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) = (η + 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.