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 p 0 = 1, and let η −1. A weight ω : D → (0, ∞) satisfies the Bekoll´ e-Bonami Bp 0 (η)-condition, denoted by ω ∈ Bp 0 (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 p 0 (1 − |z|)η dA(z) p 0 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 ∈ Bp 0 (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 Lp0 ω to Ap0 ω [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 Bp 0 (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( exp n 0 1−r ) = exp − log 1 1 − r · log n exp n 0 1 − r , n ∈ N.

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