8 1. BASIC NOTATION AND INTRODUCTION TO WEIGHTS
1.3. Bekoll´ e-Bonami and invariant weights
The Carleson square S(I) associated with an interval I T is the set S(I) =
{reit
D :
eit
I, 1 |I| r 1}, where |E| denotes the Lebesgue measure of
the measurable set E T. For our purposes it is also convenient to define for each
a D \{0} the interval Ia =
{eiθ
: |
arg(ae−iθ)|

1−|a|
2
}, and denote S(a) = S(Ia).
Let 1 p0,p0 such that
1
p0
+
1
p0
= 1, and let η −1. A weight ω : D
(0, ∞) satisfies the Bekoll´ e-Bonami Bp0
(η)-condition, denoted by ω Bp0 (η), if
there exists a constant C = C(p0,η,ω) 0 such that
S(I)
ω(z)(1
|z|)η
dA(z)
S(I)
ω(z)
−p
0
p0
(1
|z|)η
dA(z)
p0
p
0

C|I|(2+η)p0
(1.12)
for every interval I T. Bekoll´ e and Bonami introduced these weights in [16, 17],
and showed that
ω(z)
(1−|z|)η
Bp0(η) if and only if the Bergman projection
Pη(f)(z) = + 1)
D
f(ξ)
(1 ξz)2+η
(1
|ξ|2)η
dA(ξ)
is bounded from Lω0 p to Aω0 p [17]. This equivalence allows us to identify the dual
space of Aω0
p
with Aω0
p
. In the next section we will see that if ω R, then for
each p0 1 there exists η = η(p0,ω) −1 such that
ω(z)
(1−|z|)η
belongs to Bp0 (η).
However, this is no longer true if ω I.
There is one more class of weights that we will consider. To give the definition,
we need to recall several standard concepts. For a D, define ϕa(z) = (a z)/(1
az) . The automorphism ϕa of D is its own inverse and interchanges the origin and
the point a D. The pseudohyperbolic and hyperbolic distances from z to w are
defined as (z, w) = |ϕz(w)| and
h(z, w) =
1
2
log
1 + (z, w)
1 (z, w)
, z, w D,
respectively. The pseudohyperbolic disc of center a D and radius r (0, 1) is
denoted by Δ(a, r) = {z : (a, z) r}. It is clear that Δ(a, r) coincides with the
hyperbolic disc Δh(a, R) = {z : h(a, z) R}, where R =
1
2
log
1+r
1−r
(0, ∞).
The class Inv of invariant weights consists of those weights ω (that are not
necessarily radial neither continuous) such that for some (equivalently for all) r
(0, 1) there exists a constant C = C(r) 1 such that
C−1ω(a)
ω(z) Cω(a)
for all z Δ(a, r). In other words, ω Inv if ω(z) ω(a) in Δ(a, r). It is
immediate that radial invariant weights are neatly characterized by the condition
(1.2), and thus I ∩Inv = I and R∩Inv = R. To see an example of a radial weight
that just fails to satisfy (1.2), consider
ω(r) = (1
r)logn( expn 0
1−r
)
= exp log
1
1 r
· logn
expn 0
1 r
, n N.
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