By Jie Xiao

This publication records the wealthy constitution of the holomorphic Q services that are geometric within the experience that they remodel obviously lower than conformal mappings. specific emphasis is put on contemporary advancements in response to the interplay among geometric functionality / degree idea and different branches of mathematical research, together with power conception, advanced variables, harmonic research, useful research, and operator theory.

Largely self-contained, this publication may be an educational and reference paintings for complex classes and study in conformal research, geometry, and serve as areas.

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In this case, we use β(x, y) = Γ(x)Γ(y) Γ(x+y) as the Beta function, and then obtain ∞ g(z) = j=0 β(j + b + 1, m) (j + m + 1)aj+m+1 z j+m . β(j + 1, m) Writing ∞ m jaj z j−1 sm (z) = j=1 1− and h(z) = j=0 β(j + b + 1, m) aj+m+1 z j+m+1 , β(j + 1, m) we ﬁnd f (z) = g(z) + sm (z) + h (z). Note that sm is a polynomial and |g(z)|2 (1 − |z|2 )p is a p-Carleson measure. Thus, in order to prove f ∈ Qp it is enough to verify h ∈ D. To this end, we observe that the Beta function enjoys 0 < 1− β(j + b + 1, m) (b + 1)m ≤ , β(j + 1, m) j+m+1 j ∈ N ∪ {0}.

1. 1]. A good source for more on the Brownian motions associated with BMO is [Pet]. 2. 3]. 1 (iii), see the main result in [Kob1] of which a special form is: f ∈ BMOA if and only if |f − f (w)|2 has the least harmonic majorant for any w ∈ D. 1 (iii) is replaced by sup (|f − f (z)|2 )(z) = z∈D ≈ sup |f |2 (z) − |f (z)|2 z∈D sup z∈D |f |(z) 2 − |f (z)|2 < ∞. See [Leu, Section 3]. As with the limiting case p → 0, we have however noted that f ∈ D if and only if f ∈ H2 and T T |f (ζ) − f (η)|2 |dζ||dη| < ∞.

1 (ii) with F = f ◦ σw − f (w) to obtain (1 − |w|2 )|f (w)| D D 2 |f ◦ σw (z) − f (w)|2 dm(z) |f ◦ σw (z) − f (w)|2−q |(f ◦ σw ) (z)|q (1 − |z|2 )q dm(z). 40 Chapter 2. Poisson versus Berezin with Generalizations Integrating the last estimate against (1 − |σa (w)|2 )p dm(w), a ∈ D, and changing the variable z → σw (z), we ﬁnd Fp (f, a) 2 D D |f (z) − f (ζ)|2 ¯4 |1 − z ζ| Ξ(f ; ζ, z) q p 1 − |σa (z)|2 dm(z)dm(ζ), thereupon deriving f ∈ Qp . Conversely, suppose f ∈ Qp . Note that by the change of variable λ = σa (w), a ∈ D, D (1 − |σa (w)|2 )p dm(w) |1 − wu| ¯ 4 (1 − |λ|2 )p dm(λ) ¯λ − a¯ u + λ¯ u|4 D |1 − a (1 − |a|2 )2 (1 − |λ|2 )p dm(λ) 4 4 ¯ |1 − a¯ u| D |1 − λσa (u)| (1 − |a|2 )p (1 − |u|2 )p−2 , u ∈ D.