Brennan's conjecture holds for semigroups of holomorphic functions

TL;DR

The paper proves Brennan's conjecture for continuous semigroups of holomorphic self-maps of the unit disk by combining classical complex-analytic techniques with weighted Bergman space theory and integration-operator spectra. Central to the approach is the Koenigs representation $\phi_t=h^{-1}\circ\psi_t\circ h$, which reduces the problem to weighted operator estimates for composition operators on spaces with Békollé–Bonami weights. The authors establish the needed finiteness of the $p$-integral mean of $\phi_t'$ for $p\in(-2,2/3)$ by showing boundedness of $C_{\phi_t}$ on $L^2_\ abla(\omega)$ with $\omega=|h'|^p$, together with Schwarz-type and Löwner-chain techniques to control derivatives. This builds a bridge between Brennan's conjecture and semigroup dynamics, suggesting a pathway toward broader proofs via semigroup-fication or approximation of conformal maps by semigroups.

Abstract

In the present note, we give a short proof of Brennan's conjecture in the special case of continuous semigroups of holomorphic functions. We apply classical techniques of complex analysis in conjunction with recent results on Békollé-Bonami weights and spectra of integration operators.

Theorems & Definitions (15)

• Theorem 1.1
• Theorem 2.1: POM92
• Proposition 2.2
• proof
• Theorem 2.3: AC09
• Theorem 2.4: CON10
• Theorem 2.5
• Lemma 2.6
• Lemma 3.1
• proof