T
by Hölder’s inequality, with r =
p0 −1
r0
µ(y)
Close
1
λr 0
!
X
λ0 r 0
(W (µ)(y))
µ(y)
1
λ0 r 0
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y∈T
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by Lemma 3.3 and Hölder’s inequality, with λ = q 0 − p0 + 1 > 1,
p0
= CkgkLq0 (µ)
X
(W (µ)(y))
(p−1)q
p−q
p−q
! (p−1)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
=
0
0
ρ(y)1−p µ(S(y))p
y∈T
P
x∈S(y)
g(x)µ(x)
0
0
Replace g = (Mµ h) , with h ≥ 0. By Lemma 3.3, since q > p ,
q0
L p0 (µ)
≥ CkMµ hk
.
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1
p0
khk
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!p0
µ(S(y))
Carleson Measures for Analytic
Besov Spaces: The Upper
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q0
L p0 (µ)
0
0
= Ck (Mµ h)1/p kpLq0 (µ)
P
p0
1
p0 µ(x)
X
(M
h(x))
µ
x∈S(y)
 ,
≥C
e(y) 
µ(S(y))
y∈T
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0
0
where e(y) = ρ(y)1−p µ(S(y))p ,

≥C
X
P
x∈S(y)

e(y) 
p0
.
10
p
µ(x) 
t∈S(y) h(t)µ(t) µ(S(y))

µ(S(y))
P
y∈T

≥C
X
y∈T
=C
X
e(y) 
µ(S(y))

X
h(t)µ(t)
t∈S(y)
µ(t)h(t)
t∈T
X
y∈T
e(y)
χS(y)
µ(S(y))
!
(t).
Nicola Arcozzi
By duality, then, we have that
X
y∈T
Title Page
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
Carleson Measures for Analytic
Besov Spaces: The Upper
Triangle Case
! 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
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,
Page 23 of 33
x∈T
which is the desired conclusion.
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As a final remark, let us observe that the potential W admits the following,
suggestive formulation:
0
0
W (µ) = I ρ1−p (I ∗ Id)p −1 ,
where Id is the identity operator.
Condition (1.10) might, then, be reformulated as
q(p−1)
p0 −1
1−p0
∗
I ρ
(I Id)
∈ 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,
X
1
µ(E(t)) ≤
µ(S(w)) ≤ kf kL1 (µ) ,
t
w∈J
which is the desired inequality.
Carleson Measures for Analytic
Besov Spaces: The Upper
Triangle Case
Nicola Arcozzi
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4.
Equivalence of Two Conditions
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)
Carleson Measures for Analytic
Besov Spaces: The Upper
Triangle Case
Nicola Arcozzi
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and
(4.2)
Ik∗ f (α) =
X
f (β)µ(β).
β∈Sk (α)
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
Wk (µ)(x) =
ρ(y)1−p µ(Sk (y))p −1
y∈Pk (x)
and a [COV]-condition
X
q(p−1)
(4.3)
µ(x) (Wk (µ)(x)) 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
0
p
p
kf kLq (µ) ≤ C(µ)
|(1 − |z| )f (z)| ρ(z)
+ |f (0)|
(1 − |z|2 )2
D
whenever f is holomorphic on D.
Carleson Measures for Analytic
Besov Spaces: The Upper
Triangle Case
Nicola Arcozzi
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(ii) For k ≥ 2, there exists Ck > 0 such that
! p1
! 1q
(4.4)
X
|Ik ϕ(x)|q µ(x)
X
≤ Ck (µ)
x∈T
|ϕ(x)|p ρ(x)
.
x∈T
(iii) The following inequality holds,
X
(4.5)
µ(x) (Wk (µ)(x))
q(p−1)
p−q
< ∞.
Carleson Measures for Analytic
Besov Spaces: The Upper
Triangle Case
x∈T
(iv) (1.3) holds,
Nicola Arcozzi
Z
(Wco (µ)(z))
(4.6)
q(p−1)
p−q
µ(dz) < ∞.
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(v) (1.10) holds,
X
(4.7)
µ(x) (W (µ)(x))
q(p−1)
p−q
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< ∞.
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(vi) (1.9) holds for some C > 0,
Close
! 1q
(4.8)
X
x∈T
q
|Iϕ(x)| µ(x)
! p1
≤ C(µ)
X
p
|ϕ(x)| ρ(x)
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.
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x∈T
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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 k = 0. Let z ∈ D, and let α(z) ∈ G be the
box containing z. Then, it is easily checked that, if k ≥ 2,
Carleson Measures for Analytic
Besov Spaces: The Upper
Triangle Case
S(α(z)) ⊂ S(z) ⊂ Sk (α(z)).
To show the implication (iii) =⇒ (iv), observe that
Z
|dw|
0
0
ρ(w)p −1 µ(S(w))p −1
Wco (µ)(z) =
1 − |w|2
P (z)
≤C
α(z)
X
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p0 −1
ρ(β)
p0 −1
µ(S(β))
Contents
β=o
≤C
X
0
JJ
J
0
ρ(β)p −1 µ(Sk (β))p −1
β∈Pk (α(z))
= CWk (µ)(α(z)),
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hence
Close
Z
(Wco (µ)(z))
q(p−1)
p−q
µ(dz) ≤ C
D
X
α∈G
≤C
X
α∈G
sup (Wco (µ)(z))
q(p−1)
p−q
µ(α)
z∈α
(Wk (µ)(α))
Quit
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q(p−1)
p−q
µ(α)
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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
Carleson Measures for Analytic
Besov Spaces: The Upper
Triangle Case
α(z)
≥C
X
0
0
ρ(β)p −1 µ(S(β))p −1
Nicola Arcozzi
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))
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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.
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)
Carleson Measures for Analytic
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Triangle Case
]{w ∈ J : x ∈ Sk (w)} ≤ L
Nicola Arcozzi
and, for all x ∈ G,
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
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−2
w ∈ S2 (z), dG (o, w) = dG (o, z) − 1 and w ∈
/ S0 (z )
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where S0 (z) = S(z) is the same Carleson box introduced in Section 3. Then,
one can easily see that
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S2 (z) = S0 (z −2 ) ∪ S0 (z ∗ )
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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.
Carleson Measures for Analytic
Besov Spaces: The Upper
Triangle Case
Nicola Arcozzi
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References
[1] N. ARCOZZI AND R. ROCHBERG, Topics in dyadic Dirichelet spaces,
New York J. Math., 10 (2004), 45–67 (electronic).
[2] N. ARCOZZI, R. ROCHBERG AND E. SAWYER, Carleson measures for
the analytic Besov spaces, Revista Iberoam., 18(2) (2002), 443–510.
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Nicola Arcozzi
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